Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 14, 685 - 698
Some Results on Maximum Likelihood Estimators of Parameters of Generalized Half Logistic Distribution under Type-I Progressive Censoring with Changing Failure Rate Sumeet H. Arora Institute of Management, Nirma University S-G Highway, Ahmedabad- 382481, India sumeetharora @gmail.com,
[email protected] G. C. Bhimani Department of Statistics, Saurashtra University University Road, Rajkot, India
[email protected] M. N. Patel Department of Statistics, School of Sciences Gujarat University, Ahmedabad- 380009, India
[email protected]
Abstract In this paper the study of the maximum likelihood estimators of the generalized half logistic distribution under type I progressive censoring with changing failure rates is considered. The numerical evaluation of their relative performance is made for selected values of n and p. MLE and its asymptotic variance are obtained using a
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simulation study based on 1000 random samples. Further results including total expected waiting time are obtained in case of interval censoring schemes also. Keywords: Censored Samples; Progressive Censoring; Generalized Half-Logistic Distribution; Expected Waiting Time
1. Introduction In most applications, the data may be interval-censored. By interval-censored data, we mean that a random variable of interest is known only to lie in an interval, instead of being observed exactly. In such cases, the only information we have for each individual is that their event time falls in an interval, but the exact time is unknown. Generally statistician faces lot of problem in the analysis of time-to-event data such as failure time data, incubation time data etc. Such data arises in lot of fields such as medicine, engineering, economics. For example doctor may be interested to know the time of convergence to AIDS for HIV positive individual, the time to the death for cancer patients, lifetime of a device etc. The analysis to time-to-event later becomes more complicated on account of censoring. Group censoring also known as interval censoring, arises when observations occur in some interval of time a and b. Such data occurs in variety of circumstances but generally it is encountered in medical studies where patients are only monitored at regular intervals (e.g. weekly or quarterly checkup). Thus, the exact time of occurrence of some changed response may only be known to have some time between two visits. Samuelson and Kongerud (1994); Kokasa et al (1993); Farrington (1996); Odell et al (1992), Sun (1997); Lindsey and Ryan (1998) and Scallan (1999) have discussed application of interval censoring in clinical, medical, biomedical and engineering studies. Rao (1998) gave standard methods for analyzing interval censored data and discussed efficiencies of estimators derived from censoring over conventional type-I and type- II censoring schemes. In many life test studies, it is common that the lifetimes of test units may not be recorded exactly. An experimenter may terminate the life test before all n products fail in order to save time or cost. Hence, the test is said to be censored in which data collected are the exact failure times on those functional (none failed) units. Moreover, some of the test units may have to be removed at different stage(s) of censoring related study for various other reasons; which leads to progressive censoring. For example some products are withdrawn for more thorough inspection or are saved so that it can be used as test specimens in other studies, or patients who for some reasons do not turn up in a clinical study would also result in progressive removal.
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According to the current trend Type-I and Type-II progressive censoring schemes are becoming quite popular for analyzing highly reliable data. Cohen (1963) had introduced progressive Type-II censoring. Mahmond et al (2006) considered progressive type-II censoring samples for many continuous life time models. Balakrishnan and Aggarwala (2000) give an insight on this method and the applications of this scheme.
Aggarwala (2001) introduced progressive Type-I interval censoring scheme for exponential life time model. In this type of censoring n units are put on test at time 0 and each unit is kept on life test until the unit fails or is censored. All the units are observed during pre-set times T1, T2,…, Tm where m is a fixed integer. Thus the time axis is partitioned into interval Ii = (Ti-1, Ti] where i = 1, 2,…, m+1 and T0 = 0, Tm+1 = ∞, Tm is the time at which we will terminate the experiment. Let ni denote the number of units which fail in the interval Ii. The values R1, R2,…, Rm may be with pm = 100 of specified as positive integers or percentages p1, p2, …, pm remaining functional units and the number of units which are functioning at time T1, T2, …,Tm are random variables.
