Published: 14 October 2013

Electronic Journal of Alied Statistical Analysis EJASA, Electron. J. A. Stat. Anal. htt://siba-ese.unisalento.it/index.h/ejasa/index e-issn: 27-5948 DOI: 1.1285/i275948v6n213 Estimation of Parameters of Exonentiated Pareto Distribution for Progressive Tye-II Censored Data with Binomial Random Removals Scheme By Singh et al. Published: 14 October 213 This work is coyrighted by Università del Salento, and is licensed under a Creative Commons Attribuzione - Non commerciale - Non oere derivate 3. Italia License. For more information see: htt://creativecommons.org/licenses/by-nc-nd/3./it/

Electronic Journal of Alied Statistical Analysis Vol. 6, Issue 2, 213, 13-148 DOI: 1.1285/i275948v6n213 Estimation of Parameters of Exonentiated Pareto Distribution for Progressive Tye-II Censored Data with Binomial Random Removals Scheme Sanjay Kumar Singh a, Umesh Singh a, Manoj Kumar a, and G.P Singh b a Deartment of Statistics and DST-CIMS, Bananas Hindu University,Varanasi-2215 b Deartment of Community Medicine, I.M.S and DST-CIMS, Bananas Hindu University, Varanasi-2215 Published: 14 October 213 In this aer, we roose maximum likelihood estimators and Bayes estimators of arameters of exonentiated Pareto distribution under general Entroy loss function and squared error loss function for Progressive tye-ii censored data with binomial removals. The maximum likelihood estimators and corresonding Bayes estimators are comared in terms of their risks based on simulated samles from exonentiated Pareto distribution keywords: Maximum likelihood estimators, exonentiated Pareto distribution under general Entroy loss function, Progressive tye-ii censored data with binomial removals. 1 Introduction In life testing exeriments, situations do arise when units are lost or removed from the exeriments while they are still alive; i.e, we get censored data from the exeriment. The loss of units may occur due to time constraints giving tye-i censored data. In such censoring scheme, exeriment is terminated at secified time. Sometimes, the exeriment is terminated after a refixed number of observations due to cost constraints Corresonding authors: manustats@gmail.com c Università del Salento ISSN: 27-5948 htt://siba-ese.unisalento.it/index.h/ejasa/index

Electronic Journal of Alied Statistical Analysis 131 and we get tye-ii censored data. Besides these two controlled causes, units may dro out of the exeriment randomly due to some uncontrolled causes. For examle, consider that a doctor erform an exeriment with n cancer atients but after the death of first atient, some atient leave the exeriment and go for treatment to other doctor/ hosital. Similarly, after the second death a few more leave and so on. Finally the doctor stos taking observation as soon as the redetermined number of deaths (say m are recorded. It may be assumed here that each stage the articiating atients may indeendently decide to leave the exeriment and the robability ( of leaving the exeriment is same for all the atients. Thus the number of atients who leave the exeriment at a secified stage will follow binomial distribution with robability of success (. The exeriment is similar to a life test exeriment which starts with n units. At the first failure X 1, r 1 (random units are removed from the remaining (n 1 surviving units. At second failure X 2, r 2 unit from remaining n 2 r 1 units are removed, and so on, till m th failure is observed; i.e., at m th failure all the remaining r m = n m r 1 r 2 r m 1 units are removed. It may be re-emhasized that, here, m is re-fixed constant and r, is are random. Such a censoring mechanism is termed as rogressive tye-ii censoring with random removal scheme. As stated above, if we assume that robability of removal of a unit, at every stage, is for each unit then r i can be considered to follow a Binomial distribution; i.e, r i B(n m i 1 l= r l, for i = 1, 2, 3, m 1 with r =. For further details, reader are referred to Balakrishnan (27. In last few years, the estimation of arameters of different life time distribution based on rogressive censored samles have been studied by several authors such as Childs and Balakrishnan (2, Balakrishnan and Kannan (21, Mousa and Jaheen (22, Chan and Balakrishnan (22, Sarhan and Abuamooh (28, Ashour and Afify (27 and Ashour and Afify (28. The rogressive tye-ii censoring with binomial removal has been considered by Tse, Yang and Yuen (2 for Weibull distribution, Wu and Chang (22 for Exonential distribution. Under the rogressive tye-ii censoring with random removals, Wu and Chang (23 and Yuen and Tse (1996 develoed the estimation roblem for the Pareto distribution and Weibull distribution resectively, when the number of units removed at each failure time has a discrete uniform distribution, the exected time of this censoring lan is discussed and comared numerically. Let us assume that the life time of the units follow the exonentiated Pareto distribution (EPD with cumulative distribution function F (x, α, = [ 1 (1 + x α] ; x >, α >, > (1 This distribution was introduced by Guta, Guta and Guta (1998. The robability density function (df of X takes the following form with two shae arameters α and : f(x, α, = α [ 1 (1 + x α] 1 (1 + x (α+1 ; x >, α >, > (2 The corresonding survival function is S(x = 1 F (x, α, = 1 [ 1 (1 + x α] ; x >, α >, > (3

