Estimation of Population Distribution Function in the Presence of Non-Response
Mazhar Yaqub∗† and Javid Shabbir‡
Abstract This article addresses the problem of estimating the population distribution function in the presence of non-response. We suggest a general class of estimators for estimating the cumulative distribution function using the auxiliary information. Expressions for bias and mean squared error of considered estimators are derived up to the first order of approximation. The performance of estimators are compared theoretically and numerically. A numerical study is carried out to evalute the performances of estimators.
Keywords: Auxiliary variable; absolute relative bias; cumulative distribution function; mean squared error; relative efficiency. 2000 AMS Classification: Primary 62D05, Secondary 62P20.
1. Introduction It is a well established phenomenon in the theory of sample survey that the non-response is an unavoidable fact, which is devastating and almost in every surveys of human respondents, suffer from some degree of nonresponse. Non-response mainly classified as: (i), unit non-response or total failure, in which entire unit is missing, for example, a person may totally refuse or unable to participate in the survey for some specified reasons and (ii), item non-response or partial failure, in which at least one item is missing from some measurements for the given observations. For example, a household may hesitate to give information about his income. The problem of non-response has already been tackled from different ways, is common and widespread in mail surveys than in personal interviewing. The usual approach to overcome non-response problem is to contact the non-respondent and obtain maximum information as much as possible. Hansen and Hurwitz (1946) were the first to suggest a non-response technique in mail surveys, combined the advantages of mailed questionnaires and personal interviews. They plan first use the economies involved in the use of questionnaires by mailing them to a sample of population under study. After this a follow-up is carried out by interviewing a subsample of the non-respondents. ∗
Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan., Email:
[email protected]
†
Corresponding Author.
‡
Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan., Email:
[email protected]
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Consider a finite population of size N and a random sample of size m is drawn from a population by using simple random sample without replacement (SRSWOR) sampling scheme. In survey of human populations, it is often the case that mR units respond, but the remaining mM = (m − mR ) units do not. The initial survey may be conducted through the mail or by telephone, perhaps computer aided. Hansen and Hurwitz (1946) suggested a two phase sampling scheme for estimating the population mean by using the following steps. (a) a simple random sample of size m is selected and the questionnaire are mailed to the sampled units; (b) a subsample of size r =
mM k
for (k > 1) is taken from mM non-responding units.
The graphical illustration of non-response scheme is given in Figure 1. A widely debated topic in sample
Figure 1. Illustration of Hansen and Hurwitz (1946) non-response scheme
survey is the estimation of population mean for the study variable by using the auxiliary variables in the presence of non-response. Several authors including Chambers and Dunstan (1986), Rao (1986); Rao et al. (1990), Khare and Srivastava (1993, 1995, 1997), Olkin (1958) suggested different types of estimators for estimation of population mean using the auxiliary information under non-response. Okafor and Lee (2000) presented ratio and regression estimation with sub-sampling the non-respondents in estimating the population mean Y¯ . Further, Khare and Sinha (2007, 2009, 2011) proposed some classes of estimators for estimating population mean using multi-auxiliary characters in different way. For the estimating population mean under two-phase sampling scheme in presence of non-response, Singh and Kumar (2009, 2010a,b), Klein (2000), Tabasum and Khan (2006), and Shabbir and Nasir (2013) have made significant contributions. Diana and Perri (2012) suggested a class of estimators in two-phase sampling with sub-sampling of non respondents in estimating the finite population mean. For controlling the non-response bias and eliminating the need for call backs in survey sampling, John and Robert (1947), citeBS, and Dunkelberg and Goerge (1973), El-Badry
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(1956), Diana and Perri (1953), Hansen et al. (1951), and Politz and Simmons (1949), discussed some good techniques and plans. An extensive literature is available on estimation of population mean under non-response, but lesser effort has been devoted in the development of efficient methods for population cumulative distribution function by using the auxiliary information. We are often concerned with the proportion of yi values in the population. Users of sample survey data commonly need to estimate the population distribution function, or, equivalently, the proportion of units in the population with values less than or equal to a specified value ty . For example, we may be interested in the proportion of agricultural area for poisonous effect of pesticides less than zero, the proportion of filtration plants for the present of arsenic in portable water less than zero. Such a proportion is particular value of the cumulative distribution function (CDF) for the population.
FY (ty ) =
N 1 X I (yi ≤ ty ) . N i=1
Above expression is just the average of the values of Bernoulli distribution I (yi ≤ ty ) over all elements of the population, where I (yi ≤ ty ) = 1 for y ≤ ty and I (yi ≤ ty ) = 0, for y > ty . Often in survey sampling, we can only measure the study variable for those items in some sample, thus, the usual estimators of the distribution function depends exclusively on the selection of the sampling design and the sampled portion of the population. It is often seen the case, that some values of study variable are not available for non-sampled portion of the population, so we may use auxiliary information for improving the efficiency of population distribution function. Chambers and Dunstan (1986) and Chambers et al. (1992) suggested the procedure and properties for estimating the finite population distribution function and the quantiles based on use of the auxiliary information. Rao et al. (1990) used a general sampling design and proposed ratio and difference type estimators for population distribution function. Kuk (1998), presented a classical as well as a prediction approach in estimating the distribution function from survey data. Some more work is due to Woodruff (1952), Kuk and Mak (1989, 1993), Rueda et al. (2003, 2004), Rueda and Arcos (2004), Dorfman (1993), Ahmed and Abu-Dayyeh (2001), and Singh and Joarder (2002). In presence of the auxiliary information, there exist several general estimation procedures. For more details see Wang and Alan (1996), Kuk and Mak (1993), Rao et al. (1990), Rueda et al. (2004), Garcia and Cebrian (1998) and Singh et al. (2008) to obtain more efficient estimates for the population mean or totals. An extensive literature is available on estimation of population mean under non-response, but lesser effort has been devoted in the development of efficient methods for population cumulative distribution function (CDF) by using the auxiliary information. The present article focuses on the estimation of population distribution function of the study variable using the auxiliary information when data are not collected from all sampled units due to the problem of non-response.
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We organize the rest of the article as follows: Section 2 introduces the notations and symbols. Section 3 gives detailed proof for estimating the population distribution function under non-response case. Section 4 contains the expressions for the bias and mean squared error (MSE). Section 5 gives a general class of estimators to first order of approximation. A numerical study is presented in Section 6 and cost of the survey is discussed in Section 7. Section 8 gives the conclusion.
2. Notations and Symbols Consider a finite population Ω = {1, 2, ..., N } having N distinct and identifiable units. Let (y1 , y2 , ..., yN ) be the values of the study variable Y. For each index ty , (−∞ < ty < +∞), the cumulative distribution function (CDF) of Y is given by (2.1)
FY (ty ) =
N 1 X I (yi ≤ ty ) , N i=1
where I (.) is an indicator function. Then the corresponding population β quantile (0 < β < 1) is defined by (2.2)
QY (β) = inf {y|FY (y) ≥ β} = FY−1 (β),
where inf stands for infinimum. The problem is to estimate FY (ty ) for any given ty . We draw a random sample of size m from N by simple random sampling without replacement sampling scheme (SRSWOR). Then given ty , the FY (ty ) can be estimated by m
(2.3)
1 X FˆY (ty ) = I (yi ≤ ty ) . m i=1
Following Garcia and Cebrian (1998), it is easy to show that N −m (2.4) E FˆY (ty ) = FY (ty ) and V FˆY (ty ) = FY (ty ) (1 − FY (ty )) , m(N − 1) where E (.)
and
V (.) are the mathematical expectation and variance of (.), respectively. The layout of
response stratum is given in Table 1. Table 1. Layout of respondent stratum
X ≤ FX (tx ) X > FX (tx ) Total Y ≤ FY (ty )
m11 /N11
m12 /N12
N1.
Y > FY (ty )
m21 /N21
m22 /N22
N2.
Total
N.1
N.2
N
Here, N11 , N12 , N21 , and N22 be the number of units in the population in their respective cells for respondents. Similarly, m11 , m12 , m21 , and m22 be the number of units in the sample in their respective cells. Hence (m11 , m12 , m21 , m22 ) is a trivariate Hyper Geometrically(HG) distributed random variable, i.e., (m11 , m12 , m21 , m22 ) ∼ HG (N, m, N11 , N12 , N21 ).
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Also mFˆY (ty ) = m11 + m12 and mFˆX (tx ) = m11 + m21 . The non-response stratum layout is given in Table 2.
Table 2. Layout of non-response stratum (2)
(2)
X2 ≤ FX (tx2 ) X2 > FX (tx2 ) Total (2)
m11 /N11
Y2 > FY (ty2 )
(2)
m21 /N21
Total
N.1
Y2 ≤ FY (ty2 )
(2)
(2)
(2)
(2)
(2)
m12 /N12
(2)
(2)
m22 /N22
(2)
(2)
(2)
N1.
(2)
(2)
(2)
N2.
(2)
(2)
N.2
N
(2)
Here, N11 , N12 , N21 , and N22 be the number of units in the population in their respective cells for non(2)
(2)
(2)
(2)
respondents. Similarly, m11 , m12 m21 and m22 be the number of units in the sample in their respective cells. Let {I (Yi ≤ ty ) , I (Xi ≤ tx )} = 1 if ith unit possesses an attribute and {I (Yi ≤ ty ) , I (Xi ≤ tx )} = 0 otherwise, which follows the uniform probability distribution. Let sample ∗ (tx ) be the unbiased estimators of population means (FY (ty ), FX (tx )) based on m means FˆY∗ i (ty ), FˆXi observations. Let SF2 Y (ty ) = FY (ty ) (1 − FY (ty )), SF2 X (tx ) = FX (tx )(1 − FX (tx )), SF2 X2 (tx2 ) = FX2 (tx2 )(1 − FX2 (tx2 )) be the population variances and SFY X (ty , tx ) = FY,X (ty , tx ) − FY (ty )FX (tx ) be the population covariance for Stratum 1 and Stratum 2 respectively. Also CFY (ty ) =
1−FY (ty ) FY (ty ) ,
CFX (tx ) =
1−FY (tx ) FY (tx ) ,
CFX2 (tx2 ) =
1−FX2 (tx2 ) FX2 (tx2 )
be the population coefficient of
variations of X for Stratum 1 and Stratum 2 respectively. Let β1 (FX (tx )) = √
1−2FX (tx )
and β2 (FX (tx )) = of skewness and kurtosis of X. Let ρ(FY (ty ),FX (tx )) = FX (tx )(1−FX (tx ))
2 1−3FX (tx )+3FX (tx ) FX (tx )(1−FX (tx ))
SFY X (ty ,tx ) SFY (ty )SFX (tx )
be the population coefficients
be the phi-population correlation
coefficient. To obtain Bias and MSE of estimators up to first order of approximation, we define the following relative error terms. Let e∗0 =
FˆY∗ (ty )−FY (ty ) , FY (ty )
e∗1 =
∗ FˆX (tx )−FX (tx ) , FX (tx )
e0 =
FˆY (ty )−FY (ty ) FY (ty )
and e1 =
FˆX (tx )−FX (tx ) FX (tx )
such that E (e∗i ) =
E (ei ) = 0 , for i = 0, 1. To first order of approximation we have n o (2) (2) ∗ ∼ E e∗2 = F 2 1(ty ) λ1 FY (ty ) (1 − FY (ty )) + λ2 FY (ty2 ) 1 − FY (ty2 ) , = V20 0 Y n o (2) (2) 1 ∗2 ∗ ∼ E e1 = F 2 (tx ) λ1 FX (tx ) (1 − FX (tx )) + λ2 FX (tx2 ) 1 − FX (tx2 ) = V02 , X (2) (2) (2) (2) N11 N22 −N12 N21 1 ∗ 12 N21 ∼ E (e∗0 e∗1 ) = FY (ty )F λ1 N11 N22N−N + λ2 , = V11 2 (2) X (tx ) (N2 )2 E e20 = F 2 1(ty ) {λ1 FY (ty ) (1 − FY (ty ))} ∼ = V20 , Y E e21 = F 2 1(tx ) {λ1 FX (tx ) (1 − FX (tx ))} ∼ = V02 , X N N −N N 1 ∗0 , E (e∗0 e1 ) = FY (ty )FX (tx ) λ1 11 22N 2 12 21 ∼ = V11 where ∗ = Vrs
r s ∗ E {(FˆY∗ (ty )−FY (ty )) (FˆX (tx )−FX (tx )) } , (FY h (ty ))r (FXh (tx ))s
0
∗ = Vrs
r s E {(FˆY∗ (ty )−FY (ty )) (FˆX (tx )−FX (tx )) } . (FY h (ty ))r (FXh (tx ))s
λ1 =
1 m
−
1 N
, and
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λ2 =
WM (k−1) , m
with WM =
NM N
, NM be the number of units in the population corresponding to non-response
group.
3. Estimation of Population Distribution Function under Non-response (∗) In this section, we drive the expressions for mean, variance and covariance of the estimator, FˆY (ty ) = (1) (2r) wR FˆY (ty ) + wM FˆY (ty2 ) under non-response for estimating the CDF.