In case when R1, R2, …, Rm are pre- specified positive integers, the number of units removed at time Ti is Riobs = min( Ri, no. of units remaining) i = 1, 2, …, m-1 and Rmobs = all the remaining units at time Tm, when life test experiment is terminated.
2. Generalized Half Logistic Distribution Half logistic model obtained as the distribution of the absolute standard logistic variate is probability model considered by Balakrishnan (1985). Balakrishnan and Puthenpura (1986) obtained best linear unbiased estimator of location and scale parameters of the half logistic distribution through linear functions of order statistics. Balakrishnan and Wong (1991) obtained approximate maximum likelihood estimates for the location and scale parameters of the half logistic distribution with Type-II Right-Censoring. Olapade (2003) proved some theorems that characterized the half logistic distribution. The half logistic distribution has not received much attention from researchers in terms of generalization. A generalized (Type-II) version of logistic distribution is considered and some interesting properties of the distribution were derived by Balakrsihnan and Hossain (2007). The generalized versions of half logistic distribution namely Type-I and Type–II were considered along with point estimation of scale parameters and estimation of stress strength reliability based on complete sample by Ramakrishna (2008).
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3. Maximum Likelihood Estimation Let the life X of an item have the generalized half logistic distribution with cdf
( ) ( )
θ
θ θ 2e-x ⎡ 2e-x ⎤ F ( x,θ ) = 1- ⎢ , x > 0, θ > 0 . ⎥ and density function f ( x,θ ) = -x θ+1 -x ⎢⎣1 + e ⎥⎦ 1+ e
Suppose that the times of censoring are Ti , i = 1, 2, …, k-1 and the experiment is finally terminated at Tk, Tk = ∞ where Ti < Ti+1 for i = 1, 2, …, k-1. Suppose that the parameter θ of the distribution changes at T1, T2, …,Tk. If θi is the parameter in the interval [Ti-1, Ti) for i = 1, 2, …, k-1 with T0 = 0, using the lemma given by Patel and Gajjar (1995), the composite density is given by
( ) ( ) ( (
θ1 ⎧ θ1 2e-x ⎪ , 0 ≤ x < T1 ⎪f1 ( x ) = θ1 +1 ⎪ 1+e-x ⎪ ⎪ θ1 θ2 ⎪ 2e-T1 /1 + e-T1 θ 2 2e-x ⎪f 2 ( x ) = , T1 ≤ x < T2 θ θ 1 + ⎪ 2 2 f (x) = ⎨ 1+e-x 2e-T1 /1 + e-T1 ⎪ θ ⎪ θ1 θ2 ⎛ 2e-Tj /1 + e-Tj ⎞ j ⎪ ⎜ ⎟ 2e-T1 /1 + e-T1 θ 2 2e-x k −1 ⎝ ⎠ ⎪f ( x ) = , Tk-1 ≤ x < Tk = ∞ ∏ θj θ +1 ⎪k -Ti-1 -Ti-1 θi j = 2 ⎛ -Tj-1 -x 2 -Tj⎞ 1 1+ e 2e /1 e + ⎪ /1 + e ⎜ 2e ⎟ ⎝ ⎠ ⎪ ⎪⎩ k=3,4,5,........