132 Singh et al. It may be noted that for = 1, (2 reduces to f 1 (x, α, = α(1 + x (α+1 ; x >, α > (4 which is standard Pareto distribution of second kind Kotz and Balakrishnan (1994. The estimators of the arameters of exonentiated Pareto distribution have been obtained by Shawky, Abu-Zinadah and Hanna (29 under different estimation rocedures for comlete samle case. The estimation of arameters has also been attemted by Afify (21 under tye-i and tye-ii censoring scheme. However, no attemt has been made to develo estimators for the arameters of EPD under rogressive tye-ii censoring with binomial removals. Therefore, we roose to develo such an estimation rocedure. Rest of the aer is organized as follows: Section 2 rovides the likelihood function. In section 3, maximum likelihood estimator (MLE and Bayes estimators have been obtained. An algorithm for simulating the rogressive tye-ii samle with binomial removal is resented in section 4. For illustration urose estimates have been obtained for a simulated data in section 5. The comarison of MLE s and corresonding Bayes estimators are given in section 6. Comarisons are based on simulation studies of risk (average loss over samle sace of the estimators. Finally conclusions are rovided in the last section. 2 Likelihood Function Let (X 1, R 1, (X 2, R 2, (X 3, R 3,, (X m, R m, denote a random rogressive tye II censored samle from (2, where X 1 < X 2 < X 3, < X m. For given number of removals, say R 1 = r 1, R 2 = r 2, R 3 = r 3,, R m = r m, the conditional likelihood function can be written as (see Cohen (1963: m L(α, ; x R = r = c f (x i [S (x i ] r i (5 where c = n(n r 1 1(n r 1 r 2 2(n r 1 r 2 r 3 3 (n r 1 r 2 r 3,, r m m+1, and r i (n m r 1 r 2 r 3 r i 1, for i = 1, 2, 3..., m 1 and r =. Substituting (2 and (3 into (5, we get L(α, ; x R = r = c m α [ 1 (1 + x i α] 1 { 1 [ 1 (1 + x i α] } r i (1 + xi (α+1 (6 We assume that units removed from the test at i th failure, i = 1, 2,, m 1, are indeendent of each other. However, robability of removal of unit remains same (say, for all units at all failures; i.e., R i (the number of units removed at the i th failure i = 1, 2, 3,, m 1 follows binomial distribution with arameters n m i 1 l= r l and. Therefore, ( n m P (R 1 = r 1 = r 1 (1 n m r 1, (7 r 1

Electronic Journal of Alied Statistical Analysis 133 and for i = 2, 3,, m 1, P (R i = r i R i 1 = r i 1, R 1 = r 1 = ( n m i 1 l= r l r i (1 n m i 1 l= r l (8 r i We further assume that R i is indeendent of X i for all i. Hence the likelihood function takes the following form where L (α,, ; x = L (α, ; x R = r P (R = r, (9 P (R = r = P (R 1 = r 1 P (R 2 = r 2 R 1 = r 1 P (R 3 = r 3 R 2 = r 2, R 1 = r 1 P (R m 1 = r m 1 R m 2 = r m 2, R 1 = r 1. (1 Substituting (7 and (8 into (1, we get P (R = r = (n m! m 1 r i (1 (m 1(n m m 1 (m ir i ( n m i 1 l=1 r i! m 1 r i! (11 Using (6,(9 and (11, we can write the likelihood function in the following form : where A = and c (n m! (n m i 1 l=1 r i! m 1 r i! L 1 (α; = L (α,, ; x = AL 1 (α, L 2 (, (12, m α [ 1 (1 + x i α] { 1 1 [ 1 (1 + x i α] } r i (1 + xi (α+1 (13 L 2 ( = m 1 r i (1 (m 1(n m m 1 (m ir i. (14 3 Classical and Bayesian Estimation of Parameters 3.1 Maximum Likelihood Estimation In this section, we have obtained the MLE of the arameters, α and based on rogressive tye-ii censored data with binomial removals. We observe from (12, (13 and (14 that likelihood function is multilication of three terms, namely, A, L 1 and L 2. Out of these, A does not deendent on the arameters α, and ; thus, it behaves as constant for maximum likelihood estimation. L 1 does not involved and can be treated as function of α and only, where as L 2 involves only. Therefore, the MLE s of α and can be derived by maximizing L 1 with resect to α and. Similarly, the MLE of can be obtained by maximizing L 2.

