Stochastic Orderings and Parameter Estimation for the Hurwitz-Lerch Zeta Distribution

Abstract

The Hurwitz-Lerch Zeta (HLZ) family includes many well-known distributions such as the logarithmic distribution, Zipf-Mandelbrot distribution and so on. In this study, the stochastic orderings of the random variables in the HLZ family are established based on the likelihood ratio. These orderings provide a useful tool for comparing the 'magnitudes' of these random variables. The tail probability of the HLZ distribution is shown to have an interesting relation with a generalisation of the logarithmic distribution (GLD) proposed in [1]. To demonstrate the flexibility of the HLZ distribution in empirical modelling, a robust probability generating function (pgf) based estimation method using Hellinger-type divergence is implemented in data-fitting and the results are compared with various other GLD's. An augmented pgf is constructed to overcome the difficulties of this estimation method when some data are grouped. ABSTRAK Keluarga Hurwitz-Lerch Zeta (HLZ) mengandungi beberapa taburan yang ternama seperti taburan logaritma, taburan Zipf-Mandelbrot dan lain-lain. Dalam kajian ini, susunan stokastik pembolehubah-pembolehubah rawak dalam keluarga HLZ telah ditubuhkan berdasar nisbah kebolehjadian. Susunan tersebut boleh digunakan dalam perbandingan 'magnitud' antara pembolehubah-pembolehubah rawak tersebut. Hubungan antara kebarangkalian ekor bagi taburan HLZ dan taburan logaritma teritlak yang dicadangkan dalam [1] terus ditunjukkan. Demi menggambarkan kelenturan taburan HLZ, satu kaedah penganggaran teguh yang menggunakan divergen jenis Hellinger berasaskan fungsi penjana kebarangkalian (fpk) telah dilaksanakan dalam analisis data dan keputusannya dibanding dengan pelbagai generalisasi taburan logaritma yang lain. Fpk tertambah telah dibinakan untuk menyelesaikan masalah pelaksanaan kaedah ini apabila terdapat data yang terkumpul.

Sufficient Conditions for Some Stochastic Orders of Discrete Random Variables with Applications in Reliability

“…Let us point out that [22] provides several results for the comparison of two Hurwitz-Lerch distributions in the likelihood ratio order. The conditions that we have stated improved their result for the likelihood ratio order, since we have characterized the situations where this criterion holds in some sense.…”