(
) ( ) ) ( ) ) )
(
( ) ( )
(3.1) The corresponding distribution function is given by, θ ⎧ ⎛ -x ⎞ 1 ⎪F ( x ) = 1 − ⎜ 2e ⎟ ,0 ≤ x < T 1 ⎜ 1+e-x ⎟ ⎪1 ⎝ ⎠ ⎪ θ1 ⎪ -T -x -x θ 2 ⎪F x = 1 − ⎛⎜ 2e 1 ⎞⎟ ⎧⎪ 2e /1 + e ⎫⎪ , T ≤ x < T ⎨ ⎬ ⎪ 2( ) 1 2 -T ⎟ -T -T ⎜ F(x) = ⎨ ⎝ 1 + e 1 ⎠ ⎩⎪ 2e 1 /1 + e 1 ⎭⎪ ⎪ θj θ ⎪ -T -T -T1 ⎞θ1 k-1 ⎧ -x -x ⎛ ⎫⎪ k ⎪ 2e j /1 + e j ⎫⎪ ⎧⎪ 2e /1 + e ⎪F ( x ) = 1 − ⎜ 2e ,Tk-1 ≤ x < Tk = ∞ ⎟ ∏ ⎨ ⎬ ⎨ ⎬ -T -Tj-1 -Tj-1 -T ⎜ 1 + e-T1 ⎟ j =2 ⎪ k ⎪ 2e k-1 /1 + e k-1 ⎭⎪ ⎪ ⎪ + 2e /1 e ⎩ ⎝ ⎠ ⎩ ⎭ ⎪ ⎪⎩ k=3,4,5,........
(3.2)
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689
Suppose n items are placed on a life test without replacement and that ni be the (i) number of items that fail during ith stage and let x1(i) ≤ x (i) 2 ≤ .... ≤ x n i be the times of failure for i = 1, 2, …, k-1 (k>1).
Let ri be the number of items removed or censored from the test immediately after time Ti-1 , i= 2, 3,…, k. Then the likelihood function from k-stage Type-I progressive censoring is given by,
( )⎫⎪⎬⎪⎭
k ⎧ ⎪ ni L ∝ ∏ ⎨ ∏ fi x(ji ) i=1 ⎪ ⎩ j=1
k ri ∏ ⎡⎣1 − Fi ( Ti ) ⎤⎦ . (3.3) i =1
Using (3.1) and (3.2) it is easy to verify that the likelihood function L can be written as, k
L ∝ ∏ Li . i=1
Where in the case of generalized Type-II half logistic distribution
Li ∝
ni (i) - ∑ x θi j n i j=i θi e (i) ⎫θi ni ⎧ ⎪ -x j ⎪ ∏ ⎨1+e ⎬ j =1 ⎪ ⎪
⎧⎪ e-Ti-1 ⎫⎪ ⎨ -Ti-1 ⎬ ⎩⎪1+e ⎭⎪
− θi n i
(
⎧ -( Ti -Ti-1 ) 1+e-Ti-1 ⎪e ⎨ 1+e-Ti ⎪ ⎩
) (
)
θ n (i) -n i ⎫i ⎪ . ⎬ (3.4) ⎪ ⎭
⎩ ⎭ for i = 2, 3, ..., k, with T0 = 0 where n(1) = n, n(i) = n(i-1) – ni – ri for i = 1, 2, …, k The log likelihood function is given by (i) ⎛ ni ni −x ⎞ j ⎟ ⎜ lnLi = lnc +n i ln θi − ∑ x (i) θ θ 1 ln 1 − + + e ∑ ) j i ( i ⎟ j=1 j=1 ⎜ ⎝ ⎠ T − ⎛ i-1 ⎞ (i) + n iθi Ti-1 + n iθi ln ⎜1 + e ⎟ − θi n -n i ( Ti − Ti-1 ) (3.5) ⎝ ⎠ − −Ti ⎞ T ⎛ ⎛ i-1 ⎞ (i) − + +θi n (i) -n i ln ⎜1 + e θ n -n ln 1 e i ⎟ i ⎜ ⎟. ⎝ ⎠ ⎝ ⎠
(
(
)
(
)
)
Differentiating ln L with respect to θi and equating to zero we obtain
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∂ ln Li =0 ⇒ ∂θi
(i) −x n i ni (i) ni ⎛⎜ - ∑ x j − ∑ ln 1 + e j θi j=1 j=1 ⎜ ⎝ − Ti-1 ⎛ +n i ln ⎜1 + e ⎝
(
(
⎞ ⎟ + n i Ti-1 ⎟ ⎠
)
⎞ (i) ⎟ - n -n i ( Ti − Ti-1 ) ⎠
)
− Ti-1 ⎛ + n (i) -n i ln ⎜1 + e ⎝
(
(3.6)
)
− Ti ⎞ ⎞ ⎛ (i) ⎟ − n -n i ln ⎜1 + e ⎟ = 0. ⎠ ⎝ ⎠
This implies the MLE θˆ i of θi as θˆ i =
ni
(i) ⎛ ni ni −x ⎞ −Ti ⎧ ⎛ (i) j ⎟ + n (i) -n i ⎨Ti + ln ⎜1 + e ∑ x j + ∑ ln ⎜ 1 + e ⎟ j=1 j=1 ⎜ ⎝ ⎩
⎝
⎠
(
)
− Ti-1 ⎞ ⎫ (i) ⎧ ⎛ ⎟ ⎬ − n ⎨Ti-1 + ln ⎜ 1 + e ⎠⎭ ⎝ ⎩
⎞⎫ ⎟⎬ ⎠⎭
.
