This paper introduces and studies a unique probability distribution. The maximum likelihood estimation, the ordinary least squares, the weighted least squares, and the Anderson–Darling estimation methods all take into account a number of financial risk indicators, including the value-at-risk, tail-value-at-risk, tail variance, tail mean-variance, and mean excess loss function. These four approaches were used in a simulation study and an application to insurance claims data for the actuarial evaluation. The well-known Nikulin–Rao–Robson statistic is taken into consideration for distributional validation under the whole set of data. Three complete actual datasets and a simulation study are used to evaluate the Nikulin–Rao–Robson test statistic. An updated version of the Nikulin–Rao–Robson statistic is taken into consideration for censored distributional validation. Three censored actual datasets and a thorough simulation analysis are used to evaluate the novel Nikulin–Rao–Robson test statistic.
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