December, 1961 Estimating the Parameters of Negative Exponential Populations from One or Two Order Statistics
H. Leon Harter
Ann. Math. Statist. 32(4): 1078-1090 (December, 1961). DOI: 10.1214/aoms/1177704847

## Abstract

This paper discusses the use of order statistics in estimating the parameters of (negative) exponential populations. For the one-parameter exponential population, the best linear unbiased estimators, $\tilde \sigma_k = c_kx_k$ and $\tilde \sigma_{lm} = c_lm_l + c_mx_m$, of the parameter $\sigma$ are given, based on one order statistic $x_k$ and on two order statistics $x_l$ and $x_m$. For samples of any size up through $n = 100$, a table is given of $k, l$, and $m$ and of the coefficients $c_k, c_l$, and $c_m$, together with the coefficients of $\sigma^2$ in the variances $V_k$ and $V_{lm}$ of the estimators, and the corresponding efficiencies $E_k$ and $E_{lm}$ (relative to the best linear unbiased estimator based on all order statistics). For the two-parameter exponential population, the best linear unbiased estimators, $\tilde \alpha = c_{\alpha l}x_l + c_{\alpha m}x_m, \tilde \sigma = c_{\sigma l}x_l + c_{\sigma m}x_m$, and $\tilde \mu = c_{\mu l}x_l + c_{\mu m}x_m$, of the parameters $\alpha$ and $\sigma$ and the mean $\mu = \alpha + \sigma$ are given, based on two order statistics $x_l$ and $x_m$. For samples of any size up through $n = 100$, a table is given of $m(l$ is always 1 for the best estimator) and of factors $c_\alpha$ and $c_\alpha$ for computing the coefficients $c_{\alpha l} = 1 + c_\alpha, c_{\alpha m} = -c_\alpha, c_{\sigma l} = -c_\sigma, c_{\sigma m} = c_\sigma, c_{\mu l} = 1 + c_\alpha - c_\sigma$, and $C_{\mu m} = C_\sigma - C_\alpha$, together with the coefficients of $\sigma^2$ in the variances $V_{\tilde \alpha}, V_{\tilde \sigma}$, and $V_\mu$ of the estimators, and the corresponding efficiencies $E_{\tilde \alpha}, E_{\tilde \sigma}$, and $E_{\tilde \mu}$ (relative to the best linear unbiased estimators based on all order statistics).

## Citation

H. Leon Harter. "Estimating the Parameters of Negative Exponential Populations from One or Two Order Statistics." Ann. Math. Statist. 32 (4) 1078 - 1090, December, 1961. https://doi.org/10.1214/aoms/1177704847

## Information

Published: December, 1961
First available in Project Euclid: 27 April 2007

zbMATH: 0115.36403
MathSciNet: MR138156
Digital Object Identifier: 10.1214/aoms/1177704847