Suppose that the underlying population is divided into two homogeneous strata: (i) response group and (ii) non-response group. Let NR and NM be the number of units in the population that correspond to the response group and the non-response group, respectively, where NR + NM = N. Given this information, following Gross (1980), the finite population CDF, FY (ty ), can be written as (3.1)
(1)
(2)
FY (ty ) = WR FY (ty ) + WM FY (ty2 ),
where Wi = Ni /N for i = R, M. Out of m selected units, mR units respond and mM units do not respond, where mR + mM = m. In order to get response from mM , the non-respondents are contacted once again by personal interview. Then subsample of size r = mM /k for (k > 1), is obtained from mM non-responding units. It is assumed that all r (1) (2r) units respond. Let FˆY (ty ) and FˆY (ty2 ) be the CDF estimators based on mR and r responding units. On
the lines of Hansen and Hurwitz (1946), the estimator of FY (ty ) under non-response is given by (3.2)
(∗) (1) (2r) FˆY (ty ) = wR FˆY (ty ) + wM FˆY (ty2 ),
where wi = mi /m for i = R, M. Based on the estimator given in (3.2), we present the following theorem. Theorem 1 (∗) (∗) (i) FˆY (ty ) is an unbiased estimator of FY (ty ), i.e., E FˆY (ty ) = FY (ty ). (∗) (ii) V ar FˆY (ty ) h i (2) (2) WM (k−1) NM N −m = m(N . −1) FY (ty )(1 − FY (ty )) + m NM −1 FY (ty2 ) 1 − FY (ty2 ) (∗) (∗) (iii) Cov FˆY (ty ), FˆX (tx ) (2) (2) (2) (2) N11 N22 −N12 N21 ) N11 N22 −N12 N21 WM (k−1) = (1−f + . 2 (2) 2 m (N ) m (NM )
Proof (i). Taking mathematical expectation on both sides of (3.2), we have n o n o (∗) (1) (2r) E FˆY (ty ) = E wR FˆY (ty ) + E wM FˆY (ty2 ) , n o n o (∗) (1) (2r) E FˆY (ty ) = E1 wR E2 FˆY (ty )|mR + E wM E2 FˆY (ty2 )|mR , n o n o (∗) (1) (2r) E FˆY (ty ) = E1 wR FY (ty )|mR + E wM E2 E3 FˆY (ty2 )|r, mR , o n o n (1) (2) (∗) E FˆY (ty ) = E1 wR FY (ty )|mR + E wM E2 FˆY (ty2 )|mR , n o n o (∗) (1) (2) E FˆY (ty ) = E1 wR FY (ty )|mR + E wM FY (ty2 )|mR , (∗) (1) (2) E FˆY (ty ) = WR FY (ty ) + WM FY (ty2 ) = FY (ty ), which completes the proof.
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Proof (ii). From (3.2), we can write
(3.3)
(∗) (2r) (2) FˆY (t) = FˆY (ty ) + wM FˆY (ty2 ) − FˆY (ty2 ) ,
(1) (2) where, FˆY (ty ) = wR FˆY (ty ) + wM FˆY (ty2 ).
It is easy to show that (3.4)
E FˆY (ty ) = FY (ty ),
and
(3.5)
N −m FY (ty )(1 − FY (ty )) . V ar FˆY (ty ) = m(N − 1)
If we consider N − 1 ∼ = N , then we can write (3.5) as N −m V ar FˆY (t) = FY (ty )(1 − FY (ty )) . mN Applying variance on both sides of (3.3), we get N −m FY (ty ) (1 − FY (ty )) mN n o (∗) (2r) (2) (3.6) V ar FˆY (ty ) = +V ar wM FˆY (ty2 ) − FˆY (ty2 ) . n o (2r) (2) +2Cov FˆY (ty ), wM FˆY (ty2 ) − FˆY (ty2 ) From (3.6), we can write
(3.7)
o n (2) (2r) V1 E2 wM FˆY (ty2 ) − FˆY (ty2 ) n o (2r) (2) V ar wM FˆY (ty2 ) − FˆY (ty2 ) = n o . (2r) (2) +E1 V2 wM FˆY (ty2 ) − FˆY (ty2 )
Considering the terms on right hand side of (3.7), we have (3.8)
n o n o (2r) (2) (2r) (2) V1 E2 wM FˆY (ty2 ) − FˆY (ty2 ) = V1 E2 E3 wM FˆY (ty2 ) − FˆY (ty2 ) = 0.
(3.9)
n o n o (2r) (2) (2r) (2) 2 E1 V2 wM FˆY (ty2 ) − FˆY (ty2 ) = E1 wM V2 FˆY (ty2 ) − FˆY (ty2 ) .
Considering the term on right hand side of (3.9), we have n o (2r) (2) V2 FˆY (ty2 ) − FˆY (ty2 ) n o n o (2r) (2) (2r) (2) = V2 E3 FˆY (ty2 ) − FˆY (ty2 ) + E2 V3 FˆY (ty2 ) − FˆY (ty2 ) , or V2
n
(2r) (2) FˆY (ty2 ) − FˆY (ty2 )
o
= E2 V3
n o (2r) (2) FˆY (ty2 ) − FˆY (ty2 ) .
Finally, we get (3.10) V2
n
(2r) (2) FˆY (ty2 ) − FˆY (ty2 )
o
=
n o mM − r (2) (2) E2 FˆY (ty2 ) 1 − FˆY (ty2 ) . r (mM − 1)
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(2) (2) Replace the values of r = mM /k, E2 FˆY (ty2 ) = FY (ty2 ) and 2 2 (2) (2) E2 FY (ty2 ) = V2 Fˆ Y (2) (ty2 ) + FY (ty2 ) 2 (2) (2) (2) −mM F (t )(−F (t )) + F (t ) in (3.10), and after some simplifications, we get = mNMM(N y y y 2 2 2 Y Y Y −1) M (3.11) V2
n
o (2r) (2) FˆY (ty2 ) − FˆY (ty2 ) =
NM (k − 1) (2) (2) FY (ty2 ) 1 − FY (ty2 ) . mM (NM − 1)
Substituting (3.11) in (3.9), we have n o (2r) (2) 2 E1 wM V2 FˆY (ty2 ) − FˆY (ty2 ) o n 2 (2) (2) m (k−1) F (t ) 1 − F (t ) , = E1 mM2 mNMM(N y y 2 2 Y Y M −1) or n o (2r) (2) 2 E1 wM V2 FˆY (ty2 ) − FˆY (ty2 ) =
(2) WM (k−1) NM m NM −1 FY (ty2 )
(2) 1 − FY (ty2 ) .
If we consider NM − 1 ∼ = NM , then above expression can be written as n o W (k − 1) M (2r) (2) (2) (2) 2 (3.12) E1 wM V2 FˆY (ty2 ) − FˆY (ty2 ) = FY (ty2 ) 1 − FY (ty2 ) . m Substituting (3.8) and (3.12) in (3.7), we get n o W (k − 1) M (2r) (2) (2) (2) (3.13) V ar wM FˆY (ty2 ) − FˆY (ty2 ) FY (ty2 ) 1 − FY (ty2 ) . = m From (3.6), the covariance term can be written as n o (2r) (2) Cov FˆY (ty ), wM FˆY (ty2 ) − FˆY (ty2 ) n o (2r) (2) E1 Cov2 FˆY (ty ), wM FˆY (ty2 ) − FˆY (ty2 ) (3.14) = o . n (2) (2r) ˆ ˆ ˆ +Cov1 E2 FY (ty ), wM FY (ty2 ) − FY (ty2 )
n o (2r) (2) Now considering the term Cov2 FˆY (ty ), wM FˆY (ty2 ) − FˆY (ty2 ) on right hand side of (3.14), we get n o (2r) (2) E2 Cov3 FˆY (ty ), wM FˆY (ty2 ) − FˆY (ty2 ) (3.15) = n o = 0. (2r) (2) ˆ ˆ ˆ +Cov2 E3 FY (ty ), wM FY (ty y2 ) − FY (ty2 ) On similar steps, it can be shown that n o (2r) (2) (3.16) Cov1 E2 FˆY (ty ), wM FˆY (ty2 ) − FˆY (ty2 ) = 0. Substituting (3.15) and (3.16) in (3.14), we get o n (2r) (2) = 0. (3.17) Cov FˆY (ty ), wM FˆY (ty2 ) − FˆY (ty2 ) Again by using (3.13) and (3.17) in (3.6), we get N −m WM (k − 1) (2) (∗) (2) V ar FˆY (ty ) = FY (ty )(1 − FY (ty )) + FY (ty2 ) 1 − FY (ty2 ) . mN m
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This completes the proof and on the same lines we have N −m WM (k − 1) (2) (∗) (2) V ar FˆX (tx ) = FX (tx )(1 − FX (tx )) + FX (tx2 ) 1 − FX (tx2 ) . mN m Proof (iii). See Appendix A, Page 53.
4. Suggested Estimators of Population Distribution Function We suggest the following family of estimators for estimating the population distribution function. 4.1. General Family of Estimators. A general family of estimators for estimating CDF, is given by g aF (t ) + b , X x (4.1) FˆM J (ty ) = FˆY (ty ) ˆ δ aFX (tx ) + b + (1 − δ) (aFX (tx ) + b) where δ, g are suitably chosen constants and a (6= 0) , b are either real numbers or function of known parameters of the auxiliary variable X, such as standard deviation (SFX (tx )) , co-efficient of variation (CFX (tx )) , co-efficient of skewness (β1 (FX (tx ))) , co-efficient of kurtosis (β2 (FX (tx ))) , and co-efficient of correlation ρ(FY (ty ),FX (tx ) . Expressing (4.1) in terms of e0 s, we have
−g FˆM J (ty ) = FY (ty ) (1 + e0 ) (1 + δαe1 ) ,
where α =
aFX (tx ) aFX (tx )+b .
Expanding the right hand side of the above expression and retaining the terms up to power 2 in e0 s, we have g(g + 1) 2 2 2 ˆ (4.2) FM J (ty ) = FY (ty ) 1 + e0 − δαge1 + δ α e1 − δαge0 e1 . 2 ∗ (tx ). When 4.1.1. Situation I - Non-response both on the Study and the Auxiliary Variables: FˆY∗ (ty ), FˆX
non-response occurs on both the study and the auxiliary variables, and population mean FX (tx ) of the auxiliary variable X is known in advance. In agricultural survey, for instance, expenditures of fertilizer or pesticides on crop can be used as the auxiliary variable for the estimation, say, production of crop, there may be non-response on both the variables and for Situation I, Equation (4.1) can be written as, g aFX (tx ) + b . (4.3) FˆM J (ty ) = FˆY∗ (ty ) ∗ ˆ δ aF (tx ) + b + (1 − δ) (aFX (tx ) + b) X
For this, (4.2) can be written as, g(g + 1) 2 2 ∗2 (1) ∗ ∗ ∗ ∗ ∼ ˆ (4.4) FM Ji (ty ) = FY (ty ) 1 + e0 − δαge1 + δ α e1 − δαge0 e1 , 2 (1) where FˆM Ji (ty ) for case of Situation I. (1) Subtracting FY (ty ) from both sides of (4.4) and then taking expectation, we get the bias of FˆM Ji (ty ) up to
10
first order of approximation, given as 1 ∗ (1) ∗ (4.5) Bias FˆM Ji (ty ) ∼ − δgαi V11 . = FY (ty ) δ 2 g(g + 1)αi2 V02 2 (1) Squaring both sides of (4.4) and then taking expectations, we get the MSE of FˆM Ji (ty ), up to first order of
approximations, given by (4.6)
∗ (1) ∗ ∗ , + δ 2 g 2 αi2 V02 − 2δgαi V11 M SE FˆM Ji (ty ) ∼ = FY2 (ty ) V20
where αi =
aFX (tx ) aFX (tx )+b
for i = 0, 1, ..., 13.
