参数的变化对F分布密度函数之影响

该文运用对无穷级数的一些特殊处理方法,深入分析了与г函数有关的一些特殊函数 的性质,揭示了参数变化时F分布密度函数极值变化的一些深刻规律．该文指明,n增大时 F分布的密度函数fm,n(x)的极大值单调增加,而m增大时该密度函数的极大值或单调减 少,或先减后增．     

F分布密度函数之性质

本文利用特殊函数的性质,较详细地分析了F分布密度函数之性质,指出了该密度函数与相应参数之间的关系.本文主要研究第二个参数变化对密度函数的影响,证明了n增大时F(m,n)分布的密度函数极大值也越来越大,还指出了n变化时F(m,n)分布的相应密度曲线与另一特定密度曲线交点的变化规律.     

参数变化对β-分布密度函数之影响

本文通过对无穷级数的一些特殊处理方法,深入分析了与β-函数有关的一些特殊函数中所蕴含的某些单调性质,从而揭示了β-分布的密度函数极值变化的一些深刻规律性.本文证明了:当参数皆比1大时β-分布的密度函数有极大值且此极大值随着其中一个参数的逐渐变大而先减后增;当参数皆比1小时该密度函数有极小值且此极小值随着其中一个参数的逐渐变大而先增后减.     

Γ分布密度函数的性质

分析了Γ分布密度函数的性质,指出了该密度函数与相应参数之间的关系.主要研究第二个参数对密度的影响,证明了β增大时Γ(α,β)分布密度极大值也增大,还指出了β变化时Γ(α,β)分布密度与另一特定密度曲线交点的变化规律.     

Г分布密度函数的性质

分析了Г分布密度函数的性质，指出了该密度函数与相应参数之间的关系．主要研究第二个参数对密度的影响，证明了β增大时Г（α，β）分布密度极大值也增大，还指出了β变化时Г（α，β）分布密度与另一特定密度曲线交点的变化规律．     

F分布密度曲线之变化

该文较深入地分析了特殊函数的某些性质,利用这些性质分析了F分布中不同参数所对应的密度曲线之间的位置关系.此外还讨论了密度曲线的某些渐近性质并建立了一些有关的方程.作为结论的应用及验证,作者给出了一些有代表性的例子.该文的分析方法还可用来讨论其它一些分布的密度曲线的某些性质.     

对称连续分布函数的最优不变估计

该文考虑了未知对称连续分布函数的不变估计问题.连续分布函数在单调变换群下是不变的^[1], 但这个变换群不能保证对称分布函数的不变性.于是, 所要研究的判决问题在单调变换群下不再是不变的. 为了保证判决问题不变性, 考虑一个新的变换群—单调奇变换群, 它确保了所研究的判决问题的不变性.注意到对称分布函数零点的特殊性质, 即, 对任一对称分布函数F, 均有F(0)=1/2,通过视零点为一伪观察值, 得到了所有的非随机化不变估计, 并在不变估计中找到了最优不变估计.     

F分布的几个性质

F分布的概率密度函数中出现了Gamma函数和两个不同的参数.借助微积分的相关理论,固定F分布的密度函数中的一个参数,当另一个参数取不同值时,给出相应曲线的交点范围,同时研究F分布的密度函数的凸性.     

探求数列问题的函数视角

数列是一类特殊的函数,其定义域只能取正整数集(或其子集).牵涉到数列的单调性问题,或求与数列最大(小)项的问题,往往需要从函数角度去分析判断数列的特性,通过对函数定义域限定为正整数集范围内,利用函数的单调性或函数的值域来寻求.本文就数列这一特殊函数,例析在涉及到单调性问题时的一般     

定义法证函数单调性学习心得

我们在学习函数单调性时应倍感亲切,因为初中时已经接触过.当时有两句口诀人人都会讲,第一句:"y随x增大而增大".这就是高中所学的增函数.第二句:"y随x增大而减小."这就是减函数.当时没有点明函数的单调性,也没有强调单调区间,进入高中后学习函数单调性时,将上述两句语言抽象成了数学符     

Sign Regular Matrices of Order Two

A matrix is called sign regular of order k if every minor of order i has the same sign for each i = 1,2,<, k . If an m × n matrix is sign regular of order k for k = min { m,n } then it is called sign regular. This paper studies some properties of sign regular matrices of order two. Remarkable properties are proved when the row sums of these matrices form a monotone vector.     

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

In this paper we consider the probability density function (pdf) of a non-central χ² 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

In this paper our aim is to show that if a probability density function is geometrically concave (convex), then the corresponding cumulative distribution function and the survival function are geometrically concave (convex) too, under some assumptions. The proofs are based on the so-called monotone form of l'Hospital's rule and permit us to extend our results to the case of the concavity (convexity) with respect to Hölder means. To illustrate the applications of the main results, we discuss in details the geometrical concavity of the probability density function, cumulative distribution function and survival function of some common continuous univariate distributions. Moreover, at the end of the paper, we present a simple alternative proof to Schweizer's problem related to the Mulholland's generalization of Minkowski's inequality.     

Estimating means from a non-I.I.D. Mixture of Poisson samples

In this paper, a family of estimators for estimating means when mixing two independent Poisson samples is proposed. This family is based on the probability-generating function of the Poisson distribution and is offered as an alternative to the maximum likelihood estimators, which have some drawbacks. These estimators include the method of moments estimators as a special limiting case.     

On Statistical Inference Based on Record Values

We develop methodology for conducting inference based on record values and record times derived from a sequence of independent and identically distributed random variables. The advantage of using information about record times as well as record values is stressed. This point is a subtle one, since if the sampling distribution F is continuous then there is no information at all about F in the record times alone; the joint distribution of any number of them does not depend on F. However, the record times and record values jointly contain considerably more information about F than do the record values alone. Indeed, in the case of a distribution with regularly varying tails, the rate of convergence of the exponent of regular variation is two orders of magnitude faster if information about record times is included. Optimal estimators and convergence rates are derived under simple, specific models, and shown to be surprisingly robust against significant departures from those models. However, even under our special models the estimators have irregular properties, including an undefined information matrix. To some extent these difficulties may be alleviated by conditioning and by considering the relationship between maximum likelihood and maximum probability estimators.     

Generalized Bochner’s Theorem for radial function

A radial function Φ(x) can be expressed by its generator ?(·) through Φ(x)=?(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ? for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ? to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in ICM₀; a positive monotone decreasing function is in ICM₁ and a positive monotone decreasing and convex function is in ICM₂. Based on this definition we get a generalized Bochner’s Theorem for radial function: If1-dimensional Fourier transform of the generator of a radial function can be written as $F{}_1\varphi (t) = \tilde F(\frac{{t^2 }}{2})$ , then corresponding radial function Φ(x) is positive definite as a n-variate function iff $\tilde F$ is an incompletely monotone function of order α=(n-1)/2 (or simply $\tilde F \in ICM_{\frac{{n - 1}}{2}} $ .