A Goodness-of-Fit Test for the Geometric Distribution Based on a Ratio of Estimators Derived from Order Statistics

Khaleel Ali Hussein Al-Tameemi

doi:10.47134/ppm.v2i3.1937

Authors

Khaleel Ali Hussein Al-Tameemi University of Mohaghegh Ardabili

DOI:

https://doi.org/10.47134/ppm.v2i3.1937

Keywords:

Goodness-of-Fit, Geometric Distribution, Order Statistics, Ratio Estimator, Monte Carlo Simulation.

Abstract

This paper introduces a novel goodness-of-fit test for the Geometric distribution, designed to address shortcomings in detecting specific, yet common, departures from the null hypothesis, such as over-dispersion and non-constant hazard rates, the core of our methodology is the formulation of a new test statistic, Tₙ, constructed as a ratio of two distinct estimators for a function of the distribution's parameter, the first estimator is the uniformly minimum variance unbiased estimator derived from the sample mean, while the second is a novel estimator derived from the frequency of the first order statistic, we derive the asymptotic normal distribution of the standardized statistic, Zₙ, under the null hypothesis using the multivariate delta method, a comprehensive Monte Carlo simulation study reveals that our proposed test maintains excellent control over the Type I error rate. Crucially, the results demonstrate that our test possesses substantially higher statistical power than the standard Anderson-Darling test against over-dispersed alternatives like the Negative Binomial distribution and alternatives with non-constant hazard rates such as the Discrete Weibull distribution, the test also shows superior performance in detecting data contamination, making it a robust and powerful tool for practical applications.

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