Preliminary test estimation of the parameters of exponential and ...

Stat Papers (2010) 51:757–773 DOI 10.1007/s00362-008-0163-y REGULAR ARTICLE

Preliminary test estimation of the parameters of exponential and Pareto distributions for censored samples B. M. Golam Kibria · A. K. Md. E. Saleh

Received: 18 February 2008 / Revised: 9 July 2008 / Published online: 5 September 2008 © Springer-Verlag 2008

Abstract This paper deals with the preliminary test estimation (PTE) of the parameters of the exponential and Pareto distributions in censored samples. The biases, risk functions, efficiency tables and the graphs for the relative efficiency for the proposed estimators for the parameters of the exponential and Pareto distributions are given. We find that the proposed estimators dominate the corresponding unrestricted (usual) estimators in the neighborhood of null hypothesis. The range of the parameters for which the proposed estimators dominate the corresponding usual estimators for different sample sizes and level of significance are given. The findings of the paper will be useful for the practitioners who are dealing with the censored samples in life testing experiments. Keywords Dominance · Exponential distribution · MSE · Pareto distribution · Pre-test estimation · Relative efficiency 1 Introduction Censored samples is a common problem for life testing and reliability modeling. There are three types of possible censoring: right, left and double censoring. Both exponential and Pareto distributions among many, have significant applications in life testing modeling and reliability engineering modeling. It also plays a major role in quality control and stochastic process. The literature on the estimation and inference

B. M. G. Kibria (B) Department of Statistics, Florida International University, Miami, FL 33199, USA e-mail: [email protected] A. K. Md. E. Saleh Department of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada

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B. M. G. Kibria, A. K. Md. E. Saleh

procedures for the parameters of exponential distribution is extensive. Davis (1952) analyzed different types of failure data and concluded that the exponential distribution is quite well to fit most of the data. The estimation of the parameters of the two parameter exponential distribution based on the ordered statistics are considered by Kulldroff (1963), Saleh (1966), Wright et al. (1978), Piegorsch (1987) and Balakrishnan (1990) among others. For more on the parameter estimation and inference about the parameters of the exponential distribution, we refer Mann et al. (1974), Lawless (1982), Cohen and Whitten (1988), and recently Johnson et al. (1994) among others. The Pareto distribution was developed by Pareto (1897) to describe the distribution of income. However, it has a wide application in economic studies, insurance risk, business mortality, service time in queuing system among others. Estimation of the shape and scale parameters of a Pareto distribution by linear functions of order statistics have been considered by Kulldorf and Vannman (1973) and Vannman (1976) among others. The Pareto distribution with censored data have been considered by Rohatgi and Saleh (1987). For inference purposes, estimation of their parameters are very important and necessary. Several estimations techniques (for example, maximum likelihood or method of moments) exist in the literature for estimating the model parameters. However, the parameters estimation for the censored data by using the preliminary test estimator is very few. The preliminary test estimation approach has been proposed by Bancroft (1944) and Han and Bancroft (1968). Since 1944 several researchers have been used it for several purposes. Various aspects of this estimator have been discussed by many authors, such as, Bancroft (1964), Han and Bancroft (1968), Judge and Bock (1978), Kibria and Saleh (1993), Saleh and Kibria (1993), Benda (1996), Chiou and Han (1999), Han (2002) and recently Kibria and Saleh (2005, 2006) among others. A detailed review of the preliminary test estimation procedures are given by Han et al. (1977) and Giles and Gilies (1993) among others. Recently, Saleh (2006) expanded and discussed the insights of the preliminary test and the related shrinkage estimation techniques and provides a clear and balanced introduction to preliminary test and shrinkage estimation. The organization of this paper is as follows. The proposed estimators along with bias and MSE expression for the exponential distribution are given in Sect. 2 and for Pareto distribution are given in Sect. 3. An example has been considered in Sect. 4. Finally some concluding remarks are given in Sect. 5. 2 Estimation of the parameters of exponential distribution Let X be a random variable with an exponential distribution with the following pdf f (x) =

1 − x−θ e σ , x > θ, σ > 0 σ

where θ is the location parameter and σ is the scale parameter. The corresponding cdf of X is given by F(x) = 1 − e−

123

x−θ σ

, x > θ, σ > 0

Preliminary test estimation of the parameters of exponential and Pareto distributions

