Relative Increments of Pearson Distributions

This paper is a direct continuation of [6] whose results are applied to Pearson distributions, particularly to normal, gamma, beta of the rst kind, Pareto of the second kind, chi-square and other speci c distributions. Acta Mathematica Academiae Paedagogicae Ny regyh aziensis 15 (1999), 45{54 www.bgytf.hu/~amapn In this paper we investigate the hazard rate and relative increment functions of Pearson distribution functions. Let f be a (probability) density function. The corresponding distribution function is denoted by F . De nition 1. By the relative increment function [brie y, RIF] of F we mean the fraction h(x) = [F (x+ a) F (x)]=[1 F (x)]; where a is a positive constant, and F (x) < 1 for all x. Monotone properties of RIFs are important from the points of view of statistics, probability theory, in modelling bounded growth processes in biology, medicine and dental science and in reliability and actuarial theories, where the probability that an individual, having survived to time x, will survive to time x+a is h(x); \death rate per unit time" in the time interval [x; x+a] is h(x)=a, and the hazard rate (failure rate or force of mortality) is de ned to be lim a!0 h(x)=a = f(x)=[1 F (x)]: (See e.g. [3], Vol. 2, Chap. 33, Sec. 7 or [4], x 5.34 and x 5.38.) In [3], Vol. 2, Chap. 33, Sec. 7.2, some distributions are classi ed by their increasing/decreasing hazard rates. In [6], we proved Lemma 1. Let F be a twice di erentiable distribution function with F (x) < 1, f(x) > 0 for all x. We de ne the auxiliary function as follows: (x) := [F (x) 1] f (x)=f(x): If < (>)1, then the function h, the RIF of F strictly increases (strictly decreases). According to Remark 0.1 in [6], there is a connection to reliability theory: a distribution function F has IFR (increasing failure rate) i ln[1 F (x)] is concave down i.e., i (x) 1. Similarly, F has DFR (decreasing failure rate) i ln[1 F (x)] is concave up, i.e., (x) 1. In [6], we investigated the auxiliary function . In order to get rid of the inconvenient term F (x) 1 = R 1 x f(t)dt in , all problems were reduced to simple formulae containing f=f 0 only. This fact suggested to work out special methods for the family of Pearson distributions. 1991 Mathematics Subject Classi cation. .