Different estimators can be generated from proposed class of estimators by substituting the suitable choices of (δ,a,b,g). The generated estimators are listed in Table 3. Many more estimators can also be generated (1) Table 3. Some members of the suggested classes of estimators FˆM Ji (ty )
δ
a
b
g
Estimator
0
0
0
0
(1) FˆM J0 (ty ) = FˆY∗ (ty )
1
1
0
1
(1) FˆM J1 (ty ) = FˆY∗ (ty )
1
1
0
-1
(1) FˆM J2 (ty ) = FˆY∗ (ty )
1
1
CFX (tx )
1
(1) FˆM J3 (ty ) = FˆY∗ (ty )
1
1
CFX (tx )
-1
(1) FˆM J4 (ty ) = FˆY∗ (ty )
1
β2 (FX (tx ))
CFX (tx )
-1
(1) FˆM J5 (ty ) = FˆY∗ (ty )
-1
(1) FˆM J6 (ty ) = FˆY∗ (ty )
1 1 1 1 1
CFX (tx ) 1 SFX (tx ) β2 (FX (tx )) 1
β2 (FX (tx )) SFX (tx ) β2 (FX (tx )) SFX (tx ) ρ(FY (ty ),FX (tx ))
-1 -1
(1) FˆM J7 (ty ) = FˆY∗ (ty ) (1) FˆM J8 (ty ) = FˆY∗ (ty )
FX (tx ) ˆ∗ FX (tx ) ˆ ∗ (tx ) F X
FX (tx )
FX (tx )+CF (tx ) X ˆ ∗ (tx )+CF (tx ) F X
X
ˆ ∗ (tx )+CF (tx ) F X X FX (tx )+CF (tx ) X ˆ ∗ (tx )+CF β2 (FX (tx ))F
X
X
(tx ) (tx )
β2 (FX (tx ))FX (tx )+CF X ˆ ∗ (tx )+β2 (FX (tx )) CF (tx )F
CF (tx )FX (tx )+β2 (FX (tx )) X ˆ ∗ (tx )+SF (tx ) F
FX (tx )+SF (tx ) X ˆ ∗ (tx )+β2 (FX (tx )) SF (tx )F
X
X
X
SF
-1
(1) FˆM J9 (ty ) = FˆY∗ (ty )
1
(1) FˆM J10 (ty ) = FˆY∗ (ty )
1
1
ρ(FY (ty ),FX (tx ))
-1
(1) FˆM J11 (ty ) = FˆY∗ (ty )
1
1
β2 (FX (tx ))
1
(1) FˆM J12 (ty ) = FˆY∗ (ty )
1
1
β2 (FX (tx ))
-1
(1) FˆM J13 (ty ) = FˆY∗ (ty )
X
X
X
(tx )FX (tx )+β2 (FX (tx )) ˆ ∗ (tx )+SF (tx ) β2 (FX (tx ))F X
X
X
β (FX (tx ))FX (tx )+SF
2
(tx )
ˆ ∗ (tx )+ρ F (F X
Y (ty ),FX (tx ))
FX (tx )+ρ(F
Y (ty ),FX (tx ))
X
X
FX (tx )+ρ(F (ty ),F (tx )) Y X ˆ ∗ (tx )+ρ F (FY (ty ),FX (tx ))
FX (tx )+β2 (FX (tx )) ˆ ∗ (tx )+β2 (FX (tx )) F X
ˆ ∗ (tx )+β2 (Fx (tx )) F X FX (tx )+β2 (FX (tx ))
from suggested family of estimators by substituting different values of (δ,a,b,g ). (1) The biases of the suggested family of estimators FˆM Ji (ty ) up to the first order of approximations are given
below. (4.7)
(1) ∗ ∗ Bias FˆM J1 (ty ) ∼ − V11 ), = FY (ty ) (V02
(4.8)
(1) ∗ ), Bias FˆM J2 (ty ) ∼ = FY (ty ) (V11
(4.9)
(1) ∗ ∗ Bias FˆM Ji (ty ) ∼ − αi V11 , = FY (ty ) αi2 V02
11
for i = 3, 10, 12, and (1) ∗ ∗ (4.10) Bias FˆM Ji (ty ) ∼ + αi V11 , = FY (ty ) αi2 V02 for i = 4 – 9, 11, 13. The MSE of the suggested family of estimators
(1) FˆM Ji (ty )
up to first order of approximation are given
below. (1) ∗ (4.11) M SE FˆM J0 (ty ) ∼ , = FY2 (ty )V20 (1) ∗ ∗ ∗ (4.12) M SE FˆM J1 (ty ) ∼ + V02 − 2V11 ), = FY2 (ty ) (V20 (1) ∗ ∗ ∗ (4.13) M SE FˆM J2 (ty ) ∼ + V02 + 2V11 ), = FY2 (ty ) (V20 (1) ∗ ∗ ∗ (4.14) M SE FˆM Ji (ty ) ∼ + αi2 V02 − 2αi V11 , = FY2 (ty ) V20 for i = 3, 10, 12., and (1) ∗ ∗ ∗ + αi2 V02 + 2αi V11 , (4.15) M SE FˆM Ji (ty ) ∼ = FY2 (ty ) V20 for i = 4–9, 11, 13. Also here, β2 (FX (tx ))FX (tx ) β2 (FX (tx ))FX (tx )+CFX SFX FX (tx ) = SF FX (t , x )+β2 (FX (tx )) X FX (tx ) FX (tx )+ρ(FY (ty ),FX (tx )) ,
FX (tx ) FX (tx )+CFX FX (tx ) FX (tx )+SFX , α8
α1 = 0, α2 = α3 = 1, α3 = α4 =
C FX (tx ) α6 = CF FFX (t )+CFX , α7 , = X X x β2 (FX (tx ))FX (tx ) , α(10,11) α9 = β2 (F X (tx ))FX (tx )+SFX FX (tx ) α(12,13) = FX (tx )+β2 (FX (tx )) .
=
, α5 =
,
4.1.2. Situation II - Non-response only on the Study Variable: FˆY∗ (ty ). When non-response occurs only on the study variable, information on the auxiliary variable X is obtained from all sampled units and population mean FX (tx ) of the auxiliary variable X is known. In household survey, for example, by using the household size as the auxiliary variable for the estimation of family expenditures. Information can be obtained completely on family size, while there may be non-response on household expenditure. For Situation II, (4.1) can be written as g aF (t ) + b (2) . X x (4.16) FˆM Ji (ty ) = FˆY∗ (ty ) ˆ δ aFX (tx ) + b + (1 − δ) (aFX (tx ) + b)
Expression (4.16) in terms of e0 s, we have g(g + 1) 2 2 2 (2) ∗ ∗ ∼ ˆ (4.17) FM Ji (ty ) = FY (ty ) 1 + e0 − δαge1 + δ α e1 − δαge0 e1 , 2 (2) where FˆM Ji (ty ), for the case of Situation II.
Different estimators can be generated from suggested family of estimators by substituting the suitable choices of (δ,a,b,g) and are listed in Table 4. (2) The biases of the suggested family of estimators FˆM Ji (ty ) up to the first order are given below.
12 (2) Table 4. Some members of the suggested class of estimators FˆM Ji (ty )
δ
a
b
g
Estimator
0
0
0
0
(2) FˆM J0 (ty ) = FˆY∗ (ty )
1
1
0
1
(2) FˆM J1 (ty ) = FˆY∗ (ty )
1
1
0
-1
(2) FˆM J2 (ty ) =
1
1
CFX (tx )
1
(2) FˆM J3 (ty ) =
1
1
CFX (tx )
-1
(2) FˆM J4 (ty ) = FˆY∗ (ty )
1
β2 (FX (tx ))
CFX (tx )
-1
(2) FˆM J5 (ty ) = FˆY∗ (ty )
-1
(2) FˆM J6 (ty ) = FˆY∗ (ty )
1
CFX (tx )
1
1
1 1
SFX (tx ) β2 (FX (tx ))
1
1
β2 (FX (tx )) SFX (tx ) β2 (FX (tx )) SFX (tx ) ρ(FY (ty ),FX (tx ))
-1 -1
FX (tx ) ˆ FˆX (tx ) F (t ) FˆY∗ (ty ) FX (tx ) X x FX (tx )+CF (tx ) X FˆY∗ (ty ) ˆ FX (tx )+CF (tx )
(2) FˆM J7 (ty ) = FˆY∗ (ty ) (2) FˆM J8 (ty ) = FˆY∗ (ty )
X
ˆX (tx )+CF (tx ) F X FX (tx )+CF (tx ) X ˆX (tx )+CF β2 (FX (tx ))F
X
(tx ) (tx )
β2 (FX (tx ))FX (tx )+CF X ˆX (tx )+β2 (FX (tx )) CF (tx )F
CF (tx )FX (tx )+β2 (FX (tx )) X ˆX (tx )+SF (tx ) F
FX (tx )+SF (tx ) X ˆX (tx )+β2 (FX (tx )) SF (tx )F
X
X
X
SF
-1
(2) FˆM J9 (ty ) = FˆY∗ (ty )
1
(2) FˆM J10 (ty ) = FˆY∗ (ty )
1
1
ρ(FY (ty ),FX (tx ))
-1
(2) FˆM J11 (ty ) = FˆY∗ (ty )
1
1
β2 (FX (tx ))
1
(2) FˆM J12 (ty ) = FˆY∗ (ty )
1
1
β2 (FX (tx ))
-1
(2) FˆM J13 (ty ) =
(tx )FX (tx )+β2 (FX (tx )) ˆX (tx )+SF (tx ) β2 (FX (tx ))F X β2 (FX (tx ))FX (tx )+SF (tx ) X
X
FX (tx )+ρ(F (ty ),F (tx )) Y X ˆX (tx )+ρ F
ˆX (tx )+ρ F (F
Y (ty ),FX (tx ))
FX (tx )+ρ(F
Y (ty ),FX (tx ))
(FY (ty ),FX (tx ))
FX (tx )+β2 (FX (tx )) ˆ FˆX (tx )+β2 (FX (tx )) F (t )+β (F (t )) FˆY∗ (ty ) F X(t x)+β 2(F x (tx )) 2 X x X x
(2) ∗0 (4.18) Bias FˆM J1 (ty ) ∼ , = FY (ty ) V02 − V11 0 (2) ∗ (4.19) Bias FˆM J2 (ty ) ∼ , = FY (ty ) V11 h i (2) ∗0 (ty ) ∼ (4.20) Bias Fˆ , = FY (ty ) αi αi V02 − V11 M Ji
for i = 3, 10, 12, and h i (2) ∗0 (4.21) Bias FˆM Ji (ty ) ∼ , = FY (ty ) αi αi V02 + V11 for i = 4–9, 11, 13. (2) The MSE of the suggested family of estimators FˆM Ji (ty ) up to the first order are given below, (2) ∗ (4.22) M SE FˆM J0 (ty ) ∼ , = FY2 (ty )V20
(2) ∗ ∗0 (4.23) M SE FˆM J1 (ty ) ∼ + V02 − 2V11 , = FY2 (ty ) V20 (2) ∗ ∗0 (4.24) M SE FˆM J2 (ty ) ∼ + V02 + 2V11 , = FY2 (ty ) V20 h n oi (2) ∗ ∗0 (4.25) M SE FˆM Ji (ty ) ∼ + αi αi V02 − 2V11 , = FY2 (ty ) V20 for i = 3, 10, 12, and h n oi (2) ∗ ∗0 (4.26) M SE Fˆ (ty ) ∼ + αi αi V02 + 2V11 , = FY2 (ty ) V20 M Ji
for i = 4–9, 11, 13.
13
4.2. Exponential Family of Estimators. A general family of exponential estimators for estimating population distribution function is given by (cFX (tx ) + d) − cFˆX (tx ) + d , (4.27) FˆRe (ty ) = FˆY (ty )exp (cFX (tx ) + d) + cFˆX (tx ) + d or c FX (tx ) − FˆX (tx ) , (4.28) FˆRe (ty ) = FˆY (ty )exp ˆ c FX (tx ) + FX (tx ) + 2d
Expressing (4.28) in terms of e0 s, we have 1 3 2 2 ∼ ˆ (4.29) FRe (ty ) = FY (ty ) (1 + e0 ) 1 − ψe1 + ψ e1 , 2 8 with ψ =
cFX (tx ) cFX (tx )+d .
Expanding the right hand side of (4.29) and retaining the terms up to the second order of e0 s, we have 1 3 2 2 1 F (t ) 1 + e − ψe + ψ e − ψe e (4.30) FˆRe (ty ) ∼ = Y y 0 1 0 1 . 1 2 8 2 ∗ (tx ). For Situ4.2.1. Situation I - Non-response on both the Study and the Auxiliary Variables: FˆY∗ (ty ), FˆX
ation I, (4.28) can be written as
∗ (tx ) c FX (tx ) − FˆX , (4.31) FˆRe (ty ) = FˆY∗ (ty )exp ∗ (t ) + 2d c FX (tx ) + FˆX x and from (4.30), we have 1 ∗ 3 2 ∗2 1 ∗ ∗ (1) ∗ ψe + ψ e − ψe e (4.32) FˆRe (ty ) ∼ F (t ) 1 + e − . = Y y 1 0 2 1 8 2 0 1 (1) where FˆRe (ty ) for case of Situation I. (1)
The biases and MSE of the suggested family of estimators FˆRej (ty ) up to the first order are given below. 1 3 2 ∗ (1) ∗ (4.33) Bias FˆRej (ty ) ∼ F (t ) ψ V − ψ V = Y y j 11 , 8 j 02 2 1 (1) ∗ ∗ ∗ (4.34) M SE FˆRej (ty ) ∼ + ψj2 V02 − ψj V11 , = FY2 (ty ) V20 4 for (j = 1, 2, ..., 10), with FX (tx ) FX (tx ) β2 (FX (tx ))FX (tx ) FX (tx )+CFX , ψ3 = FX (tx )+β2 (FX (tx )) , ψ4 = β2 (FX (tx ))+CFx , F (t )C X (tx ) X (tx ) ψ5 = FX (tx )βX2 (FxX (tFxX))+CF , ψ6 = FX (tx )+ρF(F , ψ7 = FX (tx )+ρF(F , X Y (ty ),FX (tx )) Y (ty ),FX (tx )) FX (tx )ρ(FY (ty ),FX (tx )) X (tx ))FX (tx ) ψ8 = FX (tx )ρ(F (t ),F (t )) +CF , ψ9 = β2 (F , β2 (tx )+CFx X Y y X x FX (tx )ρ(FY (ty ),FX (tx )) ψ10 = FX (tx )ρ(F (t ),F (t )) +β2 (FX (tx )) . Y y X x
ψ1 = 0, ψ2 =
Different estimators can be generated from suggested family by substituting the suitable choices of (c, d). The generated estimators are in Table 5.