759

Consider a doubly censored sample from this distribution. Let x(a) ≤ · · · ≤ x(b) , 1 ≤ a ≤ b ≤ n be the doubly censored sample in a sample of size n. It is well known that the best linear unbiased estimator (BLUE) of (θ, σ ) are given by θˆn = x(a) − σˆ n δ1a , δ1a =

a

(n − j + 1)−1

(1)

j=1

and ⎫ ⎧ b−1 ⎬ 1 ⎨ σˆ n = x j − (n − a)x(a) + (n − b + 1)x(b) ⎭ b−a ⎩

(2)

j=a+1

respectively. The variances of these estimators are given by Var(θˆn ) = σ 2 δ2a +

σ2 2 δ1a , δ2a = (n − j + 1)−2 b−a a

j=1

and Var(σˆ n ) =

σ2 b−a

respectively. Furthermore the distribution of X (a) and σ˜ n are independent and 2(b − a)σˆ n ∼ χν2 , ν = 2(b − a) σ has a chi-square distribution with ν degrees of freedom. Let 1 − α = Hν (r2 ) − Hν (r1 ), where Hν (.) stands for the χ 2 cdf with ν degrees of freedom, 1 − α/2 = Hν (r2 ) and α/2 = Hν (r1 ). where r1 and r2 are the critical values from the central chi-square distribution with ν degrees of freedom. Our problem is to obtain a preliminary test estimators (PTE) of (θ, σ ) when it is a priori suspected that σ may be equal to σ0 . Often the information on the value of σ is available from past knowledge or previous experiment. According to Sir R. A. Fisher (Saleh 2006, page 58), this non-sample prior information can be expressed in the form of following test of the hypothesis H0 : σ = σ0 , vs. Ha : σ = σ0 Now we will choose σˆ or σ0 based on the rejection of H0 or do not reject of H0 . We reject H0 if χν2 ∈ A¯ and do not reject H0 if χν2 ∈ A, where

123

760


2 2 A = χν2 : r1 ≤ χν2 ≤ r2 , r1 = χν,α/2 , r2 = χν,1−α/2 Accordingly, the PTE of σ and θ are defined by ¯ = σˆ n − (σˆ n − σ0 )I (A) σˆ nPT = σ0 I (A) + σˆ n I ( A) and ¯ = x(a) − δ1a σˆ nPT θˆnPT = θ0 I (A) + θˆn I ( A) We may note that E(I (A)) = P(r1 ≤ χν2 ≤ r2 ) = 1 − α

2.1 Bias, MSE and relative efficiency The bias and MSE expression for the proposed estimators are given in this section. 2.1.1 Estimating σ Let us define λ = Bias =

σ0 σ ,

then the bias and MSE expressions of σˆ nPT are given by

E(σˆ nPT

1 − σ ) = σ0 (1 − α) − {Hν+2 (r2 ) − Hν+2 (r1 )} λ

(3)

and

2 MSE(σˆ nPT ) = E (σˆ n − σ ) − (σˆ n − σ0 )I (A) σ2 2 (b − a + 1) {Hν+4 (r2 ) − Hν+4 (r1 )} − σ0 = b−a (b − a)λ2 2 − × {Hν+2 (r2 ) − Hν+2 (r1 )} + (1 − α) λ 1 {Hν+2 (r2 ) − Hν+2 (r1 )} − (1 − α) −2σ0 (σ0 − σ ) × λ =

(b − a + 1) σ2 {Hν+4 (r2 ) − Hν+4 (r1 )} − σ02 b−a (b − a)λ2 +

123

2σ σ0 {Hν+2 (r2 ) − Hν+2 (r1 )} + (σ02 − 2σ0 σ )(1 − α) λ

(4)


761

respectively. The efficiency of σˆ nPT compare to σˆ n is: RE(σˆ nPT : σˆ n ) =

V (σˆ n ) = 1 + (b − a)(1 − α)(λ2 − 2λ) MSE(σˆ nPT ) +2(b − a) {Hν+2 (r2 ) − Hν+2 (r1 )} −(b − a + 1) {Hν+4 (r2 ) − Hν+4 (r1 )}

−1

(5)

Note that both MSE and efficiency functions depend on the sample size n, a, b and size of the test. Special case 1: For a = 1 and b = n. The relative efficiency in (5) becomes RE(σˆ nPT : σˆ n ) = 1 + (n − 1)(1 − α)(λ2 − 2λ) +2(n − 1) {Hν+2 (r2 , 0) − Hν+2 (r1 )} −1 − n {Hν+4 (r2 ) − Hν+4 (r1 )}

(6)

We have drawn several relative efficiency graphs which are presented in Fig. 1. The effect of n and the merits of the preliminary test estimator compared to usual estimator are evident.