134 Singh et al. Taking log of both sides of (13, we have ln L 1 (α; = m ln ( + m ln (α m + ( 1 ln [ 1 (1 + x i α] + m r i ln [1 [ 1 (1 + x i α] ] (α + 1 m ln (1 + x i Thus, the normal equations can be obtained by differentiating (15 with resect to α and and equating these to zero; i.e., MLE s ˆα and ˆ of α and resectively, can be obtained by simultaneously solving the following normal equations: and m m α + ( 1 m m (1 + x i α ln (1 + x i 1 (1 + x i α m ln (1 + x i r i [ 1 (1 + xi α] 1 (1 + xi α ln (1 + x i 1 [ 1 (1 + x i α] = m m + ln [ 1 (1 + x i α] r i [ 1 (1 + xi α] ln [ 1 (1 + xi α] 1 [ 1 (1 + x i α] = It may be noted that (16 and (17 can not be solved simultaneously to rovide a nice closed form for the estimators. Therefore, we roose to use fixed oint iteration method for solving these equations numerically. For details about the roosed method readers may refer Jain, Iyengar and Jain (1984. When rocedure for obtaining the iteration function and the choice of initial guesses, based on maximum absolute row sum norms, have been discussed. The log of L 2 ( takes the following form [ m 1 ln L 2 ( = ln r i + ln (1 (m 1 (n m m 1 r i (m i ] (15 (16 (17. (18 The first order derivative of ln L 2 ( with resect to is ln L 2 ( m 1 = r i (m 1 (n m m 1 r i (m i. (19 1

Electronic Journal of Alied Statistical Analysis 135 =, we get the normal equation for. Solving this equation for, we get the MLE of as m 1 ˆ M = r i (m 1 (n m m 1 r i (m i 1. (2 Setting ln L 2( 3.2 Bayes rocedure In this section, we have obtained the Bayes estimators of the arameters α, and based on rogressively tye-ii censored data with Binomial removals. In order to obtain the Bayes estimator, we must assume that the arameters α, and are random variables. We further assume that these are indeendently distributed. The random variables α and have non-informative rior distribution with resective rior dfs g 1 (α = 1 c ; < α < c (21 and g 2 ( = 1 ; > (22 where as has Beta distribution of first kind with known arameters a, b. The rior df of is given by g 3 ( = 1 B (a, b a 1 (1 b 1 ; < < 1, a >, b > (23 Based on the assumtions stated above, the joint rior df of α, and is g (α,, = g 1 (α g 2 ( g 3 ( ; < α < c, >, < < 1 (24 Combining the riors given by (21, (22 and (23 with likelihood given by (12, we can easily obtain joint osterior df of (α,, as π (α,, x, r = J 1 J where J 1 = α m m 1 m [ 1 (1 + xi α] { 1 1 [ 1 (1 + x i α] } r i (1 + x i (α+1 m 1 ri+a 1 (1 (m 1(n m m 1 (m ir i+b 1 and J = 1 c J 1dαdd Hence, the resective marginal osterior df s of α, and are given by π 1 (α x, r = π 2 ( x, r = 1 1 c J 1 J dd (25 J 1 J dαd (26

