Abstract
Currently, much research attention has focused on generalizations of known mathematical objects in order to obtain adequate models describing real phenomena. An important role in the applied theory of probability and mathematical statistics is the gamma class of distributions, which has proven to be a convenient and effective tool for modeling many real processes. The gamma class is quite wide and includes distributions that have useful properties such as, for example, infinite divisibility and stability, which makes it possible to use distributions from this class as asymptotic approximations in various limit theorems. One of the most important tasks of applied statistics is to obtain estimates of the parameters of the model distribution from the available real data. In this paper, we consider the gamma-exponential distribution, which is a generalization of the distributions from the gamma class. Estimators for some parameters of this distribution are given, and the asymptotic normality of these estimators is proven. When obtaining the estimates, a modified method of moments was used, based on logarithmic moments calculated on the basis of the Mellin transform for the generalized gamma distribution. On the basis of the results obtained, asymptotic confidence intervals for the estimated parameters are constructed. The results of this work can be used in the study of probabilistic models based on continuous distributions with an unbounded non-negative support.
Highlights
Estimating unknown parameters is an important problem in applied mathematical statistics
The possibility of representing the gamma-exponential distribution as a ratio of random variables having the generalized gamma distribution allows it to be used in a wide range of applied problems
Distributions from the gamma class have a rich history of modeling applications in many areas of knowledge
Summary
Estimating unknown parameters is an important problem in applied mathematical statistics. Many models are traditionally described using continuous distributions with unbounded non-negative supports For these purposes, special cases of the generalized gamma distribution and beta prime distributions are usually used. In [1] it was shown that the distribution (1) adequately describes Bayesian balance models [6] This is primarily due to the fact that the distribution with the density (1) can be represented as a scaled mixture of two random variables with generalized gamma distributions. The possibility of representing the gamma-exponential distribution as a ratio of random variables having the generalized gamma distribution allows it to be used in a wide range of applied problems. The problem of estimating unknown parameters from real data arises in the case of modeling a real process using the gamma-exponential distribution. In [18] it was proposed to estimate the parameters of the gamma-exponential distribution using a modified method based on logarithmic moments
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