14 (1) Table 5. Some members of the suggested class of estimators FˆRej (ty ) .
c
d
Estimator
1
0
1
CFX (tx )
1
β2 (FX (tx ))
β2 (FX (tx ))
CFX (tx )
CFX (tx )
β2 (FX (tx ))
1
ρ(FY (ty ),FX (tx ))
CFX (tx )
ρ(FY (ty ),FX (tx ))
ρ(FY (ty ),FX (tx ))
CFX (tx )
β2 (FX (tx ))
ρ(FY (ty ),FX (tx ))
ρ(FY (ty ),FX (tx ))
β2 (FX (t)
∗ (tx )) (FX (tx )−FˆX ∗ ˆ (FX (tx )+FX (tx )∗) (FX (t)−FˆX (tx )) (1) FˆRe2 (ty ) = FˆY∗ (ty )exp ∗ (t ) +2C ˆ F (t )+ F (t ) FX x X x ) ( X x ∗ (tx )) (FX (t)−FˆX (1) FˆRe3 (ty ) = FˆY∗ (ty )exp ˆ∗ (FX (tx )+FX (tx ))+2β2 (FX (t∗x )) ˆ (tx )) β2 (FX (tx ))(FX (tx )−F (1) X ∗ ˆ ˆ FRe4 (ty ) = FY (ty )exp ∗ ˆ β2 (FX (tx ))(FX (tx )+FX (tx∗))+2CFX (tx ) ˆ (tx )) CF (tx )(FX (tx )−F (1) X ∗ X ˆ ˆ FRe5 (ty ) = FY (ty )exp ˆ ∗ (tx ))+2β2 (FX (tx )) CF (tx )(FX (tx )+F X X ∗ (tx )) (FX (tx )−FˆX (1) FˆRe6 (ty ) = FˆY∗ (ty )exp ∗ ˆ F (t )+FX (tx ))+2ρ(F (ty ),F (tx )) Y X ( X x ˆ ∗ (tx )) CF (tx )(FX (tx )−F (1) X X FˆRe7 (ty ) = FˆY∗ (ty )exp ˆ ∗ (tx ))+2ρ CFx (FX (tx )+F (FY (ty ),FX (tx )) X ˆ ∗ (tx )) ρ(F (ty ),F (tx )) (FX (tx )−F (1) X Y X FˆRe8 (ty ) = FˆY∗ (ty )exp ˆ ∗ (tx ))+2CF (tx ) ρ F (t )+F X X (FY (ty ),FX (tx )) ( X x ˆ ∗ (tx )) β2 (FX )(FX (t)−F (1) X ∗ FˆRe9 (ty ) = FˆY (ty )exp ˆ ∗ (tx ))+2ρ β (F (t )) F (t )+F (FY (ty ),FX (tx )) X 2 X x ( X x ∗ ˆX (tx )) ρ(F (ty ),F (tx )) (FX (tx )−F (1) ∗ Y X FˆRe10 (ty ) = FˆY (ty )exp ˆ ∗ (tx ))+2β2 (FX (tx )) ρ(F (ty ),F (tx )) (FX (tx )+F X Y X (1) FˆRe1 (ty ) = FˆY∗ (ty )exp
4.2.2. Situation II - Non-response only on the Study Variable: FˆY∗ (ty ). For Situation II, (4.28) can be written as c FX (tx ) − FˆX (tx ) , (4.35) FˆRe (ty ) = FˆY∗ (ty )exp c FX (tx ) + FˆX (tx ) + 2d
and from (4.30) we have, (2) (4.36) FˆRe (ty ) ∼ = FY (ty )
1+
e∗0
1 3 1 − ψe1 + ψ 2 e21 − ψe∗0 e1 2 8 2
.
(2) where FˆRe (ty ) for case of Situation II.
(2) The biases of the suggested family of estimators FˆRej (ty ) up to first order are given below. 3 1 ∗0 (2) F (t ) ψ (4.37) Bias FˆRej (t) ∼ ψ V − V . = Y y j j 02 8 2 11 (2) The MSE of the suggested family of estimators FˆRej (ty ) up to first order are given below. 1 2 (2) ∗0 ∗ ψj V02 − ψj V11 . (4.38) M SE FˆRej (t) ∼ + = FY2 (ty ) V20 4 Different estimators can be generated from proposed class of estimators by substituting the suitable choices of c, and d are listed in Table 6.
5. Proposed Generalized Class of Exponential Ratio Type Estimators We propose a generalized class of exponential ratio type estimators, given by (5.1) FˆM JP (ty ) = K1 FˆM J (ty ) + (1 − K1 ) FˆRe (ty ) , i
j
15 (2) Table 6. Some members of the suggested class of estimators FˆRej (ty ) .
c
d
1
0
1
CFX (tx )
1
β2 (FX (tx ))
β2 (FX (tx ))
CFX (tx )
CFX (tx )
β2 (FX (tx ))
1
ρ(FY (ty ),FX (tx ))
CFX (tx )
ρ(FY (ty ),FX (tx ))
ρ(FY (ty ),FX (tx ))
CFX (tx )
β2 (FX (tx ))
ρ(FY (ty ),FX (tx ))
ρ(FY (ty ),FX (tx ))
β2 (FX (tx )
Estimator
(FX (tx )−FˆX (tx )) F (t )+Fˆ (t ) ( X x X x ) (FX (t)−FˆX (tx )) (2) FˆRe2 (ty ) = FˆY∗ (ty )exp F (t )+Fˆ (t ) +2C (t ) x FX ( X x X x ) (FX (t)−FˆX (tx )) (2) ∗ ˆ ˆ FRe3 (ty ) = FY (ty )exp F (t )+Fˆ (t ) +2β (F (t )) ( X x X x ) 2 X x β2 (FX (t))(FX (tx )−FˆX (tx )) (2) FˆRe4 (ty ) = FˆY∗ (ty )exp β (F (t )) F (t )+Fˆ (t ) +2C (t ) x FX 2 X x ( X x X x) CFX (tx )(FX (tx )−FˆX (tx )) (2) ∗ ˆ ˆ FRe5 (ty ) = FY (ty )exp C (t ) F (t )+Fˆ (t ) +2β (F (t )) x FX x ( X x X x ) 2 X (FX (tx )−FˆX (tx )) (2) ∗ ˆ ˆ FRe6 (ty ) = FY (ty )exp F (t )+Fˆ (t ) +2ρ ( X x X x ) (FY (ty ),FX (tx )) CFX (tx )(FX (tx )−FˆX (tx )) (2) ∗ ˆ ˆ FRe7 (ty ) = FY (ty )exp C F (t )+Fˆ (t ) +2ρ Fx ( X x X x ) (FY (ty ),FX (tx )) ρ(FY (ty ),FX (tx )) (FX (tx )−FˆX (tx )) (2) ∗ ˆ ˆ FRe8 (ty ) = FY (ty )exp ρ F (t )+Fˆ (t ) +2CFX (tx ) (FY (ty ),FX (tx )) ( X x X x ) ˆ β (F 2 X )(FX (t)−FX (tx )) (2) ∗ ˆ ˆ FRe9 (ty ) = FY (ty )exp β (F (t )) F (t )+Fˆ (t ) +2ρ 2 X x ( X x X x ) (FY (ty ),FX (tx )) ρ(FY (ty ),FX (tx )) (FX (tx )−FˆX (tx )) (2) FˆRe10 (ty ) = FˆY∗ (ty )exp ρ ˆ (FY (ty ),FX (tx )) (FX (tx )+FX (tx ))+2β2 (FX (tx )) (2) FˆRe1 (ty ) = FˆY∗ (ty )exp
for i = 1, 2, 3, ..., 13, j = 1, 2, 3, ..., 10 and K1 is suitably chosen constant.
∗ (tx ). 5.1. Situation I - Non-response on both the Study and the Auxiliary Variables: FˆY∗ (ty ), FˆX
For Situation I, (5.1) can be written as (1) (1) (1) (5.2) Fˆ (ty ) = K1 Fˆ (ty ) + (1 − K1 ) Fˆ (ty ) , M JP
M Ji
Rej
(1) (1) (1) where FˆM JP (ty ), FˆM Ji (ty ) and FˆM Jj (ty ) for the case of Situation I respectively.
Expressing (5.2) in terms of e0 s, we have 1 1 1 ∗ ∗ 2 2 ∗2 e0 + 2 K1 − 2 − αgK1 ψe1 + 2 g + ψ K1 gα e1 (1) ∼ ˆ (5.3) FM JP (ty ) = FY (ty ) . 3 1 1 ∗ ∗ + (1 + K1 ) ψ 2 e∗2 + + K − K αg ψe e 1 1 1 0 1 8 2 2 (1) The bias and MSE of FˆM JP (ty ), up to first order of approximation, is given by
(5.4)
1 ∗ ψ − αgψ V11 2 K1 1 1 3 (1) 2 2 2 ∗ Bias FˆM JP (ty ) ∼ = FY (ty ) + 2 gα + 2 g α − 8 V02 . 3 2 ∗ ∗ + ψ V02 − ψV11 8
16
and (5.5)
4 K1 − K12 + αgK12 αgψ 2
∗ 4V20 + F 2 (t ) y (1) MSE FˆM JP (ty ) ∼ = Y 4
∗ V02 . ∗ +4 (K1 − 1 − 8αg) ψV11 2 + (1 − ψK1 )
By differentiating (5.5) with respect to K1 , we get the optimum value as (opt)
K1
=
∗2 ∗2 2V11 −ψV02 ∗2 . (2αg−1)ψV02
(1) The minimum MSE of FˆM JP (ty ) at optimum value of K1 , up to first order of approximation is given by
(5.6)
(1) ∗ 1 − ρ2(FY (ty ),FX (tx )) , M SEmin FˆM JP (ty ) ∼ = FY2 (ty )V20
where ρ2(FY (ty ),FX (tx )) =
∗ 2 (V11 ) ∗V∗ . V20 02
5.2. Situation II - Non-response only on the Study Variable: FˆY∗ (ty ). For Situation II, (5.1) can be written as, (5.7)
(2) (2) (2) FˆM JP (ty ) = K1 FˆM Ji (t) + (1 − K1 ) FˆRej (t) ,
(2) (2) (2) where FˆM JP (ty ), FˆM Ji (ty ) and FˆM Jj (ty ) for the case of Situation II. (opt)
The bias and minimum MSE up to first order of approximation at optimum value K1
0
=
∗ 2 2 2(V11 ) −V02 2 V02
is
given by (5.8) (5.9)
0 5 1 ∗0 (opt) 3 (2) ∗ ∼ ˆ Bias FM JP (ty ) = FY (ty ) ( V02 − V11 )K1 + V02 − V11 , 8 2 8 (2) ∗ M SEmin FˆM JP (ty ) ∼ 1 − ρ2(2)(FY (ty ),FX (tx )) , = FY2 (ty )V20
0
where ρ2(2)(FY (ty ),FX (tx )) =
∗ 2 (V11 ) ∗V V20 02
for Situation II .
5.2.1. Efficiency Comparisons for General Family of Estimators. In this section, suggested estimators under Situation I are compared in terms of MSEs. Condition i: By Equation (4.11) and Equation (5.6) (1) (1) M SE FˆM J0 (ty ) − MSEmin FˆM JP (ty ) > 0,
if
∗ 2 V20 ρ(FY (ty ),FX (tx )) > 0.
Condition ii: By Equations (4.12), (4.14) and Equation (5.6) (1) (1) M SE FˆM Ji (ty ) − MSEmin FˆM JP (ty ) > 0, ∗ 2 ∗ ∗ V20 ρ(FY (ty ),FX (tx )) + αi (αi V02 − 2V11 ) > 0.
for i=1,3, 10, 12, if
17
Condition iii: By Equations (4.13), (4.15) and Equation (5.6) (1) (1) M SE FˆM Ji (ty ) − MSEmin FˆM JP (ty ) > 0, for i=2, 4-9, 11, 13, if ∗ 2 ∗ ∗ V20 ρ(FY (ty ),FX (tx )) + αi (αi V02 + 2V11 ) > 0.
(1) Note that the proposed estimator FˆM JP (ty ) is more efficient than the other suggested estimators (1) (1) FˆM J1 (ty ) , ..., FˆM J13 (ty ) , when above conditions are satisfied. The comparisons of estimators for Situation II are given below. Condition i: By Equation (4.22) and Equation (5.9) (2) (2) M SE FˆM J0 (ty ) − MSEmin FˆM JP (ty ) > 0,
if
∗ 2 V20 ρ(2)(FY (ty ),FX (tx )) > 0.
Condition ii: By Equations (4.23), (4.25) and Equation (5.9) (2) (2) M SE FˆM Ji (ty ) − MSEmin FˆM JP (ty ) > 0, for i=1,3, 10, 12, if ∗ 2 ∗ ∗0 V20 ρ(2)(FY (ty ),FX (tx )) + αi αi V02 − 2V11 > 0.
Condition iii: By Equations (4.24), (4.26) and Equation (5.9) (2) (2) M SE FˆM Ji (ty ) − MSEmin FˆM JP (ty ) > 0, for i=2, 4-9, 11, 13, if ∗ 2 ∗ ∗0 V20 ρ(2)(FY (ty ),FX (tx )) + αi αi V02 + 2V11 > 0. (2) Note that the proposed estimator FˆM JP (ty ) is more efficient than the other suggested estimators (2) (2) FˆM J1 (ty ) , ..., FˆM J13 (ty ) , when above conditions are satisfied. 5.2.2. Efficiency Comparisons for Exponential Family of Estimators. In this section, suggested estimators are compared under Situation I in terms of M SEs. Condition (i–x): By Equation (4.11) and Equation (4.34) (1) (1) M SE FˆM J0 (ty ) − M SE FˆRej (ty ) > 0, for j =1, 2,..,10, if 1 2 ∗ ∗ ψj V02 − ψj V11 > 0. 4 Condition xi: By Equations (4.34) and Equation (5.6) (1) (1) (ty ) > 0. M SE Fˆ (ty ) − MSEmin Fˆ Rej
∗ 2 V20 ρ(FY (ty ),FX (tx )) + ψi
M JP
1 ∗ ∗ ψi V02 − V11 2
> 0.
