Studying the estimates of gamma distribution parameters

Abstract

The main goal of the work is to obtain additional information about the experimentally obtained sample with a previously known theoretical distribution, the point estimates of the parameters of which are considered known. At the same time, the laws of distribution of these parameters remain unknown, whereas they could provide a researcher with additional information about both the material and technological processes. Hence, it is necessary to obtain an additional number of samples, which is not always possible experimentally. Here we used data on the service life of cutters (GOST 11.011–83 «Rules for determining estimates and confidence limits for gamma distribution parameters») as an experimental sample. The experimental sample contains the results of 50 measurements. The mean was 57.88 hours CI [50.74:65.01]. The confidence probability is taken to be 0.95. Bootstrap was used as a way to obtain additional samples. The universal mathematical package MATLAB is used in the study. Bootstrap allows generation of a large number of samples that require certain selection rules to be applied to them. The first obvious requirement is the significance of the correlation coefficient of the generated sample with the original one. Even at this stage, the bootstrap showed certain limitations in performing the task set in the study. For 1000 samples generated by the standard bootstrap routine, the mean for the population of all mean bootstrap samples was 57.80 hours, and the confidence interval was [50.59:58.08]. The result is good. Though the nonparametric hypothesis regarding an agreement between the bootstrap samples for the gamma distribution and the parameters characteristic of the original experimentally obtained sample was not rejected, the statistically significant correlation coefficient was observed only for 29 bootstrap samples. As a result of meeting these obvious requirements, less than 3% of the generated bootstrap samples remained for further consideration. This fact requires the introduction of additional conditions when using the bootstrap to obtain samples that are close to the original experimental sample, which can be rather specific. To determine the parameters of the gamma distribution for bootstrap samples, the method of moments and the one-step method were used.