10

Efficiency

0

5

10 5 0

Efficiency

15

n=20, alpha=0.05, a=2, b=8

15

n=10, alpha=0.05, a=2, b=8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

Lambda

2.0

2.5

3.0

n=100, alpha=0.05, a=2, b=8

10 0

0

5

5

Efficiency

10

15

15

n=50, alpha=0.05, a=2, b=8

Efficiency

1.5

Lambda

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Lambda

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Lambda

Fig. 1 Relative efficiency of σˆ nPT for different values of n

123

762


2.1.2 Estimating θ Let us define λ =

σ0 σ ,

then the bias and MSE expressions of θnPT are given by

1 Bias = E(θˆnPT − θ ) = δ1a σ0 (1 − α) − {Hν+2 (r2 ) − Hν+2 (r1 )} , λ

(7)

and δ2 δ1a MSE(θˆnPT ) = σ 2 δ2a + + σ 2 1a (b − a)(1 − α)(λ2 − 2λ) b−a b−a +2(b − a){Hν+2 (r2 ) − Hν+2 (r1 )} −(b − a + 1){Hν+4 (r2 ) − Hν+4 (r1 )}

(8)

respectively. The efficiency of θˆnPT compare to θˆn is: RE(θˆnPT : θˆn ) =

V (θˆn ) MSE(θˆnPT )

= 1+

(b

2 δ1a

2 − a)δ2a + δ1a

(b − a)(1 − α)(λ2 − 2λ)

+2(b − a) {Hν+2 (r2 ) − Hν+2 (r1 )} − (b − a + 1){Hν+4 (r2 ) − Hν+4 (r1 )}

−1 (9)

Special case 2: For a = 1 and b = n. The relative efficiency in (9) becomes 1 PT ˆ ˆ RE(θn : θn ) = 1 + (n − 1)(1 − α)(λ2 − 2λ) n +2(n − 1) {Hν+2 (r2 ) − Hν+2 (r1 )}

−1 −n {Hν+4 (r2 ) − Hν+4 (r1 )}

(10)

For fix n = 50, we have drawn several relative efficiency graphs which are presented in Fig. 2. The effect of a and b and the merits of the preliminary test estimator compared to unrestricted estimators are evident. The proposed σˆ nPT dominates σˆ n in the range of λ for different n and α are given in Table 1 and θˆnPT dominates θˆn in the range of λ for different n and α are given in Table 2. From these tables it is evident that the proposed PT dominates usual estimators near the null hypothesis.

123


1.5 0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

1.5

2.0

2.5

Lambda

n=50, alpha=0.05, a=2, b=5

n=50, alpha=0.05, a=2, b=8

3.0

1.5 1.0 0.5 0.0

0.0

0.5

1.0

Efficiency

1.5

2.0

Lambda

2.0

0.0

Efficiency

1.0

Efficiency

0.0

0.5

1.0 0.0

0.5

Efficiency

1.5

2.0

n=50, alpha=0.05, a=1, b=5

2.0

n=50, alpha=0.05, a=1, b=2

763

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

Lambda

1.0

1.5

2.0

2.5

3.0

Lambda

Fig. 2 Relative efficiency of θˆnPT for different values of a and b

3 Estimation of the parameters of Pareto distribution Consider a sample of size n, say x1 , x2 , . . . , xn from a Pareto distribution with density f (x) =

1 1/σ −(1/σ +1) k x , x ≥k σ

(11)

where k(>0) is the shape parameter and σ > 0 is the scale parameter. The corresponding CDF of the Pareto distribution is given by 1 k σ , k > 0, σ > 0, x ≥ k, F(x) = 1 − x

(12)

Let x(1) ≤ x(2) ≤ · · · ≤ x(n) be the order statistics of the sample. Then it is known from Johnson et al. (1994) that the r th moment about zero for the jth smallest order statistics is E(X (r j) ) =

Γ (n + 1)Γ (n − j + 1 − r σ ) r k , r = 1, 2, . . . , n Γ (n + 1 − r σ )Γ (n − j + 1)

(13)

123

764


Table 1 σˆ nPT dominates σˆ n in the range of λ for different n and α n

α 0.01

0.05

0.10

0.15

0.20

0.25

0.50

3

[0.35, 1.65]

[0.43, 1.58]

[0.48, 1.53]

[0.53, 1.48]

[0.58, 1.43]

[0.60, 1.4]

[0.73, 1.28]

4

[0.48, 1.52]

[0.48, 1.52]

[0.52, 1.48]

