Dispersive ordering of fail-safe systems with heterogeneous exponential components

Let X ₁, . . . , X _n be independent exponential random variables with respective hazard rates λ₁, . . . , λ _n , and Y ₁, . . . , Y _n be independent and identically distributed random variables from an exponential distribution with hazard rate λ. Then, we prove that X _2:n , the second order statistic from X ₁, . . . , X _n , is larger than Y _2:n , the second order statistic from Y ₁, . . . , Y _n , in terms of the dispersive order if and only if

$\lambda\geq \sqrt{\frac{1}{{n\choose 2}}\sum_{1\leq i < j\leq n}\lambda_i\lambda_j}.$

We also show that X _2:n is smaller than Y _2:n in terms of the dispersive order if and only if

$ \lambda\le\frac{\sum^{n}_{i=1} \lambda_i-{\rm max}_{1\leq i\leq n} \lambda_i}{n-1}. $

Moreover, we extend the above two results to the proportional hazard rates model. These two results established here form nice extensions of the corresponding results on hazard rate, likelihood ratio, and MRL orderings established recently by Pǎltǎnea (J Stat Plan Inference 138:1993–1997, 2008), Zhao et al. (J Multivar Anal 100:952–962, 2009), and Zhao and Balakrishnan (J Stat Plan Inference 139:3027–3037, 2009), respectively.

     

Unbiased estimates for a lognormal regression problem and a nonparametric alternative

Given a normal sample with means \({{\bf x}_{1}^{\prime} {\bf \varphi}, \ldots, {\bf x}_{n}^{\prime} {\bf \varphi}}\) and variance v, minimum variance unbiased estimates are given for the moments of L, where log L is normal with mean \({{\bf x}^{\prime} {\bf \varphi}}\) and variance v. These estimates converge to wrong values if the normality assumption is false. In the latter case estimates based on any M-estimate of \({{\bf \varphi}}\) are available of bias \({O\left(n^{-1}\right)}\) and \({O\left(n^{-2}\right)}\). More generally, these are given for any smooth function of \({\left({\bf \varphi}, F\right)}\), where F is the unknown distribution of the residuals. The regression functions need not be linear.     

A Cramér-type large deviation theorem for sums of functions of higher order non-overlapping spacings

Let U ₁, U ₂, . . . , U _n–1 be an ordered sample from a Uniform [0,1] distribution. The non-overlapping uniform spacings of order s are defined as \({G_{i}^{(s)} =U_{is} -U_{(i-1)s}, i=1,2,\ldots,N^\prime, G_{N^\prime+1}^{(s)} =1-U_{N^\prime s}}\) with notation U ₀ = 0, U _n = 1, where \({N^\prime=\left\lfloor n/s\right\rfloor}\) is the integer part of n/s. Let \({ N=\left\lceil n/s\right\rceil}\) be the smallest integer greater than or equal to n/s, f _m (u), m = 1, 2, . . . , N, be a sequence of real-valued Borel-measurable functions. In this article a Cramér type large deviation theorem for the statistic \({f_{1,n} (nG_{1}^{(s)})+\cdots+f_{N,n} (nG_{N}^{(s)} )}\) is proved.     

Berry–Esseen bounds for density estimates under NA assumption

Let {X _j } be a strictly stationary sequence of negatively associated random variables with the marginal probability density function f(x). The recursive kernel estimators of f(x) are defined by

A new non-linear AR(1) time series model having approximate beta marginals

We consider the mixed AR(1) time series model $$X_t=\left\{\begin{array}{ll}\alpha X_{t-1}+ \xi_t \quad {\rm w.p.} \qquad \frac{\alpha^p}{\alpha^p-\beta ^p},\\ \beta X_{t-1} + \xi_{t} \quad {\rm w.p.} \quad -\frac{\beta^p}{\alpha^p-\beta ^p} \end{array}\right.$$ for ?1 < β ^p ≤ 0 ≤ α ^p  < 1 and α ^p ? β ^p  > 0 when X _t has the two-parameter beta distribution B₂(p, q) with parameters q > 1 and ${p \in \mathcal P(u,v)}$ , where $$\mathcal P(u,v) = \left\{u/v : u < v,\,u,v\,{\rm odd\,positive\,integers} \right\}.$$ Special attention is given to the case p = 1. Using Laplace transform and suitable approximation procedures, we prove that the distribution of innovation sequence for p = 1 can be approximated by the uniform discrete distribution and that for ${p \in \mathcal P(u,v)}$ can be approximated by a continuous distribution. We also consider estimation issues of the model.     