(3.7)
Now again differentiating (3.6) we get
n ∂ 2 ln L i = − 2i 2 θi ∂θi Hence estimated asymptotic variance of θˆ is given by (Due to Cohen1963)
V(θˆ i )=
-1 ∂ lnL ∂θi 2 θ =θˆ 2
i
i
Illustrative example Here we generate 1000 random samples under progressive type-I censoring scheme for the distribution given in (3.2). We have considered the following parameters. Based on simulated samples MLEs and their asymptotic variances are obtained
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691
n = 35, k=3,θ1 = 2, θ 2 = 2.4, θ3 = 2.8 R1 =3, R 2 = 4, R 3 = 10, T1 = 2, T2 = 3, and finally the experiment is terminated at T3 = 4 n1 = 4, n 2 = 3, n 3 = 5
Table1: MLE and its asymptotic variance for the parameters. Parameters θ1 θ2 θ3
MLE 0.0868 0.1253 0.3248
Asymptotic Variance 0.0019 0.0052 0.0211
4. Estimation based on Interval-Censoring with different parameters at each stage It is often the practice in life testing to examine the life test periodically and the number of items that have failed in each stage of censoring (Ti-1, Ti) are counted and some fixed number of surviving items are eliminated immediately after time Ti for i = 1, 2, …, k being the times of censoring. This kind of experimentation stems from economic or practical considerations where it may not be appropriate to collect exact failure times of the items on test. Kendell and Anderson (1971) have considered the ML estimators of the scale parameter θ of the exponential distribution when items placed on test are subjected to a stress condition for a predetermined time T and the test is periodically inspected at time ti for i = 1, 2, …, k such that tk = T. Where as Patel and Gajjar (1995) have considered estimation in case of exponential life time model for k-stage progressive type-I interval censoring scheme with changing parameters at each stage.