18
(1) Note that the proposed estimator FˆM JP (ty ) is more efficient than the other suggested estimators (1) (1) FˆRe1 (ty ) , ..., FˆRe10 (ty ) , when above conditions are satisfied. Proposed and existing estimators Under Situation II are compared in terms of M SEs. Condition (i–x): By Equation (4.11) and Equation (4.38) (2) (2) M SE FˆM J0 (ty ) − M SE FˆRej (ty ) > 0, for j =1, 2,..,10, if
1 2 ∗ ∗ ψ V − ψj V11 4 j 02
> 0.
Condition xi: By Equations (4.38) and Equation (5.9) (2) (2) M SE FˆRej (ty ) − MSEmin FˆM JP (ty ) > 0. ∗ 2 V20 ρ(2)(FY (ty ),FX (tx )) + ψi
1 ∗ ∗ ψi V02 − V11 2
> 0.
(2) Note that the proposed estimator FˆM JP (ty ) is more efficient than the other suggested estimators (2) (2) FˆRe1 (ty ) , ..., FˆRe10 (ty ) , when above conditions are satisfied.
6. Numerical Study In this section, we consider the following data set for numerical comparisons of suggested estimators considered here. Population I: Source: Sarndal et al. (1992), (P-662) The CO 120 data is based on 120 countries across five continents. Let Y = P–83, 1983 population (in million) and X = P–80, 1980 population (in million). N = 120, m = 50, WM = 0.25, k = 2, 3, 4, f = 0.41667, λ1 = 0.0117, λ2 = 0.0050, FY (ty ) = 0.816667, FX (tx ) = 0.808333, N11 = 97, N12 = 01, N21 = 00, N22 = 22, ρ(FY (ty ),FX (tx )) = 0.9730, CFY (ty ) = 0.47578, CFX (tx ) = 0.48889, β1 (FX (tx )) = −1.5667, β2 (FX (tx )) = 3.454419. Let I (Yi ≤ ty ) = 1 for Y ≤ 0.816667, I (Yi ≤ ty ) = 0 for all Y > 0.816667 and I (Xi ≤ tx ) = 1 for X ≤ 0.808333 , I (Xi ≤ tx ) = 0 for all X > 0.808333 The non-response rate in the given population is considered to be 25 percent, taken as last 30 units of the population. (2)
(2)
(2)
(2)
(2)
(2)
NM = 30, FY (ty2 ) = 0.66667, FX (tx2 ) = 0.66667, N11 = 20, N12 = 00, N21 = 00, N22 = 10. (2) (2) (2) Let I Yi ≤ ty2 = 1 for Y2 ≤ 0.66667 , I Yi ≤ ty2 = 0 for all Y2 > 0.66667 and I Xi ≤ tx2 = 1 (2) for X2 ≤ 0.66667 , I Xi ≤ tx2 = 0 for all X2 > 0.66667. Population II: Source: Sarjinder Singh (2003), P.1113 Let Y = Duration of sleep (in minutes) and X = Age of old persons(≥ 50) years. N = 30, m = 12, WM = 0.25, k = 2, f = 0.400, λ1 = 0.020, λ2 = 0.0833, FY (t) = 0.5000, FX (t) = 0.5333,
19
N11 = 02, N12 = 13, N21 = 14, N22 = 01, ρ(FX ,FY ) = −0.80178, CFY (t) = 1.00, CFX (t) = 0.9355, β1 (FX (t)) = −0.1335, β2 (FX (t)) = 1.0177. Let I (Yi ≤ ty ) = 1 for Y ≤ 0.5000 , I (Yi ≤ ty ) = 0 for all Y > 0.5000 and I (Xi ≤ tx ) = 1 for X ≤ 0.5333 , I (Xi ≤ tx ) = 0 for all X > 0.5333 The non-response rate in the given population is considered to be 25 percent, taken as last 08 units of the population. (2)
(2)
(2)
(2)
(2)
(2)
NM = 08, FY (t) = 0.25, FX (t) = 0.875, N11 = 01, N12 = 01, N21 = 06, N22 = 00. We use the following expressions for Percentage Relative Efficiency (PRE). Fˆ (.) (ty ) (.) (.) J0 × 100, i : P RE FˆM J0 (ty ), FˆM Ji (ty ) = M (.) Fˆ (ty ) M Ji
Fˆ (.) (ty ) (.) (.) ii : P RE FˆM J0 (ty ), FˆRej (ty ) = M(.)J0 × 100, FˆRej (ty ) where (.) can be replaced by (1) and (2) under Situation I and Situation II respectively. We have computed the Absolute Relative Bias (ARB) for different suggested estimators by using the following expression, given by ARB (.) =
(.) |Bias FˆM Ji (ty ) |
(.) or |Bias FˆRej (ty ) |
|FY (ty )|
,
for i = 1, 2, ..., 13 and for j = 1, 2, .., 10. MSE , PRE and ARB values based on given data set under both Situations I and II are given in Tables 7–14.
0.00286 0.00009 0.01157 0.00044 0.00434 0.00990 0.00348 0.00806 0.00336 0.01017 0.00087 0.00607 0.00188 0.00405 0.00009
(1) FˆM J0 (ty )
(1) FˆM J1 (ty )
(1) FˆM J2 (ty )
(1) FˆM J3 (ty )
(1) FˆM J4 (ty )
(1) FˆM J5 (ty )
(1) FˆM J6 (ty )
(1) FˆM J7 (ty )
(1) FˆM J8 (ty )
(1) FˆM J9 (ty )
(1) FˆM J10 (ty )
(1) FˆM J11 (ty )
(1) FˆM J12 (ty )
(1) FˆM J13 (ty )
(1) FˆM JP (ty )
Estimator
MSE
3031.09
70.51
152.27
47.04
329.53
28.10
84.93
35.48
82.16
28.87
37.65
651.29
24.68
2906.35
100.00
PRE
k =2
0.00203
0.00097
0.00066
0.00284
0.00107
0.00705
0.00039
0.00482
0.00049
0.00676
0.00434
0.00102
0.00430
0.00002
-
ARB
0.00009
0.00563
0.00260
0.00844
0.00119
0.01412
0.00467
0.01112
0.00483
0.01374
0.01054
0.00059
0.01607
0.00009
0.00397
MSE
4184.14
70.49
152.48
47.03
332.66
28.11
84.91
35.48
82.14
28.88
37.65
671.22
24.70
4031.93
100.00
PRE
k =3
0.00286
0.00135
0.00092
0.00394
0.00149
0.00981
0.00055
0.00670
0.00068
0.00940
0.00604
0.00142
0.00598
0.00003
-
ARB
0.00009
0.00721
0.00333
0.01080
0.00153
0.01807
0.00598
0.01432
0.00618
0.01758
0.01349
0.00074
0.02056
0.00009
0.00508
MSE
5337.46
70.47
152.60
47.03
334.45
28.12
84.90
35.48
82.13
28.89
37.65
682.97
24.71
5155.06
100.00
PRE
k =4
-
ARB
0.00369
0.00173
0.00118
0.00505
0.00191
0.01256
0.00070
0.00858
0.00087
0.01204
0.00773
0.00182
0.00767
0.00005
(1) Table 7. MSE, PRE and ARB values of FˆM Ji (ty ) for different values of k of Population I
20
0.00286 0.00121 0.00820 0.00140 0.00575 0.00717 0.00324 0.00604 0.00317 0.00733 0.00165 0.00482 0.00226 0.00359 0.00120
(2) FˆM J0 (ty )
(2) FˆM J1 (ty )
(2) FˆM J2 (ty )
(2) FˆM J3 (ty )
(2) FˆM J4 (ty )
(2) FˆM J5 (ty )
(2) FˆM J6 (ty )
(2) FˆM J7 (ty )
(2) FˆM J8 (ty )
(2) FˆM J9 (ty )
(2) FˆM J10 (ty )
(2) FˆM J11 (ty )
(2) FˆM J12 (ty )
(2) FˆM J13 (ty )
(2) FˆM JP (ty )
Estimator
MSE
237.34
79.68
126.36
59.25
172.94
38.97
90.24
47.34
88.31
39.79
49.69
206.51
34.87
236.31
100.00
PRE
k =2
0.00121
0.00065
0.00034
0.00207
0.00031
0.00558
0.00025
0.00369
0.00031
0.00533
0.00330
0.00003
0.00262
0.00166
-
ARB
0.000231
0.00470
0.00337
0.00593
0.00276
0.00844
0.00428
0.00715
0.00435
0.00828
0.00686
0.00251
0.00931
0.00232
0.00397
MSE
171.43
84.49
117.67
66.88
143.62
47.00
92.78
55.53
91.30
47.95
57.84
158.22
42.64
171.04
100.00
PRE
k =3
0.00121
0.00071
0.00028
0.00241
0.00004
0.00686
0.00026
0.00444
0.00033
0.00653
0.00394
0.00068
0.00261
0.00333
-
ARB
0.00343
0.00581
0.00448
0.00704
0.00387
0.00956
0.00539
0.00826
0.00546
0.00939
0.00797
0.00362
0.01042
0.00343
0.00508
MSE
148.26
87.45
113.29
72.11
131.11
53.11
94.27
61.51
93.07
54.11
63.72
140.35
48.76
148.04
100.00
PRE
k =4
-
ARB
0.00121
0.00077
0.00022
0.00276
0.00038
0.00814
0.00028
0.00516
0.00035
0.00774
0.00459
0.00132
0.00262
0.00500
(2) Table 8. MSE, PRE and ARB values of FˆM Ji (ty ) for different values of k Population I
21
0.00286 0.00073 0.00136 0.00234 0.00096 0.00252 0.00171 0.00209 0.00137 0.00114 0.00235 0.00009
(1) FˆM J0 (ty )
(1) FˆRe1 (ty )
(1) FˆRe2 (ty )
(1) FˆRe3 (ty )
(1) FˆRe4 (ty )
(1) FˆRe5 (ty )
(1) FˆRe6 (ty )
(1) FˆRe7 (ty )
(1) FˆRe8 (ty )
(1) FˆRe9 (ty )
(1) FˆRe10 (ty )
(1) FˆM JP (ty )
Estimator
MSE
3031.09
121.54
250.78
208.34
136.66
167.20
113.27
299.10
122.10
210.26
389.71
100.00
PRE
k =2
0.00203
0.00074
0.00227
0.00202
0.00110
0.00161
0.00049
0.00245
0.00076
0.00203
0.00048
-
ARB
0.00009
0.00326
0.00157
0.00190
0.00290
0.00237
0.00350
0.00132
0.00325
0.00188
0.00100
0.00397
MSE
4184.14
121.60
252.13
209.06
136.78
167.52
113.30
301.45
122.16
211.01
397.76
100.00
PRE
k =3
0.00286
0.00103
0.00317
0.00281
0.00154
0.00224
0.00069
0.00342
0.00105
0.00283
0.00068
-
ARB
0.00009
0.00418
0.00201
0.00242
0.00371
0.00303
0.00448
0.00167
0.00416
0.00240
0.00128
0.00508
MSE
5337.46
121.63
252.90
209.48
136.85
167.69
113.32
302.79
122.19
211.44
397.67
100.00
PRE
k =4
-
ARB
0.00369
0.00132
0.00406
0.00361
0.00197
0.00287
0.00088
0.00439
0.00135
0.00363
0.00088
(1) Table 9. MSE, PRE and ARB values of FˆRej (ty ) for different values of k Population I
22
0.00286 0.00157 0.00195 0.00254 0.00170 0.00265 0.00216 0.00239 0.00196 0.00182 0.00255 0.00120
(2) FˆM J0 (ty )
(2) FˆRe1 (ty )
(2) FˆRe2 (ty )
(2) FˆRe3 (ty )
(2) FˆRe4 (ty )
(2) FˆRe5 (ty )
(2) FˆRe6 (ty )
(2) FˆRe7 (ty )
(2) FˆRe8 (ty )
(2) FˆRe9 (ty )
(2) FˆRe10 (ty )
(2) FˆM JP (ty )
Estimator
MSE
237.34
112.08
157.36
146.10
119.49
132.30
107.68
167.57
112.37
146.66
181.76
100.00
PRE
k =2
0.00121
0.00045
0.00137
0.00122
0.00067
0.00097
0.00030
0.00148
0.00046
0. 00283
0.00027
-
ARB
0.00231
0.00366
0.00293
0.00307
0.00350
0.00327
0.00376
0.00282
0.00365
0.00306
0.00268
0.00397
MSE
171.43
108.42
135.59
129.40
113.31
121.33
105.41
140.92
108.61
129.72
147.90
100.00
PRE
k =3
0.00121
0.00045
0.00137
0.00122
0.00067
0.00097
0.00030
0.00148
0.00046
0. 00283
0.00027
-
ARB
0.00343
0.00478
0.00404
0.00418
0.0046
0.00438
0.00488
0.00393
0.00476
0.00417
0.00379
0.00508
MSE
148.26
106.46
125.80
121.58
110.10
115.92
104.18
129.34
106.60
121.80
133.88
100.00
PRE
k =4
-
ARB
0.00121
0.00045
0.00137
0.00122
0.00067
0.00097
0.00030
0.00148
0.00046
0. 00283
0.00027
(2) Table 10. MSE, PRE and ARB values of FˆRej (ty ) for different values of k Population I
23
0.01641
0.05176
0.00693
0.02625
0.00997
0.00992
0.01043
0.00828
0.02049
0.00824
0.02295
0.02503
0.02564
0.01023
0.00670
(2) FˆM J0 (t)
(2) FˆM J1 (t)
(2) FˆM J2 (t)
(2) FˆM J3 (t)
(2) FˆM J4 (t)
(2) FˆM J5 (t)
(2) FˆM J6 (t)
(2) FˆM J7 (t)
(2) FˆM J8 (t)
(2) FˆM J9 (t)
(2) FˆM J10 (t)
(2) FˆM J11 (t)
(2) FˆM J12 (t)
(2) FˆM J13 (t)
(2) FˆM JP (t)
Estimator
MSE
244.80
160.38
63.98
65.55
71.49
199.07
80.05
198.12
157.24
165.36
164.48
62.50
236.58
31.69
100.00
PRE
k =2
0.08538
0.00765
0.02317
0.007208
0.00654
0.00554
0.00928
0.00564
0.00764
0.00761
0.00762
0.02493
0.04483
0.11045
-
ARB
0.00893
0.01311
0.03104
0.03049
0.02748
0.01078
0.02507
0.01083
0.01335
0.01275
0.01281
0.03175
0.00918
0.06133
0.02031
MSE
227.25
154.90
65.43
