Exact Distribution of Random Order Statistics and Applications in Risk Management
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Exact Distribution of Random Order Statistics and Applications in Risk Management Vasileios M. Koutras1
· Markos V. Koutras1
Received: 27 March 2018 / Revised: 7 July 2018 / Accepted: 12 August 2018 / © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract In the present work we study the exact distribution of order statistics coming from a sample of random variables (r.v.’s), with random sample size. Some new results are provided for the exact distribution of the r−th largest observation of the sample, and several interesting properties are developed when the sample size belongs to wide classes of discrete distributions such as the family of power series distributions, the Panjer Family, the class of exchangeable Bernoulli mixtures and the family of Phase-Type distributions. Finally, we illustrate how the stochastic model under study could be exploited for modeling problems arising in financial risk management (monitoring of non-performing loans and insurance portfolio surveillance). Keywords Bernoulli mixtures · Non-performing loans · Panjer family · Phase-type distributions · Power series distributions · Random order statistics · Risk management · Samples of random size Mathematics Subject Classification (2010) 62E15 · 62G30 · 60J10
1 Introduction Order statistics play a substantial role in the modeling of various problems of financial risk management, actuarial science, reliability engineering, quality control, life sciences, data mining, image processing and other applied areas. For example, in financial risk management one may be interested in the minimum and maximum loss, generated by a portfolio of Work done while VMK was a postgraduate student at the Department of Statistics and Insurance Science, Greece Vasileios M. Koutras
[email protected] Markos V. Koutras [email protected] 1
Department of Statistics and Insurance Science, School of Finance and Statistics, University of Piraeus, Piraeus, Greece
Methodology and Computing in Applied Probability
loans, securities, etc; in actuarial science the monitoring of the loss process of a portfolio of insurance contracts requires the stochastic modeling of the largest claims; in reliability engineering, the life of the k out of n system is clearly associated to the k−th largest or k−th smallest of the component lifetimes; in Statistical Quality Control, two popular classes of control charts are the X¯ − R chart and the median−R chart which make use of the smallest, largest and the median observation of a sample. Order statistics received a tremendous attention from researchers in a variety of disciplines since the early 1900s and, as a consequence, many theoretical advances have been made in this area since then. The interested reader may refer to Arnold et al. (1992), Balakrishnan and Rao (1998a) and David and Nagaraja (2003) for a detailed presentation of the theory of order statistics. Let X1 X2 , . . . , X n be a random sample of size n. Then the corresponding order statistics are obtained by arranging the n Xi sin nondecreasi
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