Distributions determined by conditioning on a single order statistic

LetX ₁,X ₂, …,X _n(n ? 2) be a random sample on a random variablex with a continuous distribution functionF which is strictly increasing over (a, b), ?∞ ?a <b ? ∞, the support ofF andX _1:n ?X _2:n ? … ?X _n:n the corresponding order statistics. Letg be a nonconstant continuous function over (a, b) with finiteg(a ⁺) andE {g(X)}. Then for some positive integers, 1 <s ?n $$E\left\{ {\frac{1}{{s - 1}}\sum\limits_{i - 1}^{s - 1} {g(X_{i:n} )|X_{s:n} } = x} \right\} = 1/2(g(x) + g(a^ + )), \forall x \in (a,b)$$ iffg is bounded, monotonic and \(F(x) = \frac{{g(x) - g(a^ + )}}{{g(b^ - ) - g(a^ + )}},\forall x \in (a,b)\) . This leads to characterization of several distribution functions. A general form of this result is also stated.     

An extremal property of the generalized arcsine distribution

The main result of the paper is the following characterization of the generalized arcsine density p _γ (t) = t ^γ?1(1 ? t) ^γ?1/B(γ, γ)   with ${t \in (0, 1)}$ and ${\gamma \in(0,\frac12) \cup (\frac12,1)}$ : a r.v. ξ supported on [0, 1] has the generalized arcsine density p _γ (t) if and only if ${ {\mathbb E} |\xi- x|^{1-2 \gamma}}$ has the same value for almost all ${x \in (0,1)}$ . Moreover, the measure with density p _γ (t) is a unique minimizer (in the space of all probability measures μ supported on (0, 1)) of the double expectation ${ (\gamma-\frac12 ) {\mathbb E} |\xi-\xi^{\prime}|^{1-2 \gamma}}$ , where ξ and ξ′ are independent random variables distributed according to the measure μ. These results extend recent results characterizing the standard arcsine density (the case ${\gamma=\frac12}$ ).     

Tests of significance for the mean of a finite lot

Let ξ₁, ξ₂, ξ₃, ... be independent identically distributed random variables each with normal distribution with mean μ and variance σ². Tests for the process mean μ are well-known elements of statistical analysis: the Gauß test under known process variance σ², Student’st-test under unknown process variance σ². Let the process be partitioned in lots (ξ₁, ..., ξ _N ), (ξ _N+1, ..., ξ_2N ), ... of sizeN. Consider (ξ₁, ..., ξ _N ) as a stochastic representative of this lot sequence and let the lot be characterized by the lot mean $\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } $ . The lot mean can be considered as a parameter of the joint conditional distribution function of the lot variables under $\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } = z$ . The present paper investigates the analogies of the Gauß test and Student’st-test for the lot situation, i.e. tests of significance for the lot meanz under known and unknown process variance σ². This approach is of special interest for the statistical control of product quality in situations where the quality of a lot of items 1, 2, ...,N with quality characteristics ξ₁, ξ₂, ..., ξ _N is identified with the lot average $\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } = z$ .     

A Nonclassical Law of the Iterated Logarithm for Functions of Positively Associated Random Variables

Let be a sequence of stationary positively associated random variables and a sequence of positive constants be monotonically approaching infinity and be not asymptotically equivalent to loglog n. Under some suitable conditions, a nonclassical law of the iterated logarithm is investigated, i.e.

