Estimates of parameters for the truncated exponential law of distribution on the basis of exponential data
Authors: Бычков Игорь Вячеславович , Зоркальцев Валерий Иванович , Мокрый Игорь Владимирович 
Published: October 2021

Рассмотрена процедура исследования свойств методов оценки параметров законов распределения случайной величины, базирующаяся на вычислительных экспериментах. Эта процедура использует многократную имитацию методом МонтеКарло реализаций набора значений случайной величины для исследуемого закона распределения при заданных его параметрах. Для каждой реализации осуществляется оценка параметров по исследуемому методу. В результате большого количества экспериментов определяются характеристики свойств методов, в том числе наличие смещений в оценках, величины среднеквадратических отклонений оценок. Данная процедура иллюстрируется на примерах трех методов оценки параметров усеченного дискретного экспоненциального закона распределения. Каждый из методов рассматривается в трех вариантах, различающихся правилами задания весовых коэффициентов.
Purpose. A comparative analysis of methods for estimating the parameters of a truncated exponential law of the distribution of a random variable on the basis of computational experiments. The problem arose in connection with the problem of estimating mortality rates, the age structure of fish in the Lake Baikal according to the experimental catch data. Methodology. The procedure for studying the properties of methods for estimating parameters laws of distribution of a random variable is based on computer experiments. This procedure uses multiple simulations based on the Monte-Carlo realizations of a set of values of a random variable. The variables are set for the given parameters under the examined law. For each implementation, the parameters are evaluated with the help of the examined method. As a result of a large number of experiments, the characteristics of the properties of the methods are determined for various sample sizes. The experiments account the presence and the magnitude of displacements in the estimates as well as the magnitude of the standard deviation of the estimates. Findings. This procedure is applied to three methods for estimating the parameters of a truncated discrete exponential distribution law. Each of the methods is considered in three versions, differing in the rules for setting weight coefficients. As a result of the research, it was found that the best results are obtained by the method based on the logarithmic transformation of the exponential distribution law into a linear regression equation. Coefficients of the regression are calculated using the method of least weighted squares. We found that the best weighing was proportional to the probability of realization of the random variable under consideration. Regarding the problem of assessing the age structure of fish the best parameter is the proportion of the number of fish of the given age. An iterative procedure of successive refinement of probabilities and weights is used. Conclusions. The studies presented in the article made it possible to selectively choose the best within the nine algorithms for estimating the parameters of the truncated normal distribution law. It seems that the technique used could be useful in a comparative analysis of other methods for estimating the parameters of the laws of distribution of random variables on the basis of their available sample values.