136 Singh et al. and π 3 ( x, r = c J 1 J dαd (27 Usually the Bayes estimators are obtained under Square Error Loss Function (SELF l 1 (φ, ˆφ = 1 (φ ˆφ 2 ; 1 > (28 Where ˆφ is the estimate of the arameter φ and the Bayes estimator ˆφ S of φ comes out to be E φ [φ], where E φ denotes the osterior exectation. This loss function is a symmetric loss function and can only be justified, if over estimation and under estimation of equal magnitude are of equal seriousness. A number of asymmetric loss functions are also available in statistical literature. Let us consider the General Entroy Loss Function (GELF, roosed by Calabria and Pulcini (1996, defined as follows : ( δ ( l 2 (φ, ˆφ = 2 ˆφ ˆφ δ ln 1 ; 2 > (29 φ φ The constant δ, involved in (29, is its shae arameter. It reflects dearture from symmetry. When δ >, it considers over estimation (i.e., ositive error to be more serious than under estimation (i.e., negative error and converse for δ <. The Bayes estimator ˆφ E of φ under GELF is given by, ˆφ E = [E φ (φ δ] ( 1 δ (3 rovided the osterior exectation exits. It may be noted here that for δ = 1, the Bayes estimator under loss (28 coincides with the Bayes estimator under SELF l 1. Exressions for the Bayes estimators ˆα E, ˆ E and ˆ E for α, and resectively under GELF can be given as [ c ] 1 ( αˆ E = α δ δ π 1 (α x, r dα (31 [ ] 1 ( ˆ E = δ δ π 1 ( x, r d (32 and [ 1 ]( 1 ˆ E = δ δ π 3 ( x, r d (33 Substituting the osterior dfs from (25, (26 and (27 in (31, (32 and (33 resectively and then simlifying, we get the Bayes Estimators ˆα E, ˆ E and ˆ E of α, and as follows : ˆα E = m 1 c αm δ m m 1 c αm m [ 1 (1 + xi α] { 1 1 [ 1 (1 + x i α] } r i (1 + xi (α+1 dαd [ 1 (1 + xi α] { 1 1 [ 1 (1 + x i α] } r i (1 + xi (α+1 dαd 1/δ (34,

Electronic Journal of Alied Statistical Analysis 137 ˆ E = and m δ 1 c αm m m 1 c αm m [ 1 (1 + xi α] { 1 1 [ 1 (1 + x i α] } r i (1 + xi (α+1 dαd [ 1 (1 + xi α] { 1 1 [ 1 (1 + x i α] } r i (1 + xi (α+1 dαd [ B( m 1 ˆ E = r i + a δ, (m 1(n m ] m 1 1/δ (m ir i + b B( m 1 r i + a, (m 1(n m m 1 (m ir. (36 i + b It may be noted that the integrals involved in the exressions for the Bayes estimators ˆα E and ˆ E are not reducible in nice closed form. Therefore, we roose the use of numerical integration method for obtaining the estimates. We have used Gauss-quadrature formula for evaluating the inner integral which has finite range and for evaluating the outer integral having range (,, we used Gauss-Laguerre formula. The evaluations were done by develoing rogram in R-language. 4 Algorithm to Simulate Progressive Tye-II Censored Samle With Binomial Removal For the study of behavior of the estimators obtained in revious sections, we need to simulate rogressive tye-ii censored samles with Binomial removals from secified EPD. To get such a samle, we roose the use of following algorithm : I. Secify the value of n. II. Secify the value of m. III. Secify the value of arameters α, and. IV. Generate a random samle (S r of size n from EP (α,. V. Generate random number r i ( from B n m i 1 l= r l,, at i th stage for i = 1, 2, 3,, m 1. (r = VI. Get ordered samle S o from S r to choose the minimum which will be first observation in desired rogressive tye-ii censored samle S. VII. Dro the observation selected at VI from S r to have a random samle S r of size n (less 1 than that of S r. VIII. Generate r i integers (at i th stage between 1 to n and observations corresonding to these numbers are droed from S r to have a random samle S r of size n = n r i and re-designate the random samle in hand as new random samle S r. 1/δ (35,

