Generalized Bayesian Parameter Estimation of Exponential Distribution Based on Selected Fuzzy Priors and Fuzzy Life Times

Statistics is the science of data based decision-making, in almost every field of life. The classical statistics require large datasets for the optimal inference about the parameters and testing. On the other hand, to obtain a suitable inference based on small samples, Bayesian statistics have been developed, and along with some pre-defined information about the parameters. The advanced science of measurement says that the data has in fact two kinds of variation: the most familiar random variation and the other, usually ignored but important, i.e., fuzziness. The classical statistics model only considers random variation among the observations but ignores the fuzziness of the single observation, which may contain useful information and, by ignoring it, will lead to misleading inference. Therefore, in this study, various priors and fuzziness are considered for the parameter estimation of the exponential distribution. The generalized Bayesian estimators are suggested in such a way that, in addition to random variation, the fuzziness encompassed in the measurements of single observations and the fuzziness involved in the hyperparameters are integrated into the estimators. In the proposed estimation, encouraging results are obtained, as the fuzzy estimates led to a high amount of fuzziness compared to the fuzziness contained in the observations and hyperparameters. Conversely, conventional estimators that are constructed only on the basis of precise or crisp measurements mainly focus on handling random variation, and they do not account for the uncertainty that may arise from imprecise information. This important observation provides a clear and strong motivation to argue that the proposed fuzzy-based estimators are more suitable in such situations. Therefore, it can be reasonably concluded that the proposed estimators should be preferred over the traditional classical ones.

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