[0.58, 1.42]

[0.65, 1.35]

[0.68, 1.33]

[0.78, 1.23]

5

[0.55, 1.45]

[0.58, 1.42]

[0.62, 1.38]

[0.65, 1.35]

[0.68, 1.30]

[0.73, 1.28]

[0.82, 1.18]

6

[0.58, 1.42]

[0.62, 1.38]

[0.68, 1.33]

[0.70, 1.3]

[0.73, 1.28]

[0.75, 1.25]

[0.85, 1.15]

7

[0.62, 1.38]

[0.65, 1.35]

[0.70, 1.30]

[0.73, 1.28]

[0.75, 1.25]

[0.78, 1.23]

[0.85, 1.15]

8

[0.65, 1.35]

[0.68, 1.30]

[0.73, 1.28]

[0.73, 1.25]

[0.78, 1.23]

[0.78, 1.20]

[0.88, 1.12]

9

[0.68, 1.33]

[0.70, 1.30]

[0.73, 1.25]

[0.75, 1.25]

[0.78, 1.23]

[0.80, 1.20]

[0.88, 1.12]

10

[0.68, 1.30]

[0.73, 1.28]

[0.75, 1.25]

[0.78, 1.23]

[0.80, 1.20]

[0.80, 1.20]

[0.88, 1.12]

15

[0.75, 1.25]

[0.78, 1.23]

[0.80, 1.20]

[0.82, 1.18]

[0.85, 1.15]

[0.85, 1.15]

[0.90, 1.10]

20

[0.78, 1.20]

[0.82, 1.18]

[0.82, 1.18]

[0.85, 1.15]

[0.85, 1.15]

[0.88, 1.12]

[0.92, 1.08]

25

[0.82, 1.18]

[0.85, 1.15]

[0.85, 1.15]

[0.88, 1.12]

[0.88, 1.12]

[0.88, 1.12]

[0.92, 1.08]

30

[0.85, 1.15]

[0.85, 1.15]

[0.88, 1.12]

[0.88, 1.12]

[0.88, 1.12]

[0.90, 1.10]

[0.95, 1.05]

35

[0.85, 1.15]

[0.88, 1.12]

[0.88, 1.12]

[0.88, 1.12]

[0.90, 1.10]

[0.90, 1.10]

[0.95, 1.05]

50

[0.88, 1.12]

[0.88, 1.12]

[0.90, 1.10]

[0.90, 1.10]

[0.92, 1.08]

[0.92, 1.08]

[0.95, 1.05]

100

[0.92, 1.08]

[0.92, 1.08]

[0.95, 1.05]

[0.95, 1.05]

[0.95, 1.05]

[0.95, 1.05]

[0.98, 1.03]

200

[0.95, 1.05]

[0.95, 1.05]

[0.95, 1.05]

[0.95, 1.05]

[0.98, 1.03]

[0.98, 1.03]

[0.98, 1.00]

Table 2 θˆnPT dominates θˆn in the range of λ for different n and α n

α 0.01

0.05

0.10

0.15

0.20

0.25

0.50

3

[0.35, 1.65]

[0.43, 1.58]

[0.48, 1.52]

[0.52, 1.48]

[0.55, 1.45]

[ 0.58, 1.42]

[0.70, 1.30]

4

[0.45, 1.55]

[0.52, 1.48]

[0.58, 1.42]

[0.60, 1.40]

[0.65, 1.35]

[0.65 ,1.35]

[0.75, 1.25]

5

[0.52, 1.48]

[0.58,1.42]

[0.62, 1.38]

[0.65, 1.35]

[0.68, 1.33]

[0.70, 1.30]

[0.78, 1.20]

6

[0.58, 1.42]

[0.62, 1.38]

[0.65, 1.35]

[ 0.68, 1.30]

[0.70, 1.30]

[0.73, 1.28]

[0.80, 1.20]

7

[0.62, 1.38]

[0.65 1.35]

[0.68, 1.30]

[0.73, 1.25]

[0.75, 1.25]

[0.75, 1.25]

[0.82, 1.18]

8

[0.65, 1.35]

[0.68, 1.33]

[0.70, 1.30]

[0.73, 1.28]

[0.75, 1.25]

[0.78, 1.23]

[0.82, 1.18]

9

[0.68, 1.33]

[0.70, 1.30]

[0.73, 1.28]

[0.75, 1.25]

[0.78, 1.23]

[0.78, 1.20]

[0.85, 1.15]

10

[0.68, 1.30]

[0.73, 1.28]

[0.73, 1.25]