138 Singh et al. IX. Reeat stes V to VIII (m 1 times. X. Set r m according to the following relation. { n m m 1 l=1 r m = r l if n m m 1 l=1 r l > otherwise and discard all the remaining r m observations. 5 A Numerical Illustration For illustration of the roosed estimation rocedures given in sections 2 and 3, we have considered below a simulated rogressive tye-ii censored samle with Binomial removals from exonentiated Pareto distribution with =.3, m = 14, α = 2, and =.5, using the above algorithm. The samle thus obtained is given below : (.241782,2, (.6227476,1, (.12238516,, (.1555823,, (.26417913,, (.276648679,, (.388598753,1, (.413575829,1, (.48234597,, (.912576221,, (1.13519893,, (1.33151842,1, (1.3752576,, (1.385353469,. Substituting value of (x i, r i i = 1,, 14 in (16 and (17, we obtain the normal equations which are solved simultaneously using fixed oint iteration rocedure to get the MLE s of α and. The MLE of is obtained from (2. The Bayes Estimates are obtained from (36, using the rocedure exlained in section 3. The results are summarized in table 1. It may be noted from the table that for the samle in hand, all the estimates are larger than the corresonding true values. However the Bayes estimates ˆα E, ˆ E and ˆ E with δ = 1.5 is nearer to the true value as comared to other estimates. For δ = 1.5, the Bayes estimate ˆα S, ˆ S and MLE ˆ M are nearer to the true as comared to other estimates. But on the basis of this articular samle, it will be illogical to conclude that under GELF, we should use ˆα E, ˆ E and ˆ E if over estimation is more serious than under estimation for their long run use. Therefore, we study the behavior of risks of the estimators in the next section. Table 1: MLE and Bayes estimate of α, and under rogressive tye-ii censored data with random removal. δ α ˆα M ˆα S ˆα E ˆM ˆS ˆE ˆ M ˆ S ˆ E 1.5 2.3947 2.5492 2.2547 1.681 1.92 1.19.5455.4211.3799-1.5 2.3947 2.3492 2.638 1.681 1.92 1.1144.5455.4211.4283

Electronic Journal of Alied Statistical Analysis 139 6 Simulation Studies The estimators ˆα M, ˆ M and ˆ M denote the MLE s of the arameters α, and resectively while ˆα S, ˆ S and ˆ S are corresonding Bayes estimators under SELF and ˆα E, ˆ E and ˆ E are the corresonding Bayes estimators under GELF. We comare the estimators obtained under GELF with corresonding MLE s and Bayes estimators under SELF. The comarisons are based on the simulated risks (average loss over samle sace under GELF. It may be mentioned here that the exact exressions for the risks can not be obtained because estimators are not in nice closed form. Therefore, the risks of the estimators are estimated on the basis of Monte-carlo simulation study of 1 samles. It may be noted that the risks of the estimators will deend on values of n, m,, α,, c, δ, a and b. In order to consider variation in the values of these, we have obtained the simulated risks for n = 2 [1] 4, m = 12 [2] 18,c = 4 [1] 8, a = 2 [2] 8, b = 4 [2] 1, =.5 [.5] 2, α =.5 [.5] 2.5, δ = ±1.5 and =.3 [.1].6. Generating the rogressive samle as mentioned in section 4, the simulated risks under GELF have been obtained for selected values of n, m,, α,, b, a, δ and c. The results are summarized in (tables (2 15. 6.1 Discussion of the results It is interesting to note that as n increases, keeing the effective samle size m fixed, the risks of all the estimators increase, in general, excet for ˆα M and ˆ M (see tables 2 and 8. On other hand, if the effective samle size m increases for given samle size n, the risks of the estimators of α and decrease as m increases, in general, excet for the estimators ˆα M and ˆα S (when δ = 1.5. We have noticed that the risks of all the estimators of increase, often, as m increases (see tables 3 and 9. Due to the increase in the value of, the risks of the estimators of α and, in general, first decrease then increase. But the risks of the estimators of do not follow a trend (see table 4 and 1. Further, for the variation in the value of arameter α (when δ = 1.5, we observe that risks of all estimators of α decrease as α increases but risk of estimators of and do not follow a definite trend (see table 5. On the other hand, for δ = 1.5, risks of the estimator ˆα S and ˆα E decrease as α increases but for the rest of the estimators no definite trend is observed (see table 11. The risk of estimators of (when δ = 1.5 often decrease as increases and the risk of estimators ˆα M and ˆ M increases as increases but no definite trend for the risks of rest of the estimators of α and is noted as increases (see table 6. For δ = 1.5, risk of estimators of and increases as increases, but the risks of estimators of α first decrease then increase when increases (see table 12. For the variation of the rior arameter c, when δ = 1.5, we observe that no definite trend is followed by the risks of the estimators of α, and excet for ˆα S which increases as c increases (see table 7. But, for δ = 1.5, the risk of estimators of α increase as c increases and for rest of estimators no definite trend is observed (see table 13.

