Unbiased Estimation of the Distribution Function of a Two-Parameter Exponential Population Using Order Statistics
09 Jul 2009-Communications in Statistics-theory and Methods (Taylor & Francis Group)-Vol. 38, Iss: 15, pp 2578-2585
TL;DR: In this article, the problem of unbiased estimation of the distribution function of a two-parameter exponential population using order statistics based on a random sample from the population was considered and necessary and sufficient conditions for the existence of an unbiased estimator based on an arbitrary set of order statistics were given.
Abstract: In this article, we consider the problem of unbiased estimation of the distribution function of a two-parameter exponential population using order statistics based on a random sample from the population. We give necessary and sufficient conditions for the existence of an unbiased estimator based on an arbitrary set of order statistics and suggest unbiased estimators in some situations where unbiased estimators exist. A few properties of the suggested estimators for some special cases have also been discussed.
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01 Jan 1994
TL;DR: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes
Abstract: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes
7,270 citations
TL;DR: In this paper, the authors present an analogue to Thompson's distribution in case the underlying distribution of a sample is exponential (the exponential model is nowadays widely used in Failure and Queuing Theories), which makes it possible to obtain minimum variance unbiased estimates of functions of the parameters of the exponential distribution.
Abstract: In case the underlying distribution of a sample is normal, a substantial literature has been devoted to the distribution of quantities such as $(X_{(i)} - u)/v$ and $(X_{(i)} - u)/w$, where $X_{(i)}$ denotes the $i$th ordered observation, $u$ and $v$ are location and scale statistics of the sample, or one is a location or scale parameter and $w$ is an independent scale statistic. The case $i = 1$ or $n$ has been frequently studied in view of the great importance of extreme values in physical phenomena and also with a view to testing outlying observations or the normality of the distribution. Bibliographical references will be found in Savage [10] and, as far as the general problem of testing outliers is concerned, in Ferguson [4]; references to recent literature include Dixon [1], [2], Grubbs [5], Pillai and Tienzo [9]. Thompson [12] has studied the distribution of $(X_i, - \bar{X})/s$ where $X_i$ is one observation picked at random among the sample, and this statistic has been used in the study of outliers; Laurent has generalized Thompson's distribution to the case of a subsample picked at random among a sample [7], then to the multivariate case and the general linear hypothesis [8]. Thompson's distribution is not only the marginal distribution of $(X_i - \bar{X}/s$ but its conditional distribution, given the sufficient statistic $(\bar{X}, s)$, hence it provides the distribution of $X_i$ given $\bar{X}, s$, and, using the Rao-Blackwell-Lehmann-Scheffe theorem, gives a way of obtaining a minimum variance unbiased estimate of any estimable function of the parameters of a normal distribution for which an unbiased estimate depending on one observation is available, a fact that has been exploited in sampling inspection by variable. The present paper presents an analogue to Thompson's distribution in case the underlying distribution of a sample is exponential (the exponential model is nowadays widely used in Failure and Queuing Theories). Such a distribution makes it possible to obtain minimum variance unbiased estimates of functions of the parameters of the exponential distribution. Here an estimate is provided for the survival function $P(X > x) = S(x)$ and its powers. As an application of these results the probability distribution of the "reduced" $i$th ordered observation in a sample and that of the reduced range are derived. For possible applications to testing outliers or exponentially the reader is invited to refer to the bibliography.
76 citations
"Unbiased Estimation of the Distribu..." refers background in this paper
...…size n from the population, i.e., n independent observations X1 X2 Xn on X, the uniformly minimum variance unbiased estimator (UMVUE) of F t is (see Laurent, 1963): T ∗ = 1 if X 1 ≥ t n− 1 n [ 1− t − X 1 S ]n−2 + if X 1 t (1.3) where X r is the rth order statistic based on the random sample, S…...
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TL;DR: In this paper, the authors considered the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample and presented a (unique) unbiased estimator based on single, say ith, order statistic and study some properties of the estimator for i = 2.
Abstract: In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.
9 citations
"Unbiased Estimation of the Distribu..." refers background in this paper
...…independently of X 1 and the distribution of (X r − X 1 ) is same as the distribution of the (r − 1)th order statistic obtained from a random sample of size (n− 1) from an exp(0 ) population, from Theorem 2.6 of Sinha et al. (2006) it follows that E Tr X 1 = T ∗′ where T ∗′ is defined by (1.4)....
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...6 of Sinha et al. (2006) it follows that E Tr X 1 = T ∗′ where T ∗′ is defined by (1....
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...From Theorem 2.2 of Sinha et al. (2006) we have, for x 1 < t, E [ h23 X 1 Z3 X 1 = x 1 ] = 1 n− 1 [ e− t−x 1 / + ∑ k=1 { 1 2k−1 + 1 2k } e− k t−x 1 / ] (3.9) when it is easy to verify (3.8)....
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...The uniqueness follows by completeness of X 1 for given and of Zr (see Theorem 2.1 in Sinha et al., 2006)....
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...Only recently the problem has been studied in Sinha et al. (2006) for a one-parameter exponential, i.e., exp(0 ) population....
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