[0.75, 1.25]

[0.78, 1.23]

[0.80, 1.20]

[0.85, 1.15]

15

[0.75, 1.25]

[0.78, 1.23]

[0.78, 1.20]

[0.80, 1.20]

[0.82, 1.18]

[0.82, 1.18]

[0.88, 1.12]

20

[0.78, 1.23]

[0.80, 1.20]

[0.82, 1.18]

[0.82, 1.18]

[0.85, 1.15]

[0.88, 1.15]

[0.88, 1.12]

25

[0.80, 1.20]

[0.82, 1.18]

[0.85, 1.15]

[0.85, 1.15]

[0.85, 1.15]

[0.88, 1.12]

[0.88, 1.12]

30

[0.82, 1.18]

[0.85, 1.15]

[0.85, 1.15]

[0.85, 1.15]

[0.88, 1.12]

[0.88, 1.12]

[0.88, 1.12]

50

[0.85, 1.15]

[0.88, 1.12]

[0.88, 1.12]

[0.88, 1.12]

[0.88, 1.12]

[0.90, 1.10]

[0.90, 1.10]

100

[0.88, 1.12]

[0.90, 1.10]

[0.90, 1.10]

[0.90, 1.10]

[0.90, 1.10]

[0.90, 1.10]

[0.90, 1.10]

200

[0.92, 1.08]

[0.92, 1.08]

[0.92, 1.08]

[0.92, 1.08]

[0.92, 1.08]

[0.92, 1.09]

[0.90, 1.10]

123


765

Further we notice that, Y = ln(X ) has an exponential distribution with location parameter θ = ln(k) and scale parameter σ . Thus following Johnson et al. (1994) we have E(Y(i) ) = ln(k) + σ δi1 , δ1i =

i (n − j + 1)−1

(14)

j=1

and V (Y(i) ) = Cov(Y (i, Yi+t )) = σ 2 δ2i , δ2i =

i (n − j + 1)−2

(15)

j=1

Our problem is to obtain an improved estimator of (k, σ ) when a priori σ may be equal σ0 is suspected. 3.1 Bias, MSE and relative efficiency The bias and MSE of the proposed estimators are given in this section. 3.1.1 Estimating σ Here we are testing H0 : σ = σ0 vs. Ha : σ = σ0 From Sect. 2, we note that the PTE of σ is given by σˆ bPT = σ˜ b − (σ˜ b − σ0 )I (A)

(16)

The bias of σ˜ bPT is given by B(σ˜ bPT )

= σ0

σ {Hν+2 (r2 ) − Hν+2 (r1 )} ; ν = 2(b − 1) (1 − α) − σ0

(17)

Following (4), the MSE of σ˜ bPT is given by MSE(σ˜ bPT ) =

b σ2 {Hν+4 (r2 ) − Hν+4 (r1 )} − σ2 (b − 1) (b − 1) + 2σ 2 {Hν+2 (r2 ) − Hν+2 (r1 )} + (σ02 − 2σ σ0 )(1 − α)

(18)

The Bias and MSE expressions of the σ˜ bPT for Pareto distributions are identical to those in exponential distribution which are given in Sect. 2. Thus, the efficiency tables are also appropriate for Pareto distribution.

123

766


3.1.2 Estimating k Defining θ = ln k as in Sect. 2. Then based on the censored sample Y(a) < · · · < Y(b) , the BLUE of θ and σ are given, respectively θñ = Y(a) − δ1a σ˜ n , and σ˜ n =

b−1 1 Y( j) − (n − a)Y(a) + (n − b + 1)Y(b) b−a

(19)

j=1

where Y( j) = ln X ( j) , j = a, a + 1, . . . , b. Thus, PTE of θ is given by θˆnPT = Y(a) − δ1a σ˜ nPT The bias and MSE expression of θˆnPT are given in (7) and (8), respectively. Thus the efficiency expression for θˆnPT is given (9). But, what about the direct estimation of k. Since, θ = ln k, we have k = eθ . Hence, the unrestricted estimator of k is given by ˜ kñ = eθn = eln X (a) −δ1a σ˜ n = X (a) e−δ1a σ˜ n

(20)

It is known that X (a) and σ˜ n are independent with the pdf’s given by g X (a) (y) =

n! a!(n − a)!σ

k

n−a+1 σ

(y)

+1 − n−a+σ σ

,

y>k

(21)

and 1 g(σ˜ n ) = Γ (b − a)

b−a σ

b−a e

− b−a σ˜ n σ

(σ˜ n )b−a−1

(22)

respectively. The moments of X (a) are given by r E[X (a) ]=

Γ (n + 1)Γ (n − a + 1 − r σ ) r k ; Γ (n + 1 − r σ )Γ (n − a + 1)

r = 1, 2, . . .

