Abstract
When modeling real phenomena, special cases of the generalized gamma distribution and the generalized beta distribution of the second kind play an important role. The paper discusses the gamma-exponential distribution, which is closely related to the listed ones. The asymptotic normality of the previously obtained strongly consistent estimators for the bent, shape, and scale parameters of the gamma-exponential distribution at fixed concentration parameters is proved. Based on these results, asymptotic confidence intervals for the estimated parameters are constructed. The statements are based on the method of logarithmic cumulants obtained using the Mellin transform of the considered distribution. An algorithm for filtering out unnecessary solutions of the system of equations for logarithmic cumulants and a number of examples illustrating the results obtained using simulated samples are presented. The difficulties arising from the theoretical study of the estimates of concentration parameters associated with the inversion of polygamma functions are also discussed. The results of the paper can be used in the study of probabilistic models based on continuous distributions with unbounded non-negative support.
Highlights
Accepted: 14 February 2022Gamma and beta classes of distributions play an important role in applied probability theory and mathematical statistics and have proven to be convenient and effective tools for modeling many real processes
To define the sample logarithmic cumulants, we introduce a notation for the sample logarithmic moments of the random variable ζ: Lm ( X ) =
To describe the solution of this system, we introduce a number of functions of sample logarithmic cumulants with the arguments k = (k1, k2, k3, k4 ): ψ(m) ( t )
Summary
Gamma and beta classes of distributions play an important role in applied probability theory and mathematical statistics and have proven to be convenient and effective tools for modeling many real processes. [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 class of distributions (3) is wide enough and includes exponential distribution; χ2 -distribution; Erlang distribution; gamma distribution; half-normal distribution (the distribution of the maximum of the Brownian motion process); Rayleigh distribution; Maxwell–Boltzmann distribution; χ-distribution; Nakagami m-distribution; Wilson–Hilferty distribution; Weibull–Gnedenko distribution, and many others, including scaled and inverse analogs of the above The paper contains the sections of discussions and conclusions
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