14 Singh et al. It is worth while to remark here that a variation in the values of hyer arameters a and b do not effect the risks of the estimators of α and. It is further seen that for variation in hyer arameter a, the risks of estimators of do not have a definite trend, excet for the risk of ˆ S (when δ = 1.5 and ˆ E (when δ = 1.5 which increase as a increases (see table 14. For the variation of arameter b, the risk of estimators of, excet for ˆ M (for δ = 1.5 and ˆ S (for δ = 1.5 decrease for increase in the value of b in the beginning and then increase for further increase in the value of b (see table 15. It may further be noted from the tables that for δ = 1.5, the risk of ˆ E is smaller than the risks of ˆ M and ˆ S. However, for δ = 1.5, the risk of ˆ M is smaller than those of ˆ E and ˆ S. Among the estimator of α, ˆα E has smaller risks than those of others for δ = 1.5, but for δ = 1.5, none of the estimators ˆα M, ˆα S and ˆα E outerforms the other although there is little variation in the risk. The risk of ˆ E when δ = 1.5 whereas ˆ S when δ = 1.5 is smaller than the risks of other estimators of. 7 Conclusions It is exected that the estimator obtained under a articular loss function shall in general erform better than the estimators obtained under other loss functions, but this could not be established in the resent study. We have seen above that risk under GELF for the estimators ˆα E, ˆ E and ˆ E is not always less than those of ˆα M, ˆ M, ˆ M, ˆα S, ˆ S, ˆ S. Although the risks associated with ˆ E is smaller than the risk associated with other estimators in most of the cases. The risks associated with ˆα E and ˆ E are smaller than those of other estimators when δ = 1.5. On other hand if δ = 1.5 risk associated with ˆα M and ˆ S are noted to be smaller than other estimators. Therefore, for the use of the roosed estimators, we may recommend the following: 1. ˆE may be used as an estimator of when over estimation is more serious than under estimation. On the other hand, if under estimation is more serious than over estimation, ˆ M may be used. 2. ˆα E may be used as an estimator of α when under estimation is more serious than over estimation. Otherwise, ˆα M may be used. 3. ˆ E may be used an estimator of when under estimation is more serious than over estimation; Otherwise, ˆ S may be used. Acknowledgements The authors are grateful to the editor and the learned referee for making useful comments which resulted in much imrovement in the aer. The second author is also grateful to the UGC-BHU, Varanasi-2215, India for roviding financial assistance.

Electronic Journal of Alied Statistical Analysis 141 Table 2: Risk of Estimators of α, and under GELF for fixed, m = 14, α = 2, =.5, =.3, c = 5, a = 2, b = 6, δ = 1.5. RE (ˆαM RE (ˆαS RE (ˆαE RE RE RE RE (ˆM RE (ˆS RE (ˆE n α (ˆM (ˆS (ˆE 2.1459191.148331.1988515.2812842.3645443.233535 1.297787.5848324.7549444 3.391645.388912.3911926.7785958.7743881.771871 3.248732 3.231744 3.231876 4.3932742.3932742.3932742.788764.788764.788764 3.279847 3.279847 3.279847 Table 3: Risk of Estimators of α, and under GELF for fixed, n = 2, α = 2, =.5, =.3, c = 5, a = 2, b = 6, δ = 1.5. RE (ˆαM RE (ˆαS RE (ˆαE RE RE RE RE (ˆM RE (ˆS RE (ˆE m α (ˆM (ˆS (ˆE 12.1878715.157183.2716257.2613284.4195576.256414 1.3946.598246.7412193 14.1373194.1472565.1849168.2697733.375292.2349265 1.36697.654113.849189 16.128441.1487981.1536917.223128.3193126.253112 1.453691.665792.12373 18.1385783.1558678.1458456.163872.2368763.1577898 1.61478.6775665.17254

142 Singh et al. Table 4: Risk of Estimators of α, and under GELF for fixed, m = 14, α = 2, n = 2, =.3, c = 5, a = 2, b = 6, δ = 1.5. RE (ˆαM RE (ˆαS RE (ˆαE RE RE RE RE (ˆM RE (ˆS RE (ˆE α (ˆM (ˆS (ˆE.5.139676.1478296.1881797.277512.385586.2426155 1.31221.6329389.86758 1.214863.918265.1127921.2169895.33732.186352 1.31558.67841.8571692 1.5.4232116.831724.13758.234791.479166.211936 1.297559.6274592.7876336 2.4479795.313224.317672.3618798.413243.3544649 2.784513 2.478634 2.483556 Table 5: Risk of Estimators of α, and under GELF for fixed, n = 2, m = 14, =.5, =.3, c = 5, a = 2, b = 6, δ = 1.5. α α RE (ˆαM RE (ˆαS RE (ˆαE RE (ˆM RE (ˆS RE (ˆE RE (ˆM RE (ˆS RE (ˆE.5.5892752.5881838.5814836.6142454.656499.647511 2.61751 2.179152 2.186224 1.182674.199438.212454.2763414.43654.2514949 1.31142.6497218.857856 1.5.1599294.1784462.247829.2844317.412996.2583423 1.323.6375178.89425 2.1423826.1473353.19152.2658764.3853314.241666 1.314761.6488765.8264993 2.5.1117617.166854.168186.257236.359291.23814 1.3948.6693853.8716