(23)

For a = 1, we have r E[X (1) ]=

123

n kr ; n − rσ

r = 1, 2, . . .

(24)


767

We then have E(kñ ) = E(X (a) )E e−δ1a σ˜ =

Γ (n + 1)Γ (n + 1 − a − σ ) k E e−δ1a σ˜ Γ (n + 1 − σ )Γ (n − a + 1)

(25)

But, E e−δ1a σ˜ =

1 b−a

∞ 0

b−a σ

b−a

e−σ˜ n [δ1a +

b−a σ ]

[σ˜ n ]b−a−1 d σ˜ n

1 b − a −1 b − a b−a b − a −(b−a−1) δ1a + δ1a + = b−a σ σ σ ∞ × z b−a−1 e−z dz

0

= Γ (b − a)(b − a)b−a−1 [δ1a σ + b − a]−(b−a)

(26)

Now the PTE of k is defined as ¯ kñPT = k0 I (A) + kñ I ( A) ¯ = X (a) e−δ1a σ0 I (A) + X (a) e−δ1a σ˜ n I ( A)

¯ = X (a) e−δ1a σ0 I (A) + e−δ1a σ˜ n I ( A)

(27)

The bias of the estimator, kñPT is obtained as

¯ −k Bias = E(kñPT ) − k = E(X (a) )E e−δ1a σ0 I (A) + e−δ1a σ˜ n I ( A)

(28)

We observe from (2.8) that the bias and MSE expressions of the estimator kñPT are not simple to evaluate. Instead, we may obtain approximate expression by writing kˆn = X (a) (1 − δ1a σ˜ n )

(29)

Then we have the following information E(kˆn ) = E(X (a) ) {1 − δ1a E(σ˜ n )} =

Γ (n + 1)Γ (n + 1 − a − σ ) k(1 − δ1a σ ) Γ (n + 1 − σ )Γ (n − a + 1)

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768


2 2 2 b−a+1 E(kˆn2 ) = E(X (a) ) 1 − 2δ1a σ + δ1a σ b−a Γ (n + 1)Γ (n + 1 − a − 2σ ) 2 2 2 b−a+1 k 1 − 2δ1a σ + δ1a = σ Γ (n + 1 − 2σ )Γ (n − a + 1) b−a 2 2 ˜ ˜ ˆ (30) V (kn ) = E(kn ) − [E(kn )] Similarly the PTE of k may be written approximately as kˆnPT = X (a) (1 − δia σ˜ nPT )

(31)

For computational convenience, based on the right censored sample given by X (1) < · · · < X (b) , we define the unrestricted and PTE of k are given, respectively kˆb = X (1)

σ˜ b 1− n

and kˆbPT = X (1)

σ˜ PT 1− b n

(32)

The estimator kˆb and σ˜ b are unbiased with variances 2σ −1 bσ 2 2 ˆ V (k b ) = k 1 − n n 2 (b − 1) 2 bσ 2 k and = n − 2σ n(b − 1) σ2 V (σ˜ b ) = b−1

(33)

respectively. The bias and MSE of kˆbPT are, respectively k B(σ˜ bPT ) and n−σ MSE(kˆbPT ) = V (k˜bPT ) + B 2 (k˜bPT ) B(kˆbPT ) = −

(34)

where

PT σ ˜ V (kˆbPT ) = V X (1) ) 1 − b n 2 PT 2 PT σ ˜ σ ˜ 2 2 )E 1 − b − E(X (1) ) E 1− b = E(X (1) n n σ 2 1 σ 2σ −1 2 1− 1− Bias(σ˜ bPT ) = k2 1 − + 2 MSE(σ˜ bPT ) − n n n n n 2 σ 1 σ −2 1− − Bias(σ˜ bPT ) (35) − k2 1 − n n n

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769

The MSE of kˆbPT is MSE(kˆbPT ) = k 2

MSE(σ˜ bPT ) (n − σ )2 + n(n − 2σ ) n(n − 2σ ) 2 ⎫ 2(B(σ˜ bPT )) ⎬ 2(n − σ )B(σ˜ bPT ) − 1− − ⎭ n(n − 2σ ) (n − σ )

(36)

3.2 Relative efficiency The relative efficiency (RE) of kˆbPT compared to kˆb is given by RE(kˆbPT : kˆb ) =

V (kˆb ) MSE(kˆ PT )