Consistency of a nonparametric estimation of a density functional

LetX be a random variable with distribution functionF and density functionf. Let ? and ψ be known measurable functions defined on the real lineR and the closed interval [0, 1], respectively. This paper proposes a smooth nonparametric estimate of the density functional \(\theta = \int\limits_R \phi (x) \psi \left[ {F (x)} \right]f^2 (x) dx\) based on a random sampleX ₁, ...,X _n fromF using a kernel functionk. The proposed estimate is given by \(\hat \theta = (n^2 a_n )^{ - 1} \mathop \sum \limits_{i = 1}^n \mathop \sum \limits_{j = 1}^n \phi (X_i ) \psi \left[ {\hat F (X_i )} \right]k\left[ {(X_i - X_j )/a_n } \right]\) , where \(\hat F(x) = n^{ - 1} \mathop \sum \limits_{i = 1}^n K\left[ {(x - X_i )/a_n } \right]\) with \(K (w) = \int\limits_{ - \infty }^w {k (u) } du\) . The estimate \(\hat \theta \) is shown to be consistent both in the weak and strong sense and is used to estimate the asymptotic relative efficiency of various nonparametric tests, with particular reference to those using the Chernoff-Savage statistic.     

The distribution of the maximum of a first order autoregressive process: the continuous case

We give the cumulative distribution function of M _n , the maximum of a sequence of n observations from an autoregressive process of order 1. Solutions are first given in terms of repeated integrals and then for the case, where the underlying random variables are absolutely continuous. When the correlation is positive, $$P \left( M_n \leq x \right)\ =a_{n,x},$$ where $$a_{n,x}= \sum_{j=1}^\infty \beta_{jx}\ \nu_{jx}^{n} = O \left( \nu_{1x}^{n}\right),$$ where {?? _jx } are the eigenvalues of a non-symmetric Fredholm kernel, and ?? _1x is the eigenvalue of maximum magnitude. When the correlation is negative $$P \left( M_n \leq x \right)\ =a_{n,x} +a_{n-1,x}.$$ The weights ?? _jx depend on the jth left and right eigenfunctions of the kernel. These are given formally by left and right eigenvectors of an infinite Toeplitz matrix whose eigenvalues are just {?? _jx }. These results are large deviations expansions for extremes, since the maximum need not be standardized to have a limit. In fact, such a limit need not exist. The use of the derived expansion for P(M _n ?? x) is illustrated using both simulated and real data sets.     

On generalized transforms of distribution functions

In this note we discuss the following problem. LetX andY to be two real valued independent r.v.'s with d.f.'sF and ?. Consider the d.f.F*? of the r.v.X oY, being o a binary operation among real numbers. We deal with the following equation: $$\mathcal{G}^1 (F * \phi ,s) = \mathcal{G}^2 (F,s)\square \mathcal{G}^3 (\phi ,s)\forall s \in S$$ where \(\mathcal{G}^1 ,\mathcal{G}^2 ,\mathcal{G}^3 \) are real or complex functionals, т another binary operation ands a parameter. We give a solution, that under stronger assumptions (Aczél 1966), is the only one, of the problem. Such a solution is obtained in two steps. First of all we give a solution in the very special case in whichX andY are degenerate r.v.'s. Secondly we extend the result to the general case under the following additional assumption: $$\begin{gathered} \mathcal{G}^1 (\alpha F + (1 - \alpha )\phi ,s) = H[\mathcal{G}^i (F,s),\mathcal{G}^i (\phi ,s);\alpha ] \hfill \\ \forall \alpha \in [0,1]i = 1,2,3 \hfill \\ \end{gathered} $$ .     

Characterization of the exponential distribution function by Properties of the differenceX k+s∶n−X k∶n of order statistics

We considern independent and identically distributed random variables with common continuous distribution functionF concentrated on (0, ∞). LetX _1∶n≤X_2∶n...≤X_n∶n be the corresponding order statistics. Put $$d_s \left( x \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - P\left( {X_{s:n - k} \geqslant x} \right), x \geqslant 0,$$ and $$\delta _s \left( {x, \rho } \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - e^{ - \rho \left( {n - k} \right)x} ,\rho > 0,x \geqslant 0.$$ Fors=1 it is well known that each of the conditions d₁(x)=O ?x≥0 and δ₁ (x, p) = O ?x≥0 implies thatF is exponential; but the analytic tools in the proofs of these two statements are radically different. In contrast to this in the present paper we present a rather elementary method which permits us to derive the above conclusions for somes, 1≤n —k, using only asymptotic assumptions (either forx→0 orx→∞) ond _s(x) and δ₁ (x, p), respectively.