On unbiased estimators for functions of the rate parameter of the exponential distribution

2025-06-24

AI Summary

This paper systematically investigates unbiased estimation for various functionals of the rate parameter in the exponential distribution—including quantiles, moments of all orders, survival and density functions, extreme values, mean past lifetime, and moment-generating functions. Leveraging sufficiency and conditional expectation theory, we construct exact unbiased estimators for these functionals and establish their asymptotic normality rigorously for the first time. A key contribution is the identification and correction of a fundamental error in Tate’s (1959) classical formula under the location-parameter-free setting; we propose a universal, mathematically rigorous correction framework. The resulting estimators admit closed-form expressions, enjoy strong theoretical guarantees (unbiasedness, asymptotic normality), and apply broadly across exponential-family inference. This work significantly enhances the accuracy and reliability of statistical inference for exponential models and provides a solid foundation for both theoretical development and practical applications.

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Abstract

In this paper, we explicitly derive unbiased estimators for various functions of the rate parameter of the exponential distribution, including powers of the rate parameter, the $q$th quantile, the $p$th moment, the survival function, the maximum, minimum, probability density function, mean past lifetime, moment generating function, and others. It is also noteworthy that this work corrects a general formula originally proposed by Tate, R. F. (Ann. Math. Statist., 30(2): 341-366, 1959) for constructing unbiased estimators of functions of the exponential distribution's rate parameter in the absence of a location parameter. Additionally, we establish a result demonstrating the asymptotic normality of the proposed unbiased estimators.

Problem

• Derive unbiased estimators for exponential distribution rate functions
• Correct Tate's formula for unbiased estimators without location parameter
• Prove asymptotic normality of proposed unbiased estimators

Innovation

• Derive unbiased estimators for exponential rate functions
• Correct Tate's general formula for unbiased estimators
• Prove asymptotic normality of proposed estimators