(37)

b

From Eq. 37 we observed that the RE does not depend on the value of k. The relative efficiency of the proposed estimator compared to unrestricted estimator for various values of n, α and b are presented in Figs. 3, 4, and 5. The effect of sample size, censoring and α is evident from these figures. It appeared that near the null hypothesis the proposed estimator is more efficient. The range of λ for which the proposed estimator dominates the corresponding unrestricted estimator are presented in Table 3. 4 An application In this section, we consider an example to illustrate the performance of the proposed PT estimator. Thirty seven different categories of contract lens wearers were analyzed for cataract mortality (Williams et al. 2002, pg. 129; Table 1); the following ordered data represent the number of deaths from cataracts for each category: 24, 54, 60, 67, 82, 99, 108, 111, 126, 146, 166, 212, 247, 262, 282, 284, 319, 360, 362, 390, 425, 438, 438, 445, 469, 478, 480, 501, 517, 520, 560, 767, 769, 1021, 1109, 1269, 1281 We want to estimate the scale parameter σ . The sample is positively skewed with skewness = 1.27, and mean = 412.11. The estimated value of θ is θˆn = 24. Then the estimated value of σ is σˆ n = 388.108. Suppose we doubt that the true σ could be close to 400. That means, we want to test the following hypothesis H0 : σ = 400, vs. Ha : σ = 400 The value of the test statistic is χ02 =

2(n − 1)σˆ = 69.85946 σ0

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770

B. M. G. Kibria, A. K. Md. E. Saleh n=20, b=8, k=1, sig0=1, alpha=0.05

1.5 1.0

Efficiency

1.0

0.0

0.5

UE PT

0.5

1.5

UE PT

0.0

Efficiency

2.0

2.0

n=10, b=8, k=1, sig0=1, alpha=0.05

0

1

2

3

4

0

1

Sigma

3

4

n=100, b=8, k=1, sig0=1, alpha=0.05

2.0

2.0

n=50, b=8, k=1, sig0=1, alpha=0.05

1.0 0.0

0.5

1.0

Efficiency

1.5

UE PT

0.5

1.5

UE PT

0.0

Efficiency

2

Sigma

0

1

2

3

4

Sigma

0

1

2

3

4

Sigma

Fig. 3 Relative efficiency of kˆbPT for different values of n

The p-value with 2(37 − 1) = 72 degrees of freedoms is 0.55. Therefore, we cannot reject the null hypothesis. In this case the PT for σ is σˆ nPT = 400. The estimated value of λˆ = σσˆ0 = 1.031. From Table 1 we observed that for sample size (35) and for all α, the estimated value 1.031 falls within the limits of the λ for which σˆ nPT is more efficient compare to σˆ . From Table 1, we also observed that at 5% level of significance, the σˆ nPT will dominate σˆ n for any value σ0 between 340 and 440.

5 Concluding remarks In this paper, we have proposed the preliminary test estimators for estimating the location and scale parameters of the exponential distribution and shape and scale parameters of the Pareto distribution when data are censored. It is evident that all proposed estimators dominate corresponding usual estimators in the neighborhood of null hypothesis. We also observed that the proposed estimators based on the censored data are more efficient than the estimators based on the complete data set. A real life example has been analyzed to illustrate the performance of the PT estimator. We believe that the findings of the paper will be useful for the practitioners who deal with the censored data.

123

Preliminary test estimation of the parameters of exponential and Pareto distributions n=20, b=8, k=1, sig0=1, alpha=0.05

1.5 1.0

Efficiency

1.0

0.0

0.5

UE PT

0.5

1.5

UE PT

0.0

Efficiency

2.0

2.0

n=20, b=8, k=1, sig0=1, alpha=0.01

0

1

2

3

4

0

1

Sigma

2

3

4

Sigma n=20, b=8, k=1, sig0=1, alpha=0.20

2.0

2.0

n=20, b=8, k=1, sig0=1, alpha=0.10

1.0 0.0

0.5

1.0

Efficiency

1.5

UE PT

0.5

1.5

UE PT

0.0

Efficiency

771

0

1

2

3

0

4

1

Sigma

2

3

4

Sigma

Fig. 4 Relative efficiency of kˆbPT for different values of α n=50, b=20, k=1, sig0=1, alpha=0.05