66.610
73.91
188.37
81.03
187.57
152.13
159.27
158.51
63.97
221.23
33.12
100.00
PRE
k =3
0.09956
0.00832
0.02754
0.00749
0.006769
0.00511
0.01093
0.00525
0.00836
0.00820
0.00822
0.02964
0.05215
0.13340
-
ARB
0.01117
0.01600
0.03645
0.0.599
0.03202
0.01332
0.02964
0.01338
0.01627
0.01558
0.01565
0.03725
0.01143
0.07090
0.02422
MSE
216.75
151.39
66.45
67.29
75.64
181.75
81.71
181.03
148.85
155.40
154.70
65.01
211.92
34.15
100.00
PRE
k =4
0.11374
0.00900
0.03190
0.00821
0.00732
0.00469
0.01258
0.00487
0.00908
0.00878
0.00882
0.03437
0.05948
0.15635
-
ARB
0.01341
0.01888
0.04185
0.04080
0.03655
0.01587
0.03421
0.01593
0.01919
0.01842
0.01850
0.04275
0.01367
0.08048
0.02812
MSE
(1) Table 11. Values of MSE, PRE and ARB, of FˆM Ji (t) for different values of k of Population II
209.76
148.96
67.20
68.92
76.94
177.26
82.21
176.59
146.57
152.72
152.06
65.79
205.66
34.95
100.00
PRE
k =5
0.12792
0.00967
0.03626
0.00881
0.00881
0.00426
0.01424
0.00448
0.00980
0.00936
0.00942
0.03908
0.06680
0.17930
-
ARB
24
0.01641
0.04610
0.00859
0.02466
0.01104
0.01099
0.01142
0.00964
0.01983
0.00960
0.02232
0.02405
0.02415
0.01125
0.00837
(1) FˆM J0 (t)
(1) FˆM J1 (t)
(1) FˆM J2 (t)
(1) FˆM J3 (t)
(1) FˆM J4 (t)
(1) FˆM J5 (t)
(1) FˆM J6 (t)
(1) FˆM J7 (t)
(1) FˆM J8 (t)
(1) FˆM J9 (t)
(1) FˆM J10 (t)
(1) FˆM J11 (t)
(1) FˆM J12 (t)
(1) FˆM J13 (t)
(1) FˆM JP (t)
Estimator
MSE
196.00
145.81
67.94
69.22
73.50
170.79
82.73
170.22
143.64
149.20
148.61
66.54
190.90
35.59
100.00
PRE
k =2
0.07121
0.00514
0.02065
0.00540
0.18444
0.00172
0.00806
0.00186
0.00523
0.00492
0.00496
0.02227
0.03750
0. 10313
-
ARB
0.01228
0.01516
0.02850
0.02934
0.02623
0.01351
0.02374
0.01354
0.01533
0.01490
0.01495
0.02856
0.01250
0.05000
0.02031
MSE
165.45
134.00
72.41
69.22
77.45
150.33
85.57
149.97
132.52
136.30
135.90
71.11
162.50
40.62
100.00
PRE
k =3
0.07121
0.00329
0.02250
0.00589
0.24609
0.00252
0.00849
0.00231
0.00354
0.00282
0.00290
0.02433
0.03750
0.11875
-
ARB
0.01618
0.01906
0.03196
0.03396
0.03013
0.01742
0.02764
0.01745
0.01923
0.01881
0.01885
0.03247
0.01641
0.05391
0.02422
MSE
149.65
127.03
75.78
71.32
80.37
139.04
87.61
138.78
125.91
128.76
128.46
74.59
147.62
44.92
100.00
PRE
k =4
0.07121
0.00144
0.02434
0.00601
0.30774
0.00676
0.00893
0.00448
0.00185
0.00071
0.00084
0.02639
0.03750
0.13438
-
ARB
0.02008
0.02297
0.03586
0..03806
0.03404
0.02132
0.03155
0.02136
0.02314
0.02271
0.02276
0.03637
0.02031
0.05781
0.02812
MSE
(2) Table 12. Values of MSE, PRE and ARB, of FˆM Ji (t) for different values of k of Population II
140.00
122.44
78.42
73.89
82.62
131.89
89.14
131.69
121.54
123.82
123.58
77.32
138.46
48.65
100.00
PRE
k =5
0.07121
0.00040
0.02619
0.00620
0.36939
0.01100
0.00936
0.01065
0.00016
0.00140
0.00121
0.02845
0.03750
0.15000
-
ARB
25
0.01641
0.03085
0.02090
0.02064
0.02096
0.02056
0.00691
0.00672
0.00926
0.00713
0.00998
0.00670
(2) FˆM J0 (t)
(2) FˆRe1 (t)
(2) FˆRe2 (t)
(2) FˆRe3 (t)
(2) FˆRe4 (t)
(2) FˆRe5 (t)
(2) FˆRe6 (t)
(2) FˆRe7 (t)
(2) FˆRe8 (t)
(2) FˆRe9 (t)
(2) FˆRe10 (t)
(2) FˆM JP (t)
Estimator
MSE
244.80
164.31
230.13
177.08
243.95
237.95
79.81
78.28
79.48
78.49
53.18
100.00
PRE
k =2
0.08538
0.00605
0.03826
0.00511
0.01575
0.03207
0.00978
0.01085
0.01000
0.01000
0.04182
-
ARB
0.00894
0.01283
0.00940
0.01198
0.00897
0.00916
0.02514
0.02560
0.02524
0.02554
0.03709
0.02031
MSE
227.25
158.36
216.16
169.49
226.35
221.84
80.80
79.34
80.49
79.53
54.77
100.00
PRE
k =3
0.09056
0.00712
0.04377
0.00606
0.01787
0.03665
0.01135
0.01260
0.01162
0.01162
0.04849
-
ARB
0.01117
0.01567
0.01166
0.01470
0.01122
0.01140
0.02972
0.03025
0.02983
0.03018
0.04332
0.02422
MSE
216.75
154.56
207.62
164.71
215.81
212.43
81.49
80.06
81.18
80.26
55.90
100.00
PRE
k =4
0.11374
0.00820
0.04929
0.00702
0.01999
0.04123
0.01293
0.01434
0.01323
0.01323
0.05516
-
ARB
0.01341
0.01851
0.01393
0.01742
0.01347
0.01364
0.03430
0.03489
0.03443
0.03481
0.04956
0.02812
MSE
(2) Table 13. Values of MSE, PRE and ARB, of FˆRej (t) for different values of k of Population II
209.76
151.93
201.87
161.42
208.78
206.11
81.89
80.60
81.69
80.89
56.74
100
PRE
k =5
0.12792
0.00928
0.05480
0.00797
0.02211
0.04581
0.01451
0.01609
0.01484
0.01484
0.06183
-
ARB
26
0.01641
0.02852
0.02017
0.01995
0.02022
0.01988
0.00857
0.00838
0.01045
0.00877
0.01105
0.00837
(2) FˆM J0 (t)
(1) FˆRe1 (t)
(1) FˆRe2 (t)
(1) FˆRe3 (t)
(1) FˆRe4 (t)
(1) FˆRe5 (t)
(1) FˆRe6 (t)
(1) FˆRe7 (t)
(1) FˆRe8 (t)
(1) FˆRe9 (t)
(1) FˆRe10 (t)
(1) FˆM JP (t)
Estimator
MSE
196.00
148.49
187.12
156.97
195.71
191.36
82.52
81.15
82.22
81.34
57.53
100.00
PRE
k =2
0.07121
0.00497
0.03274
0.00416
0.01363
0.02750
0.00820
0.00910
0.00839
0. 00897
0.03516
-
ARB
0.01228
0.01495
0.01267
0.01436
0.01229
0.01248
0.02379
0.02412
0.02386
0.02408
0.03242
0.02031
MSE
165.45
135.82
160.27
141.47
165.28
162.77
85.39
84.20
85.13
84.36
62.65
100.00
PRE
k =3
0.07121
0.00497
0.03274
0.00416
0.02750
0.00038
0.00820
0.00910
0.00839
0. 00897
0.03516
-
ARB
0.01618
0.01886
0.01658
0.01826
0.01619
0.01638
0.02769
0.02803
0.02776
0.02798
0.03632
0.02422
MSE
149.65
128.04
146.07
132.60
149.54
147.80
87.45
86.40
87.22
86.55
66.66
100.00
PRE
k =4
0.07121
0.00497
0.03274
0.00416
0.02750
0.00038
0.00820
0.00910
0.00839
0. 00897
0.03516
-
ARB
0.02008
0.02277
0.02049
0.004752217
0.02010
0.02029
0.03160
0.03193
0.03167
0.03188
0.04023
0.02812
MSE
(1) Table 14. Values of MSE, PRE and ARB, of FˆRej (t) for different values of k of Population II
140.00
123.53
137.29
126.86
139.91
138.60
89.00
88.07
88.80
88.19
69.90
100
PRE
k =5
0.07121
0.00497
0.03274
0.00416
0.02750
0.00038
0.00820
0.00910
0.00839
0. 00897
0.03516
-
ARB
27
0.00286 0.00073 0.00136 0.00234 0.00096 0.00252 0.00171 0.00209 0.00137 0.00114 0.00235 0.00009
(1) FˆM J0 (ty )
(1) FˆRe1 (ty )
(1) FˆRe2 (ty )
(1) FˆRe3 (ty )
(1) FˆRe4 (ty )
(1) FˆRe5 (ty )
(1) FˆRe6 (ty )
(1) FˆRe7 (ty )
(1) FˆRe8 (ty )
(1) FˆRe9 (ty )
(1) FˆRe10 (ty )
(1) FˆM JP (ty )
Estimator
MSE
3031.09
121.54
250.78
208.34
136.66
167.20
113.27
299.10
122.10
210.26
389.71
100.00
PRE
m=50
0.00203
0.00074
0.00227
0.00202
0.00110
0.00161
0.00049
0.00245
0.00076
0.00203
0.00048
-
ARB
0.00007
0.00179
0.00086
0.00104
0.00159
0.00130
0.00192
0.00072
0.00178
0.00103
0.00055
0.00217
MSE
3223.24
121.55
251.07
208.50
136.69
167.27
113.28
299.61
122.11
210.42
390.80
100.00
PRE
m=60
0.00155
0.00056
0.00133
0.00173
0.00084
0.00122
0.00038
0.00186
0.00057
0.00154
0.00036
-
ARB
0.00005
0.00138
0.00067
0.00081
0.00123
0.00101
0.00149
0.00056
0.00138
0.00080
0.00043
0.00168
MSE
3492.27
121.57
251.43
208.69
136.72
167.35
113.29
300.23
122.13
210.62
392.13
100.00
PRE
m=70
0.00121
0.00044
0.00132
0.00119
0.00065
0.00095
0.00029
0.00145
0.00044
0.00120
0.00028
-
ARB
(1) Table 15. MSE, PRE and ARB values of FˆRej (ty ) for different values of m of Population I
28
0.00286 0.00157 0.00195 0.00254 0.00170 0.00265 0.00216 0.00239 0.00196 0.00182 0.00255 0.00120
(2) FˆM J0 (ty )
(2) FˆRe1 (ty )
(2) FˆRe2 (ty )
(2) FˆRe3 (ty )
(2) FˆRe4 (ty )
(2) FˆRe5 (ty )
(2) FˆRe6 (ty )
(2) FˆRe7 (ty )
(2) FˆRe8 (ty )
(2) FˆRe9 (ty )
(2) FˆRe10 (ty )
(2) FˆM JP (ty )
Estimator
MSE
237.34
112.08
157.36
146.10
119.49
132.30
107.67
167.57
112.37
146.66
181.76
100.00
PRE
m=50
0.00121
0.00045
0.00137
0.00122
0.00067
0.00097
0.00030
0.00148
0.00046
0. 00203
0.00027
-
ARB
0.00100
0.00195
0.00143
0.00153
0.00184
0.00167
0.00203
0.00135
0.00195
0.00152
0.00125
0.00217
MSE
219.04
111.26
152.05
142.11
118.09
129.75
107.17
160.96
111.53
142.61
173.14
100.00
PRE
m=60
0.00086
0.00032
0.00098
0.00087
0.00048
0.00070
0.00021
0.00105
0.00033
0. 00154
0.00019
-
ARB
0.00084
0.00153
0.00115
0.00122
0.00145
0.00133
0.00158
0.00110
0.00152
0.00122
0.00103
0.00168
MSE
200.32
110.29
146.08
137.56
116.43
126.78
106.57
153.60
110.53
138.00
163.74
100.00
PRE
m=70
0.00062
0.00023
0.00070
0.00062
0.00034
0.00050
0.00015
0.00075
0.00023
0. 00120
0.00014
-
ARB
(2) Table 16. MSE, PRE and ARB values of FˆRej (ty ) for different values of m of Population I
29
0.00286 0.00073 0.00136 0.00234 0.00096 0.00252 0.00171 0.00209 0.00137 0.00114 0.00235 0.00009
(1) FˆM J0 (ty )
(1) FˆRe1 (ty )
(1) FˆRe2 (ty )
(1) FˆRe3 (ty )
(1) FˆRe4 (ty )
(1) FˆRe5 (ty )
(1) FˆRe6 (ty )
(1) FˆRe7 (ty )
(1) FˆRe8 (ty )
(1) FˆRe9 (ty )
(1) FˆRe10 (ty )
(1) FˆM JP (ty )
Estimator
MSE
3031.09
121.54
250.78
208.34
136.66
167.20
113.27
299.10
122.10
210.26
389.71
100.00
PRE
WM =0.25
0.00203
0.00074
0.00227
0.00202
0.00110
0.00161
0.00049
0.00245
0.00076
0.00203
0.00048
-
ARB
0.00009
0.00258
0.00125
0.00150
0.00229
0.00187
0.00276
0.00104
0.00256
0.00149
0.00080
0.00313
MSE
3314.18
121.56
251.20
208.56
136.70
167.30
113.28
299.83
122.12
210.49
391.27
100.00
PRE
WM =0.30
0.00224
0.00081
0.00249
0.00221
0.00121
0.00176
0.00054
0.00269
0.00083
0.00223
0.00053
-
ARB
0.00009
0.00267
0.00129
0.00155
0.00237
0.00194
0.00286
0.00108
0.00265
0.00154
0.00083
0.00324
MSE
3427.33
121.56
251.35
208.65
136.72
167.34
113.28
300.09
122.13
210.58
391.83
100.00
PRE
WM =0.35
0.00232
0.00084
0.00258
0.00229
0.00125
0.00183
0.00056
0.00278
0.00086
0.00231
0.00055
-
ARB
(1) Table 17. MSE, PRE and ARB values of FˆRej (ty ) for different values of WM of Population I
30
0.00286 0.00157 0.00195 0.00254 0.00170 0.00265 0.00216 0.00239 0.00196 0.00182 0.00255 0.00120
(2) FˆM J0 (ty )
(2) FˆRe1 (ty )
(2) FˆRe2 (ty )
(2) FˆRe3 (ty )
(2) FˆRe4 (ty )
(2) FˆRe5 (ty )
(2) FˆRe6 (ty )
