Mills' ratio: Monotonicity patterns and functional inequalities

In this paper we study the monotonicity properties of some functions involving the Mills' ratio of the standard normal law. From these we deduce some new functional inequalities involving the Mills' ratio, and we show that the Mills' ratio is strictly completely monotonic. At the end of this paper we present some Turán-type inequalities for Mills' ratio.     

A Turán-type inequality for the gamma function

We prove that the following Turán-type inequality holds for Euler's gamma function. For all odd integers n?1 and real numbers x>0 we have

α?Γ(n−1)(x)Γ(n+1)(x)−Γ(n)²(x),     

A note on Turán type and mean inequalities for the Kummer function

Turán-type inequalities for combinations of Kummer functions involving Φ(a±ν,c±ν,x) and Φ(a,c±ν,x) have been recently investigated in [Á. Baricz, Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math. 26 (3) (2008) 279-293; M.E.H. Ismail, A. Laforgia, Monotonicity properties of determinants of special functions, Constr. Approx. 26 (2007) 1-9]. In the current paper, we resolve the corresponding Turán-type and closely related mean inequalities for the additional case involving Φ(a±ν,c,x). The application to modeling credit risk is also summarized.     

Bounds for ratios of modified Bessel functions and associated Turán-type inequalities

New sharp inequalities for the ratios of Bessel functions of consecutive orders are obtained using as main tool the first order difference-differential equations satisfied by these functions; many already known inequalities are also obtainable, and most of them can be either improved or the range of validity extended. It is shown how to generate iteratively upper and lower bounds, which are converging sequences in the case of the I-functions. Few iterations provide simple and effective upper and lower bounds for approximating the ratios Iν(x)/Iν−1(x) and the condition numbers for any x,ν?0; for the ratios Kν(x)/Kν+1(x) the same is possible, but with some restrictions on ν. Using these bounds Turán-type inequalities are established, extending the range of validity of some known inequalities and obtaining new inequalities as well; for instance, it is shown that Kν+1(x)Kν−1(x)/(Kν²(x))<|ν|/(|ν|−1), x>0, ν∉[−1,1] and that the inequality is the best possible; this proves and improves an existing conjecture.     

Functional inequalities involving Bessel and modified Bessel functions of the first kind

In this paper, we extend some known elementary trigonometric inequalities, and their hyperbolic analogues to Bessel and modified Bessel functions of the first kind. In order to prove our main results, we present some monotonicity and convexity properties of some functions involving Bessel and modified Bessel functions of the first kind. We also deduce some Turán and Lazarević-type inequalities for the confluent hypergeometric functions.     

Computing the probability density function of the stable Paretian distribution

The practical implementation of a stable Paretian model is a nontrivial task, because—with the exception of a few special cases—its probability density function cannot be expressed analytically. Here, we present an algorithm for calculating the probability density function of the asymmetric stable Paretian distribution. Due to the use of the Fast Fourier Transform, the algorithm is computationally efficient, easily implemented, and of similar accuracy as existing algorithms.     

关于Smarandache二重阶乘函数的值分布问题

对任意正整数n,著名的Smarandache二重阶乘函数SDF(n)定义为最小的正整数m使得m!!能够被n整除,其中二重阶乘函数m!!=1·3·5…m,如果m是奇数;m!!=2.4.6…m,如果m是偶数.本文的主要目的是利用初等方法研究函数SDF(n)的值分布性质,并给出一个有趣的均值公式.     

A shorter proof of Kanter’s Bessel function concentration bound

We give a shorter proof of Kanter’s (J. Multivariate Anal. 6, 222–236, 1976) sharp Bessel function bound for concentrations of sums of independent symmetric random vectors. We provide sharp upper bounds for the sum of modified Bessel functions I₀(x) + I₁(x), which might be of independent interest. Corollaries improve concentration or smoothness bounds for sums of independent random variables due to Čekanavičius & Roos (Lith. Math. J. 46, 54–91, 2006); Roos (Bernoulli, 11, 533–557, 2005), Barbour & Xia (ESAIM Probab. Stat. 3, 131–150, 1999), and Le Cam (Asymptotic Methods in Statistical Decision Theory. Springer, Berlin Heidelberg New York, 1986).      

Asymptotic distribution of the discriminant function

This paper describes discrimination among multivariate autoregressive processes by the Bayes method. The asymptotic distribution of the discriminant function and estimation of the probability of misclassification are investigated.     

On the distribution of the sequence of fractional parts of a slowly increasing exponential function

In this paper for a slowly increasing exponential function we study the degree of deviation of the distribution of the sequence of its fractional parts from the “ideal” uniform distribution. An estimate is given which can appropriately be called the measure of irregularity of the distribution of a given sequence. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 148–152, January, 1999.     

Tail approximations to the density function in EVT

Let X ₁, X ₂, ...,X _n be independent identically distributed random variables with common distribution function F, which is in the max domain of attraction of an extreme value distribution, i.e., there exist sequences a _n > 0 and b _n ∈ ℝ such that the limit of exists. Assume the density function f (of F) exists. We obtain an uniformly weighted approximation to the tail density function f, and an uniformly weighted approximation to the tail density function of under some second order condition.Partially supported by a grant of the Swiss National Science Foundation.     

On the limit behaviour of the joint distribution function of order statistics

Fork ₀ fixed we consider the joint distribution functionF _n ^k of then-k smallest order statistics ofn real-valued independent, identically distributed random variables with arbitrary cumulative distribution functionF. The main result of the paper is a complete characterization of the limit behaviour ofF _n ^k (x ₁,,x _n-k) in terms of the limit behaviour ofn(1-F(x _n)) ifn tends to infinity, i.e., in terms of the limit superior, the limit inferior, and the limit if the latter exists. This characterization can be reformulated equivalently in terms of the limit behaviour of the cumulative distribution function of the (k+1)-th largest order statistic. All these results do not require any further knowledge about the underlying distribution functionF.     

长方体上均匀分布的密度函数

讨论了长方体上均匀分布密度函数问题,得到了长方体体积的估计量、估计量的点估计及估计量的密度函数.     

Error Bounds for Uniform Asymptotic Expansions—Modified Bessel Function of Purely Imaginary Order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals. By using a class of rational functions, they express these quantities in terms of Cauchy-type integrals; these expressions are natural generalizations of integral representations of the coefficients and the remainders in the Taylor expansions of analytic functions. By using the new representation, a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.     

Asymptotic approximation for the probability density function of an arbitrary sequence of random variables

We establish a large deviation approximation for the density function of an arbitrary sequence of random variables. The results are analogous to those obtained by Chaganty and Sethuraman (1985). We apply our theorems to the sample variance and the Mann–Whitney two-sample statistic.     

Derivation of the probability distribution functions for succession quota random variables

The probability distribution functions (pdf's) of the sooner and later waiting time random variables (rv's) for the succession quota problem (k successes and r failures) are derived presently in the case of a binary sequence of order k. The probability generating functions (pgf's) of the above rv's are then obtained directly from their pdf's. In the case of independent Bernoulli trials, expressions for the pdf's in terms of binomial coefficients are also established.     

A uniform bound for the deviation of empirical distribution functions

If X₁, …, X_n are independent R^d-valued random vectors with common distribution function F, and if F_n is the empirical distribution function for X₁, …, X_n, then, among other things, it is shown that P{sup_x F_n(x) ε} 2e²(2n)^de^−2nε2 for all nε² ≥ d². The inequality remains valid if the X_i are not identically distributed and F(x) is replaced by Σ_iP{X_i ≤ x}/n.