Electronic Journal of Alied Statistical Analysis 143 Table 6: Risk of Estimators of α, and under GELF for fixed, n = 2, m = 14, α = 2, =.5, c = 5, a = 2, b = 6, δ = 1.5. RE (ˆαM RE (ˆαS RE (ˆαE RE RE RE RE (ˆM RE (ˆS RE (ˆE α (ˆM (ˆS (ˆE.3.3963992.3918289.39237.7858412.7881818.7874846 3.269148 3.263524 3.263574.4.565922.3677883.385516.98182.7768941.75993 1.585762 1.428244 1.429113.5.591798.3361664.3548922.957484.7888174.746422 1.36194.633212.6326188.6.71341.348167.3589572 1.113673.818969.783772.6681636.332889.3339984 Table 7: Risk of Estimators of α, and under GELF for fixed, n = 2, m = 14, α = 2, =.5, =.3, a = 2, b = 6, δ = 1.5. c α RE (ˆαM RE (ˆαS RE (ˆαE RE (ˆM RE (ˆS RE (ˆE RE (ˆM RE (ˆS RE (ˆE 4.133713.171883.1742957.252952.3382173.2123965 1.3623.641544.835438 5.1418948.1522755.19128.2535542.3714355.2322441 1.35162.6795876.8826163 6.1427238.1751292.1972878.299998.4186578.268431 1.3711.6882673.8422973 7.136681.1847313.288432.2654237.43864.2547928 1.31522.635916.8282253 8.1357412.1856477.267242.26589.3967457.245948 1.3425.646348.756216

144 Singh et al. Table 8: Risk of Estimators of α, and under GELF for fixed, m = 13, α = 2, =.5, =.3, c = 5, a = 2, b = 6, δ = 1.5. n α RE (ˆαM RE (ˆαS RE (ˆαE RE (ˆM RE (ˆS RE (ˆE RE (ˆM RE (ˆS RE (ˆE 2.256455.1569264.1429979.143696.1966461.29318 5.47759.7577261.774239 3.813742.786426.7857344.398113.3917175.3918124 1.7595.962792.9626953 4.788764.788764.788764.3932742.3932742.3932742.97276.97276.97276 Table 9: Risk of Estimators of α, and under GELF for fixed, n = 2, α = 2, =.5, =.3, c = 5, a = 2, b = 6, δ = 1.5. m α RE (ˆαM RE (ˆαS RE (ˆαE RE (ˆM RE (ˆS RE (ˆE RE (ˆM RE (ˆS RE (ˆE 12.458337.219515.194377.15845.21653.2246941 3.85492.687333.655247 14.284325.167437.1475378.1441369.1955944.27674 5.442995.6789313.6376622 16.1391414.1343914.1288445.1218889.1658354.175268 7.23498.893988.8219142 18.136912.131792.1292712.958756.1275673.133658 9.159985.9593564.8259461

Electronic Journal of Alied Statistical Analysis 145 Table 1: Risk of Estimators of α, and under GELF for fixed, m = 14, α = 2, n = 2, =.3, c = 5, a = 2, b = 6, δ = 1.5. α RE (ˆαM RE (ˆαS RE (ˆαE RE (ˆM RE (ˆS RE (ˆE RE (ˆM RE (ˆS RE (ˆE.5.1761425.157288.1456272.1536995.246119.2164952 5.513191.7776.7222737 1 1.47646.139942.9782153.167857.1765228.1917982 5.441861.729523.6773793 1.5 1.692799.5481985.5459216.1987279.2165787.222747 2.552693.654576.6535196 2 1.188187.723466.7229112.7325644.7359981.7376849 1.45152.8862815.885823 Table 11: Risk of Estimators of α, and under GELF for fixed, n = 2, m = 14, =.5, =.3, c = 5, a = 2, b = 6, δ = 1.5. α α RE (ˆαM RE (ˆαS RE (ˆαE RE (ˆM RE (ˆS RE (ˆE RE (ˆM RE (ˆS RE (ˆE.5.2447477.22669.219723.2298253.2641567.2734419 4.32987.3541624.3512526 1.276529.1897896.1761434.152419.248875.2173689 5.452927.82817.746755 1.5.22915.1719469.16675.162865.215632.2285961 5.461925.78225.673688 2.1978864.168681.155574.152385.24692.2168534 5.54892.794477.7453942 2.5.1845989.1455462.1299962.14392.19388.2596 5.487717.6629183.624312

