Sampling theorem and efficiency comparison of three local minimum variance unbiased estimators of the mean and variance of the exponential distribution

This article continues the works of references to improve and perfect the sampling theorem of exponential distribution. First, the distribution of the sample range of exponential distribution is derived, and that the sample range ismutually independent of the sample minimum is proven. Then, this article derives the distribution of the difference between sample maximum and mean and demonstrates that the difference of these two statistics ismutually independent of the sampleminimum. Thus, three local minimum variance unbiased estimators of the mean could be constructed. The estimator built by sampleminimum and the difference between samplemean andminimum is precisely the uniformly minimum-variance unbiased estimator (UMVUE) of the mean. Similarly, three local minimum variance unbiased estimators of the variance are derived. At last, the efficiency comparison is made among the above three local minimum variance unbiased estimators of mean and variance of the exponential distribution. Subjects: Mathematical Statistics; Statistics & Computing; Statistical Theory & Methods


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What is the sampling theorem of the exponential distribution? It includes the content about the distributions of the sample mean, sample maximum, sample minimum and their differences. It also includes the content of whether their differences are mutually independent of sample minimum. What is the local minimum variance unbiased estimation? Based on two mutually independent unbiased estimators, a kind of weighted linear unbiased estimators could be constructed, among which the one with the minimum variance is the local minimum variance unbiased estimation. One should remember that three local minimum variance unbiased estimators of mean and variance are not substituted for uniformly minimum variance unbiased estimators of mean and variance, respectively, but only rich in natural estimators.

Introduction
Sample minimum, sample maximum and sample mean are important statistics in exponential distribution. Sample minimum has an exponential distribution, and sample mean has a gamma distribution or Chi-square distribution with degree freedom of n. The difference of sample mean and minimum has a gamma distribution or Chi-square distribution with degree freedom of n-1. The difference between sample mean and minimum is mutually independent of the sample minimum (Arnold, 1968;Gupta & Kundu, 0000;Marshall & Olkin, 1967).
This article derives the distribution of the sample range and demonstrates that the sample range is mutually independent of sample minimum. Then, the distribution of the difference between sample maximum and mean is derived, and that the difference of these two statistics is mutually independent of the sample minimum is demonstrated (Cohen and Helm, 1973;Kundu & Gupta, 2009;Lawrance & Lewis, 1983;Nie, Sinha, & Hedayat, 2017).
Thus, the sampling theorem is improved. As natural corollary of the sampling theorem of the exponential distribution, a first local minimum variance unbiased estimators of expectation could be constructed by sample minimum and the difference between sample mean and minimum, which is precisely the UMVUE of the expectation. A second local minimum variance unbiased estimators of expectation could be constructed by sample minimum and sample range. A third local minimum variance unbiased estimators of expectation could be constructed by sample minimum and the difference between sample maximum and mean; similarly, three local minimum variance unbiased estimators of the variance are derived. At last, the efficiency comparison is made among the above three local minimum variance unbiased estimators of mean and variance of the exponential distribution (Al-Saleh & Al-Hadhrami, 2003;Baklizi & Dayyeh, 2003;Dixit & Nasiri, 2008;Guoan, Jianfeng, & Lihong, 2017;Li, 2016).

Three local minimum variance unbiased estimators of expectation
Theorem 3.1. If X,EðαÞ, X 1 ; :::; X n is the sample from X,EðαÞ with sample size n , X ð1Þ ; :::; X ðnÞ are the order statistics, then the local minimum variance unbiased estimator, which is based on X ð1Þ and X À X ð1Þ , is the UMVUE of expectation.
Proof. From Theorem 2.1: is mutually independent of X ð1Þ , we obtain n X ð1Þ and nð XÀX ð1Þ Þ nÀ1 are both unbiased estimator of α, and the effective unbiased (3:1) Plug in and getα 0 ¼ X, which is the UMVUE of the expectation.

Efficiency comparison of three local minimum variance unbiased estimators of expectation and variance
Remark 5.1. The efficiency comparison of three local minimum variance unbiased estimators is the comparison of variances.
Proof. Based on comparison among scatter plot with regression line 4 or 5 as well as 6, we can obtainDλ 0 < Dλ 1 < Dλ 2 .

Discussion and conclusion
This article continues the works of references, to improve and perfect the sampling theorem of the exponential distribution. As natural corollary of the sampling theorem of the exponential distribution, one can obtain three local minimum variance unbiased estimators of mean and variance of the exponential distribution, respectively. We know that the sample mean is the UMVUE of expectation and n 1þn X 2 is the UMVUE of variance. Therefore, three local minimum variance unbiased estimators of mean and variance are not substituted for uniformly minimum variance unbiased estimators of mean and variance, respectively, but only rich in natural estimators. From Tables 1-2 and scatter plots with regression lines 1-6, we can draw a conclusion that Dα i ði ¼ 0; 1; 2Þ and Dλ i ði ¼ 0; 1; 2Þ are strictly monotonous decreasing as nincreases; moreover, they are all convergent to zero, hence, they are all consistent.
Remark 6.1. The advantages of those estimators are as follows: If sample is not complete or the record value of the sample mean is not given, and the record value of the difference between sample maximum and mean and the sample minimum are known, then the local minimum variance unbiased estimator of expectation, which is based on X ð1Þ and ðX ðnÞ À XÞ is a practical estimator; similarly, if sample is not complete or the record value of the sample mean is not given, and the record value of the sample maximum and the sample minimum are known, then the local minimum variance unbiased estimator of expectation, which is based on X ð1Þ and ðX ðnÞ À X ð1Þ Þ is a recommendable estimator. If sample is not complete or the record value of the sample mean is not given, and the record value of ð X À X ð1Þ Þ 2 andðX ð1Þ Þ 2 are known, then the local minimum variance unbiased estimator of variance, which is based on ðX ð1Þ Þ 2 and ð X À X ð1Þ Þ 2 is a practical estimator, similarly, under different sample condition, If sample is not complete or the record value of the sample mean is not given, and the record value of ðX ðnÞ À XÞ 2 and ðX ð1Þ Þ 2 are known, or the record value of ðX ðnÞ À X ð1Þ Þ 2 and ðX ð1Þ Þ 2 are known, then the local minimum variance unbiased estimator of variance, which is based on ðX ðnÞ À XÞ 2 and ðX ð1Þ Þ 2 or which is based on ðX ðnÞ À X ð1Þ Þ 2 and ðX ð1Þ Þ 2 is a recommendable estimator, respectively. Funding