Size Biased Fréchet Distribution: Properties and Statistical Inference

Fréchet distribution, initially introduced by (Fréchet, M. (1927) Ann de la Soc), is usually used for modeling several extreme random phenomena, while several generalizations of this distribution are available in the statistical literature (Phaphan, W., Ibrahim, A., Wirawan, P. (2023) Symmetry 15(7):1380, Alzeley, O., Almetwally, E.M., Gemeay, A.M., Alshanbari, H.M., Hafez, E.H., Abu-Moussa, M.H (2021) Comput Intell Neurosci 2021(1):2167670). In this paper, a generalization of the Fréchet distribution is considered as a method of describing samples that cannot be considered as a random sample from Fréchet, since the sampling mechanism selects units with probability proportional to some measure of the unit size (Mudasir, S., Ahmad, S.P (2021) J Stat Theory Appl 20:395–406). In this frame, motivated by (Fisher, R.A. (1934) Ann Eugen 6:13–25) and (Rao, C.R. (1965) J Stat Ser A 311–324), the $r-$ $r-$ size-biased version of the Fréchet distribution is presented and studied. In this paper, it is proved that the maximum likelihood estimators of the unknown parameters of the $r$ $r$ -size-biased Fréchet distribution, with $r$ $r$ known, always exist and they are unique. Moreover, it discusses the impact on the estimation of the unknown parameters via the maximum likelihood method when someone ignores the bias or misspecifies the value of $r$ $r$ . Finally, using a real data set we also highlight that ignoring the bias affects the estimation of quantities such as the mean value and the quantiles.