1.0

1.5

UE PT

0.5

1.0

Efficiency

1.5

UE PT

0.0

0.0

0.5

Efficiency

2.0

2.0

n=50, b=10, k=1, sig0=1, alpha=0.05

0

1

2

3

4

0

1

Sigma

3

4

n=50, b=40, k=1, sig0=1, alpha=0.05

1.5

UE PT

0.0

0.0

0.5

0.5

1.0

Efficiency

1.5

UE PT

1.0

2.0

2.0

n=50, b=30, k=1, sig0=1, alpha=0.05

Efficiency

2

Sigma

0

1

2

3

4

Sigma

0

1

2

3

4

Sigma

Fig. 5 Relative efficiency of kˆbPT for different values of b

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772


Table 3 kˆbPT dominates kˆb in the range of λ for different n, b, and α n

b

α 0.01

0.05

0.10

0.20

0.50

10

10

[0.18, 1.43]

[0.33, 1.35]

[0.40, 1.28]

[0.53, 1.20]

[0.68, 1.10]

10

8

[0.15, 1.53]

[0.30, 1.43]

[0.38, 1.35]

[0.50, 1.25]

[0.68, 1.13]

10

7

[0.15 ,1.60]

[0.28, 1.48]

[0.38, 1.40]

[0.48, 1.28]

[0.68, 1.15]

10

5

[0.13 ,1.85]

[0.28, 1.68]

[0.35, 1.55]

[0.45, 1.38]

[0.63, 1.20]

20

20

[0.23, 1.45]

[0.40, 1.35]

[0.50, 1.28]

[0.63, 1.20]

[0.78, 1.10]

20

18

[0.23, 1.50]

[0.40, 1.40]

[0.50, 1.30]

[0.63, 1.20]

[0.78, 1.10]

20

15

[0.20, 1.6]

[0.38, 1.45]

[0.48, 1.35]

[0.60, 1.25]

[0.78, 1.13]

20

10

[0.18, 1.90]

[0.35, 1.65]

[0.45, 1.50]

[ 0.58, 1.33]

[0.73, 1.15]

20

5

[0.15, 2.50]

[0.28, 2.33]

[0.38, 1.95]

[0.48, 1.58]

[0.68, 1.25]

50

50

[0.33, 1.53]

[0.53, 1.38]

[0.63, 1.28]

[0.73, 1.20]

[0.83, 1.13]

50

40

[0.30, 1.63]

[0.53, 1.45]

[0.63, 1.33]

[0.73, 1.20]

[0.83, 1.13]

50

30

[0.28, 1.80]

[0.50, 1.55]

[0.60, 1.40]

[0.70, 1.25]

[0.83, 1.13]

50

20

[0.25, 2.15]

[0.45, 1.75]

[0.58, 1.53]

[0.68, 1.33]

[0.83, 1.15]

50

10

[0.20, 2.50]

[0.38, 2.45]

[0.48, 1.93]

[0.60, 1.50]

[0.78, 1.20]

100

100

[0.38, 1.60]

[0.63, 1.40]

[0.70, 1.30]

[0.78, 1.20]

[0.88, 1.13]

100

80

[0.38, 1.70]

[0.60, 1.45]

[0.70, 1.33]

[0.78, 1.20]

[0.85, 1.15]

100

60

[0.35, 1.88]

[0.58, 1.53]

[0.68, 1.35]

[0.78, 1.23]

[0.85, 1.15]

100

40

[0.33, 2.20]

[0.55, 1.70]

[0.65, 1.45]

[0.75, 1.28]

[0.85, 1.15]

100

20

[0.28, 2.50]

[0.48, 2.23]

[0.58, 1.73]

[0.70, 1.4o]

[0.83, 1.18]

200

200

[0.45, 1.75]

[0.68, 1.43]

[0.75, 1.30]

[0.83, 1.20]

[0.88, 1.15]

200

160

[0.45, 1.85]

[0.68, 1.45]

[0.75, 1.30]

[0.83, 1.20]

[0.88, 1.15]

200

120

[0.43, 2.00]

[0.65, 1.50]

[0.73, 1.35]

[0.80, 1.23]

[0.88, 1.15]

200

80

[0.40, 2.30]

[0.63, 1.63]

[0.73, 1.40]

[0.80, 1.25]

[0.88, 1.15]

200

40

[0.35, 2.50]

[0.58, 2.00]

[0.68, 1.58]

[0.75, 1.33]

[0.85, 1.15]

Acknowledgments The authors are thankful to the Editor and two anonymous referees for the careful reading and constructive suggestions which improved the presentation and quality of the paper. The first author is thankful to the Dean of the College of Arts and Sciences of Florida International University for awarding him the summer research award 2007.

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