146 Singh et al. Table 12: Risk of Estimators of α, and under GELF for fixed, n = 2, m = 14, α = 2, =.5, c = 5, a = 2, b = 6, δ = 1.5. α RE (ˆαM RE (ˆαS RE (ˆαE RE (ˆM RE (ˆS RE (ˆE RE (ˆM RE (ˆS RE (ˆE.3.1926444.1636426.1491117.1456174.1972236.29648 5.49238.639176.643838.4.161832.144839.136481.1785616.23336.246882 9.641454.68714.6125533.5.147586.1357172.1294271.1887159.2459813.26811 14.5875.9219164.814336.6.173985.14333.1355625.283287.2658582.2811275 2.15317.1459551.131492 Table 13: Risk of Estimators of α, and under GELF for fixed, n = 2, m = 14, α = 2, =.5, =.3, a = 2, b = 6, δ = 1.5. c α RE (ˆαM RE (ˆαS RE (ˆαE RE (ˆM RE (ˆS RE (ˆE RE (ˆM RE (ˆS RE (ˆE 4.1798442.135267.128644.15842.1891916.25542 5.458144.6754857.63981 5.1694883.158413.1398245.153352.24592.216421 5.493616.75737.6637562 6.1768365.1655576.154674.1458951.22437.214919 5.492914.849661.7514648 7.1982377.1677997.158899.1527374.2163435.2294159 5.483429.795818.667329 8.21565.1978975.1834582.1585488.2157997.2289757 5.49652.7621357.7157

Electronic Journal of Alied Statistical Analysis 147 Table 14: Risk of Estimators of α, and under GELF for fixed, n = 2, m = 14, α = 2, =.5, =.3, c = 5, b = 6, δ = ±1.5. a δ = 1.5 δ = 1.5 R E (ˆ M R E (ˆ S R E (ˆ E R E (ˆ M R E (ˆ S R E (ˆ E 2 1.36358.669435.84493 5.398939.6622675.623254 4 1.295787.8748874.6644892 5.459162.642437.6633571 6 1.3361.1449965.164253 5.42943.9412163.9857798 8 1.3485.229179.1793196 5.45997.1275746.1326883 Table 15: Risk of Estimators of α, and under GELF for fixed, n = 2, m = 14, α = 2, =.5, =.3, c = 5, a = 2, δ = ±1.5. b δ = 1.5 δ = 1.5 R E (ˆ M R E (ˆ S R E (ˆ E R E (ˆ M R E (ˆ S R E (ˆ E 4 1.38718.8872191.878298 5.451559.767966.758928 6 1.3794.6696796.8681964 5.5135.7914773.7417736 8 1.313298.589652.9528147 5.499888.865787.7742965 1 1.39155.6726842.124681 5.484824.11221.987816 References Afify, W. M (21. On estimation of the exonentiated areto distribution under different samle scheme. Alied Mathematical Sciences, 4(8, 393 42. Ashour, S. K. and Afify, W. M. (27. Statistical analysis of exonentiated Weibull Family under tye-i Progressive Interval Censoring with random removals. Journal of Alied Sciences Research, 3(12, 1851 1863. Ashour, S. K. and Afify, W. M. (28. Estimations of The Parameters of Exonentiated Weibull Family with tye-ii Progressive Interval Censoring with random removals. Journal of Alied Sciences Research, 4(11, 1428 1442. Balakrishnan, N (27. Progressive methodology : An araisal (with discussion. Test, 16 (2, 211 259. Balakrishnan, N. and Kannan, N. (21. Point and Interval Estimation for Parameters of the Logistic Distribution Based on Progressively tye-ii Censored Samles, in Handbook of Statisticsm N. Balakrishnan and C. R. Rao, volume 2. Eds. Amsterdam, North- Holand. Calabria, R. and Pulcini, G. (1996. Point estimation under - asymmetric loss functions for life - truncated exonential samles. Commun. statist. Theory meth., 25(3, 585 6.

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