Estimators And Tests Of Two Ordered Exponential Means Based On Type Ⅱ Censoring

Estimators and tests of two ordered exponential means based on typeâ…¡censoringStatistical inference under order restriction has been a important topic of statistical analysis since it first be studied in the 60th of this century, the reason of this is that it is very popular in reality. With respect to the topic of Statistical inference under order restriction of multivariate populations, a lot of scientist has studied it and has got a lot of useful results. We can refer to Lee(1981),Kaur and Singh(1991),Vijayasree and Singh(1993),Katz(1963),Kumar and Sharma(1988),Zhao Shishun, Wang Dehuiå’ŒSong Lixin(2001) and so on. Lee(1981) considered the maxium likelihood estimator of k ordered normal means and showed that the MLE's obtained under the order restrition have pointwise smaller mean square error than the usual estimators i.e. sample means. Similar results were established by [2] for the case of samples of equal size from two exponential populations having ordered means. [3] got more popular results of this question and showed the admissible class mixed estimators. Katz(1963) showed that the mixed estimators of the binomial probabilities beat the sample means for general convex loss functions with include the sum of the mean squared errors. [5] obtained a class of minimax estimators and a class of estimators amissible among mixed estimators. [6] obtained similar results as Kaur and Singh(1991) of two exponential means under cone restrictions.Exponential distributions are widely used in the reliability of a production. So statistical inference of two exponential means under restriction is very useful in reliability. For example, there situations when one is interested in the estimation of the mean lives of two mechanical devices having exponential life distributions, of which one is a improvement of the other, and naturally the improved device should have a mean life length less than that of the original device. In yet other situation, the interest may be in the situation of two components having exponential life distributions, in which one is produced by a standered company whereas the other is manufactured by a local company. and where it is known, a prior, that the mean life of the componentof the standared company is not less than that of the component produced by the local company.There two parts of this paper.In the first part of this paper, Based on the sufficiency and complete statisticswe give the mixed estimators of two ordered exponential means under typeâ…¡censoringδ_1α=min(S₁,αS₁+(1-α)S₂),0≤α＜1,δ_2α=max(S₂,αS₁+(1-α)S₂),0≤α＜1.DenoteR(δ_iα)=E(δ_iα-λ_i)²,i=1,2.R(S_i)=E(S_i-λ_i)²=λ_i²/r,i=1,2.e(δ_iα,S_i)=R(S_i)/R(δ_iα),i=1,2. We also denote z₁=λ₁/(λ₁+λ₂), z₂=λ₂/(λ₁+λ₂), y₁=λ₂/λ₁, y₂=1/y₁.Because 0＜λ₁≤λ₂, we have y₁≥1, 0＜y₂≤1, 0＜z₁≤1/2, 1/2≤z₂＜1. In the next three theorem, the probability of the mixed estimators are given. First we proved that a subclass of the mixed estimator ofδ_iα has a smaller mean square error than the usual estimator S_i, The asymptotic efficiency ofδ_iα relative to S_i has also been obtained, i=1,2. Now Let's see theorem (2.2.1) and (2.2.2)Theorem2.2.1(a) R(δ_1α)＜R(S₁).(b)(c) when c₀(constant more than zero), we haveTheorem2.2.2(a) whenr=1,λ₁=λ₂,α=0orα=1/2,R(δ_2α)=R(S₂):whenr≥1,λ₁≤λ₂,α∈(0,1/2),R(δ_2α)＜R(S₂).(b)(c) when c₁(constant more than zero), we haveSimulations of the power are also given when the size of sampling are different.Then we give the admissible class of mixed estimators. This is what theorem (2.2.3) says. Theorem2.2.3Letα^*=(1/2)-(1/2)^2r((?)), we have(a) whenα∈[0,α^*],δ_1α is admissible.(b) whenα∈[α^*,1],δ_2α is admissible.(c) whenα∈[0, a^*),R(δ_2α^*)＜R(δ_2α),(?)λ₁≤λ₂.In the last part of this chapter, we give hypothesis testingH₀:λ₁=λ₂ H₁:λ₁＜λ₂.We give the rejection region of the UMPUT tests, Simulations of the power of the test are also given. The conclusion as follows:When the sample size is 200, when the censoring size is 5% of the sample size, Letλ₁=1,λ₁=2,λ₁=3 three cases, which are given in the first colum of "parameter valueλ₁". When levels are 5% and 10%.Δsatisfiesλ₂-λ₁=Δ, It is said that 1/2^1/2Δdenote the distants ofλ₂-λ₁=0 andλ₂-λ₁=Δ, LetΔ=2,Δ=4,Δ=6,Δ=8,Δ=10 respectly. The number of the table are the power of the test. we can have the reality as follows: for every fixedλ₁, The power of the test grows asΔgrows. For every fixedΔ, the power of the test decrease whenλ₁ increase, so we can differs H₀ and H₁ whenλ₁ large enough.In the second part of this paper, we give the maxium likelihood estimator of two exponential means under cone restriction when sampling are typeâ…¡censoring. The estimators areWe also denoteIn the next two theorem, we give the probability of the maxium likelihood estimator under cone restriction, We have proved that the maxium likelihood estimator under cone restriction has a smaller mean square error than the usual estimator, We also give the asymptotic efficiency of (?)₁ relative to S_i,i=1,2. Now Let's see the containt of this two theorem.Theorem3.2.1(1) R((?)₁)＜R(S₁).(2) when c₀(constant more than zero), we have(3) whena₂/a₁→+∞, we have Theorem3.2.2(1) R((?)₂)＜R(S₂).(2) when c₀(constant more than zero), we have(3) whena₂/a₁→+∞, we have...