Maximum Likelihood Estimation under Interval-Censoring In this sub-section we consider the ML estimation of the parameters of the generalized half logistic distribution based on k-stage Type-I progressively interval censored samples under the assumption that the parameter θ changes at each stage of censoring. Under this scheme the likelihood function becomes n ⎫⎪ i k k ⎧ r ⎪ Ti L ∝ ∏ ⎨ ∫ fi ( x ) dx ⎬ ∏ 1 − Fi ( Ti ) i (4.1) i=1 ⎪ ⎪⎭ i=1 ⎩Ti-1
(
)
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Using the probability model (3.2), the likelihood L given by (4.1) can be written as k
L ∝ ∏ Li . i=1
⎡ ⎧ -Ti ⎢ ⎪ 2e -Ti ⎢ ⎪ where Li = ⎢1 − ⎨ 1 + e -T ⎢ ⎪ 2e i-1 ⎢ ⎪⎩ 1 + e-Ti-1 ⎣⎢ ⎧ 2e-Ti ⎪ -Ti ⎪ Let w i = ⎨ 1 + e -T ⎪ 2e i-1 ⎪⎩ 1 + e-Ti-1
θi ⎤ n i
⎫ ⎪ ⎪ ⎬ ⎪ ⎪⎭
⎥ ⎥ ⎥ ⎥ ⎥ ⎦⎥
⎧ 2e-Ti ⎪ -Ti ⎪ × ⎨ 1+ e -T ⎪ 2e i-1 ⎪⎩ 1 + e-Ti-1
(
)
θ n (i) − n i ⎫i ⎪ ⎪ ⎬ ⎪ ⎪⎭
⎫ ⎪ ⎪ θ ⎬ , and pi = 1-w i i ⎪ ⎪⎭
Hence Li can be rewritten as n n (i) -ni Li = ( pi ) i (1 − pi )
The log likelihood function is given by
(
)
ln Li = n i lnpi + n (i) -n i ln (1-pi ) Differentiating ln L with respect to θ and equating to zero we obtain ∂ ln Li =0⇒ ∂pi
pi =
ni
n (i)
(
)
n (i) -n i ni − =0 pi 1-pi
Using pi = 1-w iθi , we get θˆ i =
(
ln n(i) -ni / n(i)
The equation (4.2) can be rewritten as,
lnwi
)
(4.2)
Results on maximum likelihood estimators
θˆ i θi
693
⎛ n (i) -n ⎞ ln ⎜ (i) i ⎟ ⎜ n ⎟ ⎠ = ⎝ ln (1-pi )
Differentiating ln Li again with respect to θi we get, ∂ 2 ln Li ∂θi 2
∂ ⎛ ∂ ln Li ⎞ ∂pi ⎜ ⎟ ∂pi ⎝ ∂θi ⎠ ∂θi
=
2
θ
− ( ln w i ) w i i n i = pi2
(
Since n i ~ b n (i) , pi
⎛ ∂ 2 ln L i E⎜ ⎜ ∂θ 2 i ⎝
) 2
( )
(i) ⎞ − ( ln w i ) (1-pi ) E n ⎟= ⎟ pi ⎠
Hence the asymptotic variance is given by,
( )
AsyV θˆ i =
−1 ⎛ ∂ 2 ln L ⎞ i⎟ E⎜ ⎜ ∂θ 2 ⎟ i ⎝ ⎠
=
( )=
AsyV θˆ i θi2
Table 2:
pi
( lnw i ) (1 − pi ) E ( n (i) ) 2
pi
( lnpi )2 (1 − pi ) E ( n (i) )
, i= 1,2,...k
( ) for different values of n(i) and p
AsyV θˆ i θi2
i
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S. H. Arora, G. C. Bhimani and M. N. Patel
10
15
0.1
0.002096
0.001397 0.001043 0.000699 0.00524
0.2
0.009651
0.006434 0.004826 0.003217 0.002413 0.00193
0.3
0.029566
0.019711 0.014783 0.009855 0.007391 0.005913 0.003696 0.002957
0.4
0.079404
0.052936 0.039702 0.026468 0.019851 0.015881 0.009925 0.00794
0.5
0.208137
0.138758 0.104068 0.069379 0.052034 0.041627 0.026017 0.020814
0.6
0.574839
0.383226 0.287419 0.191613 0.14371
0.7
1.834136
1.222757 0.917068 0.611379 0.458534 0.366827 0.229267 0.183414
0.8
8.033251
5.3555
0.9
81.07496
54.04997 40.53748 27.02499 20.26874 16.21499 10.13437 8.107496
0.99
98010.82
65340.55 49005.41 32670.27 24502.71 19602.16 12251.35 9801.082
n (i)
20
30
40
50
80
100
0.00419
0.000262 0.00021
pi
4.016625 2.67775
0.114968 0.071855 0.057484
2.008313 1.60665
(i)
From above table we conclude that for fixed pi as n increases and for fixed n(i)
0.001206 0.000965
( ) increases with p .
