Properties of the probability density function of the non-central chi-squared distribution

Abstract

In this paper we consider the probability density function (pdf) of a non-central χ 2 distribution with arbitrary number of degrees of freedom. For this function we prove that can be represented as a finite sum and we deduce a partial derivative formula. Moreover, we show that the pdf is log-concave when the degrees of freedom is greater or equal than 2. At the end of this paper we present some Turán-type inequalities for this function and an elegant application of the monotone form of l'Hospital's rule in probability theory is given.

Geometrically concave univariate distributions

“…This result has been used recently by András and Baricz [5] in order to present a simple and elegant alternative proof of Theorem 1 and parts (a) and (b) of Theorem 2, given below. It is worth mentioning here that in [5] we tacitly assumed that the values f (a) and f (b) exist. However, many statistical distributions are defined on the whole real line or on the interval (0, ∞) and the value f (0) does not exist always.…”

“…The proof given in [2] uses also classical analysis, but is somewhat different to the proof given in [6]. Recently, András and Baricz [5] pointed out that in fact Theorem 1 is an immediate consequence of the so-called monotone form of l'Hospital's rule (see Lemma 1 below). It is also worth mentioning here that in general the reverse implications in Theorem 1 does not hold.…”

“…For a proof of Theorem 1 we refer to the papers [2,5,6]. The proof of part (a) of Theorem 1, presented by Bagnoli and Bergstrom in [6], is attributed to Dierker [11] and uses classical calculus.…”

On New Formulas for the Cumulative Distribution Function of the Noncentral Chi-Square Distribution

“…A r.v. with such distribution is usually denoted by χ 2 n (a) [19, p. 433] and the appropriate probability density function (PDF) f n,a : [0, ∞) → [0, ∞) can be defined in terms of the modified Bessel function of the first kind I ν , of order ν, as [5, p. The noncentral χ 2 distribution has a wide range of application, for example it frequently occurs in finance, estimation and decision theory, time series analysis [5,30] and it can be regarded as a generalized Rayleigh distribution [19, p. 435] in which case it is used in mathematical physics. It is also important to mention, considering the results given in this article, that the noncentral χ 2 distribution is frequently used in communication theory in which context it is called the generalized Marcum Q-function and the noncentrality parameter is interpreted as a signal-to-noise ratio [19,34].…”

Testing behavior of the reversed hazard rate

“…The decreasing behavior of the RHR is equivalent to the logconcavity of the df F. This notion has wide applications in applied mathematics, reliability, economics among other areas. For definitions, characterizations and further details we refer to [9][10][11][12]. It is thus seen that the results which discuss the behavior of the RHR function can be very useful.…”