(2) FˆRe7 (ty )
(2) FˆRe8 (ty )
(2) FˆRe9 (ty )
(2) FˆRe10 (ty )
(2) FˆM JP (ty )
Estimator
MSE
237.34
112.08
157.36
146.10
119.49
132.30
107.68
167.57
112.37
146.66
181.76
100.00
PRE
WM =0.25
0.00121
0.00045
0.00137
0.00122
0.00067
0.00097
0.00030
0.00148
0.00046
0. 00123
0.00027
-
ARB
0.00148
0.00282
0.00209
0.00223
0.00266
0.00243
0.00293
0.00198
0.00282
0.00222
0.00184
0.00313
MSE
211.94
110.91
149.86
140.45
117.49
128.67
106.96
158.24
111.17
140.93
169.66
100.00
PRE
WM =0.30
0.00121
0.00045
0.00137
0.00122
0.00067
0.00097
0.00030
0.00148
0.00046
0. 00123
0.00027
-
ARB
0.00159
0.00293
0.00220
0.00234
0.00278
0.00254
0.00304
0.00209
0.00293
0.00233
0.00196
0.00324
MSE
204.07
110.50
147.33
138.52
116.79
127.41
106.70
155.13
110.74
138.97
165.68
100.00
PRE
WM =0.35
0.00121
0.00045
0.00137
0.00122
0.00067
0.00097
0.00030
0.00148
0.00046
0. 00123
0.00027
-
ARB
(2) Table 18. MSE, PRE and ARB values of FˆRej (ty ) for different values of WM of Population I
31
32
(2) (1) From Tables 7–8, it is observed that M SE FˆM Ji (ty ) and M SE FˆM Ji (ty ) increases at increasing rate of (2) (1) (k) for the given data set. Percentage Relative Efficiencies of FˆM J(i=1−3,5,9,10,12,P ) (ty ) and FˆM J(i=2,4−9,11,13) (ty ) (2) (1) increases by increasing values of (k) and decreases for FˆM J(i=6,8,11,13) (ty ) , FˆM J(i=1,3,10,12,P ) (ty ) , but hav (2) ing same values for FˆM J(i=4,7) (ty ) . The PREs of ratio type of estimators having more values in comparison with product type of estimators as there is positive correlation between study and auxiliary variables here, (.) but our proposed estimator FˆM JP (ty ) is more efficient from all other suggested estimators considered here. (2) (1) As for as ARBs of FˆM J(i=1−P ) (ty ) and for FˆM J(i=1,3−13) (ty ) increases at increasing rates of inverse sampling (1) rate (k) but having same values for FˆM J(i=2,P ) (ty ). (2) (1) From Tables 9–10, we examined that M SE FˆM Ji (ty ) and M SE FˆM Ji (ty ) increases at increasing rate (2) of (k) for the given data set. By increasing values of k the PREs of FˆRe(j=1−P ) (ty ) increases, and decreases (1) (2) (1) for FˆRe(j=1−P ) (ty ) . The ARBs of FˆRe(j=1−P ) (ty ) increases and having same values for FˆRej (ty ) at
increasing rates of inverse sampling rates. (1) (2) From Tables 11–12, we observed that M SE FˆM Ji (ty ) decreases, as compared to M SE FˆM Ji (ty ) , at fixed values of (k), and increases, at increasing rate of (k) for Population II. Efficiencies increases for (1) (2) (1) FˆM J(i=1,3,8,10−12) (ty ) and FˆM J(i=1,3,8,10−12) (ty ) by increases (k) and decreases for FˆM J(i=2,4,7,9,13,P ) (ty ) , (2) FˆM J(i=2,4−7,9,13,P ) (ty ) . Here, efficiencies of product type of estimators having more values as compared to ratio type of estimators because of negative correlation between study and auxiliary variables here, (.) but our proposed estimator FˆM JP (ty ) is more efficient from all suggested estimators. As for as ARB of (1) (2) (1) (2) FˆM J(i=1−6,8,10−P ) (ty ) and (FˆM J(i=1,3,7−11) (ty ) increases, decreases for FˆM J(i=7,9) (ty ) and (FˆM J(i=4−6,12,13) (ty ) (2) but having same values for FˆM J(i=2,P ) (ty ).
(1) (2) From Tables 15–16, we studied that M SE FˆM Ji (ty ) and M SE FˆM Ji (ty ) decreases by increasing values of (m) for the given data set. For increasing the sample size (m), the Percentage Relative Efficiencies (2) (1) of FˆRe(j=0−P ) (ty ) increases, and decreases for FˆRe(j=0−P ) (ty ) . The ARBs decreases under both the (2) (1) situations of FˆRej (ty ) and FˆRej (ty ) by increasing the sample size (m) respectively. (1) (2) From Tables 17–18, we see that M SE FˆRej (ty ) increases, as compared to M SE FˆRej (ty ) , at fixed values of (WM ) for given Population, but increases under both the situations at increasing rates of (WM ). (2) (1) PREs of FˆRe(j=0−P ) (ty ) increases and decreases for FˆRe(j=0−P ) (ty ) at increasing rates of (WM ). The (2) (1) ARBs of FˆRe(j=1−P ) (ty ) increases and having same values for FˆRej (ty ) at increasing values of (WM ). (.) Overall our proposed class of estimators FˆM JP (ty ) is more efficient than all other suggested estimators
considered here.
6.1. Comparison Through Percentage Loss in Efficiency. To judge the effect of non-response in (2) simple random sampling, we obtain the percent relative loss in efficiency of proposed estimator FˆRej (ty ) with (1) respect to the same situation as discussed earlier but without of non-response. For this we modify FˆRej (ty )
33 0
0
() () as FˆRej (ty ) for estimating population distribution function. The MSE of FˆRej (ty ) is given by 0 1 2 () ∗0 (6.1) M SE FˆRej (ty ) = FY2 (ty ) V20 + ψj V02 − ψj V11 . 4 0
() (1) The Percentage Relative Loss in Precision (PRLP) of FˆRej (ty ) with respect to FˆM J0 (ty ) is given by 0 () (1) ˆ 0 MSE FˆM (t ) − MSE F (t ) J0 y Rej y () (6.2) P RLP FˆRej (ty ) = × 100 (1) MSE FˆM J0 (ty )
m
50
60
70
WM
0.25
0.30
0.35
Estimator →
3
2
3
2
3
2
PRLP
PRE
70.57
100.00
54.52
100.00
PRE
PRLP
64.90
100.00
PRLP
PRE
48.04
100.00
PRE
PRLP
55.99
100.00
38.88
100.00
PRLP
PRE
PRLP
PRE
k Case
P
92.23
399.17
87.99
394.33
90.73
397.44
86.28
392.40
88.38
397.76
83.86
389.71
85.89
211.65
78.20
210.95
83.17
211.40
75.09
210.66
78.90
211.01
70.70
210.86
75.87
122.21
62.72
122.16
71.22
122.19
57.40
122.13
63.92
122.16
49.89
122.10
89.99
303.48
84.53
301.25
88.06
302.69
82.32
300.36
85.02
301.45
79.20
299.10
74.00
113.33
59.83
113.30
68.99
113.32
54.10
113.29
61.12
113.30
46.01
113.27
82.32
167.78
72.69
167.49
78.92
167.68
68.80
167.37
73.57
167.52
63.29
167.20
78.42
136.88
66.66
136.77
74.27
136.85
61.90
136.73
67.73
136.78
55.19
136.66
85.76
209.69
78.00
209.00
83.02
209.44
74.86
208.73
78.71
209.06
70.43
208.34
88.12
253.30
81.64
252.01
85.83
252.84
79.03
251.50
82.24
252.13
75.33
250.78
75.76
121.64
62.55
121.59
71.09
121.63
57.20
121.57
63.75
121.60
49.66
121.54
98.43
6222.85
97.58
4050.60
98.13
5228.86
97.23
3553.50
97.66
4184.14
96.74
3031.09
(0 ) (0 ) (0 ) (0 ) (0 ) (0 ) (0 ) (0 ) (0 ) (0 ) (0 ) (0 ) FˆM J0 (ty ) FˆRe1 (ty ) FˆRe2 (ty ) FˆRe3 (ty ) FˆRe4 (ty ) FˆRe5 (ty ) FˆRe6 (ty ) FˆRe7 (ty ) FˆRe8 (ty ) FˆRe9 (ty ) FˆRe10 (ty ) FˆM J (ty )
(.) Table 19. PRE, PRLP, of FˆRej (ty ) at different values of WM , m and k
34
35
From Table 15, it is observed that, if non-response is considered, there is loss in precision. The percent relative loss in precision increases by increasing the values of k and m respectively. It is also observed that for fixed values of k and m with the increasing values of WM , PRLP also increase, so more values of k and m, taken more loss in precision is to be observed due to presence of non-response.
7. Cost of the Survey Following Hansen and Hurwitz (1946), for attaining the better precision at minimum cost, we consider here the case, for determining the number of questionnaires to be sent out and the personal interviews to take in follow ups for non-responses to the mail questionnaires. For this, we assume that questionnaires are sent to 30 people, randomly drawn from 120 countries of a given data set. Further assume that 50 percent or 15 respond, and from other 15 which are non-respondents, 10 percent or 3 visited for insuring some representation of the class of non-respondents. An unbiased estimate is given by: (7.1)
(∗) (1) (2r) FˆX (tx ) = wR FˆX (tx ) + wM FˆX (tx ),
where wi = mi /m for i = R, M. N = 120 = Total number of people in population; m =30 = Total mailed out questionnaires; (1) FˆX (tx )= The average of people to the mailed out questionnaires;
mR = 15 = The number of respondents; (2r) FˆX (tx ) = The average number of peoples for personal interviewing;
mM = 3 = Not reply through mailed questionnaires which are personally interviewed. (∗) It is noted that the actual processed sample size would be 15 +3 = 18. The sample variance of FˆX (t), is
given by (7.2)
(∗) V FˆX (tx ) =
N −m ˆ FX (tx )(1 − FX (tx )) m(N ˆ − 1) , WM (k − 1) NM (2) (2) FX (tx ) 1 − FX (tx ) + m NM − 1
where FX (tx )(1 − FX (tx )) is the variance of whole population and (2) (2) FX (tx ) 1 − FX (tx ) is the variance from not respondents. NM is the number of people in the population having no response; r is personally visited numbers; and k =
mM r
, mM is the number of non-respondents in
the sample. By using Equation (7.2), we can see that there are wide range of different sample sizes which will give us the same reliability and finally we reached at that point the sample size m alone give us the poor (2) (2) indicator for sampling reliability. For example, assume that FX (tx )(1 − FX (tx )) = FX (tx ) 1 − FX (tx ) and that N and NM are so large that is
N N −1
and
NM NM −1
tends to one. Further assume that the accuracy
we required is such that, the average value of ( = standard error) would be given by m = 30 people when WR = 100%. If questionnaires were mailed to a random sample of m people with WR = 100%, the variance
36
of the auxiliary variable FX (tx ) estimated from sample would be N −m (∗) (7.3) V FˆX (tx ) = FX (tx )(1 − FX (tx )), m(N − 1) Thus, (7.4)
2 =
N − 30 FX (tx )(1 − FX (tx )). 30(N − 1)
By substituting different numerical values at different response rate of mailed returns along with personal interviews in (7.2), we see that, although m which differs in size but each one will give us same reliability.