1.004156 0.803325
( )
AsyV θˆ i θi2
decreases
AsyV θˆ i θi2
i
5. Expected Duration of the life test (EDLT) In case of a life test with k-stage interval Type-I progressive censoring the expected duration of the test can be obtained using the method suggested by Kendall and Anderson (1971). The expected duration of the life test (EDLT) is given by, EDLT = E [D({ti}, Tk, θ]
Results on maximum likelihood estimators
k-1
n
=T1 p 1 +
∑T
i
i=2
+Tk
⎡ ⎢⎣1- ( p
⎡( p ⎢⎣
695
i-1
+.....+p i ) 1
n-
∑ - ( p +.....+p j=1
1
) ∑ ⎤⎥ ⎦ i-2
Rj
n-
i-1
Rj
j=1
∑R ⎤ , ⎥ k-2
+.....+p k-1 ) 1
n-
j
j=1
⎦
w here p i = Fi − Fi-1
i-1
⎡ ⎛ 2e 1 ⎞ 1 ⎤ ⎢1- ⎜ ⎟ ⎥ 1 ⎢⎣ ⎝ 1 + e ⎠ ⎥⎦ -T
= T1
θ
n
-T
k -1
+
∑
Ti
i= 2
⎛ ⎛ 2 e -Ti ⎞ θ i ⎞ ⎜ 1- ⎜ ⎟ ⎜ ⎝ 1 + e -Ti ⎟⎠ ⎟ ⎝ ⎠
i-2
⎛ ⎛ 2e ⎞ − ⎜ 1- ⎜ ⎜ ⎝ 1 + e -Ti-1 ⎟⎠ ⎝ -Ti-1
= Tk −
( T2
θ i-1
⎞ ⎟ ⎟ ⎠
n-
∑
R j
∑
+ Tm
n
−
k −1 ∑ i=2
( Ti+1
R j
j= 1
k -2
-T
j= 1
⎡ ⎛ 2e -T1 ⎞ θ1 ⎤ − T1 ) ⎢ 1 − ⎜ ⎟ ⎥ ⎢ ⎝ 1+e -T1 ⎠ ⎥ ⎣ ⎦
⎡ ⎢ ⎛ ⎛ 2e ⎢ 1 - ⎜⎜ 1 - ⎜ ⎢ ⎝ ⎝ 1+e ⎢⎣
n-
k -1
-T
k -1
⎞ ⎟ ⎠
θ
k -1
⎞ ⎟ ⎟ ⎠
n-
⎡ ⎛ 2e -Ti ⎞ θ i ⎤ − Ti ) ⎢ 1 − ⎜ ⎟ ⎥ ⎢ ⎝ 1+e -Ti ⎠ ⎥ ⎣ ⎦
∑
j= 1
R
j
⎤ ⎥ ⎥ ⎥ ⎥⎦
i-1 n- ∑ R j j=1
.
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F o r eq u al len g th in tervals Ti = it, i.e Ti − Ti-1 = t , i= 1 ,2 ,...,k -1
E D L T red u ces to ⎡ ⎛ 2 e -t E D L T1 = k t-t ⎢ 1 - ⎜ ⎢ ⎜⎝ 1 + e -t ⎣
⎞ ⎟⎟ ⎠
θ1
n
i-1 n- ∑ R j θi ⎤ -t j= 1 2e ⎞
⎤ k -1 ⎡ ⎛ ⎥ - ∑ t ⎢1 - ⎜ ⎟ ⎥ i= 2 ⎢ ⎜⎝ 1 + e -t ⎟⎠ ⎦ ⎣
⎥ ⎥ ⎦
Illustrative example n = 35, k=3,θ1 = 2, θ 2 = 2.4, θ3 = 2.8 R1 =3, R 2 = 4, R 3 = 10, T1 = 2, T2 = 3, and finally the experiment is terminated at T3 = 4 n1 = 4, n 2 = 3, n 3 = 5 t 2 3 4 5 6 7 8
EDLT 5.036356 4.132488 4.220808 5.036364 6.005695 7.000873 8.000132
REFERENCES 1) Aggarwala R. : Progressive interval censoring, Some mathematical results with applications to inference, Communication in Statistics- Theory and Methods, 30(8&9), (2001), 1921-1935. 2) Balakrishnan N. : Order statistics from the Half Logistic Distribution, Journal of Statistics and Computer Simulation, 20, (1985), 287-309.
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3) Balakrsihnan N. and Hossain A. : Inference for the Type II generalized logistic distribution under progressive Type II censoring, Journal of Statistical Computation and Simulation, 77 (12) (2007), 1013-1031. 4) Balakrishnan N. and Puthenpura S. : Best Linear Unbiased Estimators of Location and Scale Parameters of the Half Logistic Distribution, Journal of Statistical Computation and Simulation, 25, (1986), 193-204. 5) Balakrishnan N. and Leung M.Y. : Order statistics from the Type I generalized Logistic Distribution, Communications in Statistics - Simulation and Computation, 17(1), (1988), 25-50. 6) Balakrishnan N. and Wong K.H.T. : Approximate MLEs for the Location and Scale Parameters Of the Half-Logistic Distribution with Type-II RightCensoring. IEEE Transactions on Reliability, 40(2), (1991) 140-145. 7) Balakrishnan N. and Aggarwala R. : Progressive Censoring, Theory Methods and applications, Birkhauser publishers, Boston, 2000. 8) Cohen A.C. : Progressive censored samples in life testing, Technometrics, 5, (1963), 327-329. 9) Farrington C.P. : Interval censored survival data, A generalized linear modeling approach, Statistics in medicine, 15, (1996), 283-292. 10) Kendall P.J. and Anderson R.L. : An estimation problem in life testing, Technometrics,13(2), (1971), 289-301. 11) Kokosha S.M; Hardin M and Grubbs C.J; Hsu, C. : The statistical analysis of cancer inhibition promotion experiments, Anticancer Research, 13(4), (1993), 1357-1363. 12) Lindsey J.C and Ryan L.M. : Tutorial in biostatistics methods for interval censored data, Statistics in Medicine, 17, (1998), 219-238. 13) Mahmond M.R; Sultan K.S and Saleh H.M. : Progressively censored data from the linear exponential distribution, Moments and Estimation, Metron, LXIV (2), (2006), 199-215. 14) Odell P.M; Anderson K.M and Dagostino R.B. : Maximum likelihood estimation for interval censored data using a Weibull based all iterated failure time model, Biometrics, 48, (1992), 951-959. 15) Olapade A.K. : On Characterizations of the Half Logistic Distribution, InterStat, 2, (2003).
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16) Patel M.N. and Gajjar A.V: Some results on maximum likelihood estimators of parameters of exponential distribution under type-I progressive censoring with changing failure rates, Commun Statist.-Theory METH, 24(9), (1995), 2421-2435. 17) Ramakrsihnan V : Generalizations to half logistic distribution and related inference, PhD thesis, Acharya Nagarjuna University (AP), India, 2008. 18) Rao M.B. : Interval censored type-II Man-a-knell of the traditional type-II plan, In IISA International conference, 1998. Abstracts, McMaster University, 1998, 55. 19) Samuelson S.O and Kongerud J. : Interval censoring in longitudinal data of respiratory symptoms in aluminium potroom workers, A comparison of methods, Statistics in Medicine, 13, (1994), 1771-1780. 20) Scallan A.J. : Regression modeling of interval censored failure time data using the Weibull distribution, Journal of Applied statistics, 26(5),( 1999), 613-618. 21) Sun J.: Regression analysis of interval censored failure time data, Statistics in Medicine, 16, (1997), 497-504.
Received: September, 2009