Table 20. Some Cost under different sample sizes at WR = 50% for Situation I.
m
mR
mM
r=
mM k
Scheduled Tabulated
Cost=c0 m + c1 mR + c2 mM
(mR + r) 16
8
8
8
16
1040
20
10
10
5
15
800
26
13
13
5
18
890
30
15
15
4
19
850
40
20
20
4
24
1000
60
30
30
3
33
1200
Let c0 = Rs 5= Overhead cost, c1 = Rs 20 = Cost per unit for responding stratum, and c2 = Rs 100, = Cost per unit for non-responding stratum. Generally c2 having more values than c1 , as extra effort is required for making contact with non-respondents and obtained responses from them. From Table 20, Column 5 shows that for different sample sizes m each one give us the same required precision. For instance, sending 20 questionnaires, obtaining (10 by mail and 05 by personal interviewing) give us the same () as for sending out 60 questionnaires and obtaining a total of 33 questionnaire (30 by mail and 03 by visited personally). Therefore at some point it would be at a loss to put extra money for having additional mail returns. Sensibly, it will be better to spend extra effort for those which are non-respondents. Column 6 gives us the total cost for each of the sample sizes under the unit cost. Since in Table 20, for different schedules tabulated all give us same precision, so logically it will be better for us to choose that particular value of m which would give us minimum cost. Consequently, by sending 20 schedules, 10 of them are returned by mail (at 50 per cent response rate) and 5 are personally interviewed which were non-respondents. Now instead of suggested procedure for Table 20, we obtain an optimum number of schedules mailed out and choose personal interviews accordingly, the optimum values of m and r are given by: (7.5)
m(opt) = m ˆ {1 + (k − 1) WM } ,
37
and (7.6)
mM k q
r=
where, k =
c2 WR c0 +c1 WR
and (7.7)
m ˆ =
N T1 , (N − 1) 2 + T1
where T1 = FY (ty )(1 − FY (ty )) + FX (tx )(1 − FX (tx )) − 2
N11 N22 −N12 N21 N2
,
WR is the response rate obtained through mailed questionnaire, WR = 1−WM . Expressions in (7.5) and (7.6) (2) (2) M ∼ are obtained under the assumptions that FX (t)(1 − FX (t)) = FX (t) 1 − FX (t) and NN−1 = NN = 1. M −1 But when the above assumption is eliminated the optimum values of m and r are given by T2 (7.8) m(opt) = m ˆ 1 + (k − 1) WM , T1 where (2)
(2)
(2)
(2)
T2 = FY (ty )(1 − FY (ty )) + FX (tx )(1 − FX (tx )) − 2
(2)
(2)
(2)
(2)
N11 N22 −N12 N21 (2)
(N2 )2
.
and (7.9)
mM , k r
r=
where, k =
c2 WR c0 +c1 WR
n
(N −m)T1 N WM T2
o −1 .
Of course, at different response rate mR and r varies for achieving the specified precision. In practice we do not know what will be the approximate response rate but for estimating optimum values and by using (7.8) and (7.9), there must be an approximate known value of WR in advance. For the case, when WR is not known in advance, suggesting to design the survey for achieving at least definite specified precision at minimal cost, and parallel to these must know about total cost of the survey. Under such circumstances, it is possible for obtaining optimum values of mR and k. For example, instead of using 50% response rate in Table 20, we compute optimum values of mR and r at different values of WR (10% to 90%) respectively for achieving same precision. The optimum values of Equations (7.5) and (7.6) along with their cost are given in Table 21. Table 21, Column 6 gives us the optimum cost at specified response rate, but for unknown response rate, we give Strategy 1 by sending 30 questionnaires and follow up on all non-respondents whatever the value of WR is. Therefore for specified response rate WR , the cost for Strategy 1 will always be more than the optimum cost as shown in Table 21. It is interesting to note that how costly this strategy is, as compared to the optimum cost method by using different response rates WR . The comparison between Column 6 and Column 7 is presented in Column 8 which shows that at smaller response rates give low cost since less questionnaires have been received. For at least 30% response rate, increase in cost from 09% to 22% is to be expected for this strategy.
38
Table 21. Comparison of minimum cost for various response rates of WR
WR
m
mR
mM
r=
mM k
Optimum
Cost of
Increase
Cost
Strategy 1
in Cost
0.10
35
4
31
26
2855
2910
055
0.20
42
8
34
23
2670
2710
040
0.30
44
13
31
19
2380
2430
050
0.40
44
18
26
15
2080
2190
110
0.50
42
21
21
12
1830
1950
120
0.60
40
24
16
9
1600
1710
110
0.70
38
27
11
6
1330
1470
140
0.80
35
28
7
4
1135
1230
095
0.90
33
30
3
2
965
990
025
When an approximate value of WR is not known in advance, the Strategy 2 is preferable as compared to Strategy 1. There are two steps involved in Strategy 2 as: (i) Determine the maximum number of m, whatever the size for WR , (ii) Determine r for achieving the required precision and value of WR is actually determined from the sample results. Hence r will change its value with the actual WR ; In Table 21 as there are maximum number (44) questionnaires to be sent out for WR = 40% then by using formula, r =
mM k
, we get r = 15.
Table 22. Comparison between optimum cost for known values of WR and Strategies 1 and 2.
WR
m
mR
mM
r=
mM k
Cost of
Cost of
Optimum
Strategy 2
Strategy 1
cost
0.10
44
4
40
33
3600
2910
2855
0.20
44
9
35
23
2700
2710
2670
0.30
44
13
31
19
2380
2430
2380
0.40
44
18
26
15
2080
2190
2080
0.50
44
22
22
12
1860
1950
1830
0.60
44
26
18
10
1740
1710
1600
0.70
44
31
13
7
1540
1470
1330
0.80
44
35
9
5
1420
1230
1135
0.90
44
40
4
2
1220
990
965
39
Table 22, show the number of mailed questionnaires and number of personal interviewed r at varying values of WR for achieving the required precision, along with their total cost of the survey. The optimum costs are also given at known values of WR . Of course at high value of WR , the optimum cost of any survey will give us the small values accordingly. Thus from the above discussions, we conclude that it is not necessarily found that an optimum value of m and r but an optimum procedure (Strategy) is also vital even when we have nothing in hand about values of WR in advance and consequently it will give us in any case at least a precise procedure at slightly low cost.
8. Conclusion (.) In this article, we proposed an improved generalized class of ratio type exponential estimators FˆM JP (ty ). (.) Expressions for bias and MSE of the proposed class of estimators FˆM JP (ty ) are compared with two suggested
families of estimators theoretically and numerically under Situations I and II. From Tables 7–15, it is observed (.) that proposed class of estimators FˆM JP (ty ) is preferable in both the Situations and is recommended for precise
estimation for population distribution function in the presence of non-response.
Acknowledgments The authors are very grateful to the unknown referees and editor (Prof. Dr. Cem Kadilar) for their constructive and encouraging suggestions which led to the present improved version of the manuscript. The first author would like to thank to Dr. Abdul Haq (Department of Statistics, Quaid-i-Azam Univesity, Islamabad, Pakistan) for his valuable suggestions.
40
Appendix A (∗) (∗) For finding the Cov FˆY (ty ), FˆX (tx ) , we have
(8.1)
(1) (2r) (1) (2r) = Cov wR FˆY (ty ) + wM FˆY (ty2 ), wR FˆX (tx ) + wM FˆX (tx2 ) .
(8.1) can also be written as (8.2)
n o (2r) (2) (2r) (2) Cov FˆY (ty ) + wM FˆY (ty2 ) − FˆY (ty2 ) , FˆX (tx ) + wM FˆX (tx2 ) − FˆX (tx2 ) .
By applying covariance on (8.2), we have (2r) (2) Cov FˆY (ty ), FˆX (tx ) + Cov wM (FˆY (ty2 ) − FˆY (ty2 )), FˆX (tx ) (2r) (2) ˆ ˆ ˆ +Cov FY (ty ), wM (FX (tx2 ) − FX (tx2 )) (8.3) = . (2r) (2) (2r) (2) +Cov wM (FˆY (ty2 ) − FˆY (ty2 )), wM (FˆX (tx2 ) − FˆX (tx2 ) Since (2r) (2) Cov FˆY (ty ), wM (FˆX (tx2 ) − FˆX (tx2 )) (2) (2r) = Cov wM (FˆY (ty2 ) − FˆY (ty2 )), FˆX (tx ) = 0. Hence (8.3) becomes (8.4)
Cov FˆY (ty ), FˆX (tx ) = . (2r) (2) (2r) (2) +Cov wM FˆY (ty2 ) − FˆY (ty2 ), wM FˆX (tx2 ) − FˆX (tx2 )
Following Garcia and Cebrian (1998), it is easy to obtain N − m m N N − N N 11 22 12 21 . Cov FˆY (ty ), FˆX (tx ) = N − 1 N2 m2 If we consider N − 1 ∼ = N , then we can write above as
(8.5)
1 − f N N − N N 11 22 12 21 , Cov FˆY (ty ), FˆX (tx ) = m N2
where f = m N and now consider last term of (8.3), (2r) (2) (2r) (2) Cov wM (FˆY (ty2 ) − FˆY (ty2 )), wM (FˆX (tx2 ) − FˆX (tx2 ))
(8.6)
(2r) (2) (2r) (2) E1 Cov2 wM (FˆY (ty2 ) − FˆY (ty2 )), wM (FˆX (tx2 ) − FˆX (tx2 )) = . (2r) (2) (2r) (2) ˆ ˆ ˆ ˆ +Cov1 E2 wM (FY (ty2 ) − FY (ty2 )), wM (FX (tx2 ) − FX (tx2 ))
Since, (2r) (2) (2r) (2) E2 w2 (FˆY (ty2 ) − FˆY (ty2 )), w2 (FˆX (tx2 ) − FˆX (tx2 )) = 0. Hence (8.6), becomes (8.7)
h i (2r) (2) (2r) (2) = E1 Cov2 wM (FˆY (ty2 ) − FˆY (ty2 )), wM (FˆX (tx2 ) − FˆX (tx2 )) .
41
Now consider (2r) (2) (2r) (2) Cov2 wM (FˆY (ty2 ) − FˆY (ty2 )), wM (FˆX (tx2 ) − FˆX (tx2 )) ,
(8.8)
(2r) (2) (2r) (2) E2 Cov3 wM (FˆY (ty2 ) − FˆY (ty2 )), wM (FˆX (tx2 ) − FˆX (tx2 )) = . (2r) (2) (2r) (2) +Cov2 E3 wM (FˆY (ty2 ) − FˆY (ty2 )), wM (FˆX (tx2 ) − FˆX (tx2 ))
h i (2r) (2) (2r) (2) = E2 Cov3 wM (FˆY (ty2 ) − FˆY (ty2 )), wM (FˆX (tx2 ) − FˆX (tx2 )) . (2r) (2) (2r) (2) Consider Cov3 wM (Fˆ (ty ) − Fˆ (ty )), wM (Fˆ (tx ) − Fˆ (tx )) , (8.9)
Y
Y
2
X
2
2
X
2
(2r) (2r) (8.10) = Cov3 FˆY (ty2 ), FˆX (tx2 ) .
By taking r sample points out of mM and after some simplifications, (??) becomes ! (2) (2) (2) (2) (k − 1) N11 N22 − N12 N21 (2r) (2r) ˆ ˆ (8.11) Cov3 FY (ty2 ), FX (tx2 ) = . (2) mM − 1 (mM )2 (2r) (2) (2r) (2) E2 Cov3 wM FˆY (ty2 ) − FˆY (ty2 ), wM FˆX (tx2 ) − FˆX (tx2 ) ,
m2 (k − 1) E2 (8.12) = 2 M m (mM − 1)
"
ˆ (2) ˆ (2) − N ˆ (2) N ˆ (2) N N 21 22 12 11 (2)
(mM )2
!# .
Applying expectations on (8.12), we have (k − 1)WM (8.13) = m
(2)
(2)
(2)
(2)
N11 N22 − N12 N21 NM (NM − 1)
! .
For considering, NM − 1 ∼ = NM , Equation (8.13) becomes, WM (k − 1) (8.14) = m
(2)
(2)
(2)
(2)
N11 N22 − N12 N21 (2)
(NM )2
! .
Using (8.5) and (8.14) in (8.2) we have following result given as (1 − f ) N11 N22 − N12 N21 m (N )2 (∗) (∗) ˆ ˆ ! (8.15) Cov FY (ty ), FX (tx ) = (2) (2) (2) (2) . + WM (k − 1) N11 N22 − N12 N21 (2) m (NM )2
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