Multivariate likelihood ratio orderings between spacings of heterogeneous exponential random variables

Let X ₁, X ₂, ..., X _n be independent exponential random variables such that X _i has failure rate λ for i = 1, ..., p and X _j has failure rate λ^* for j = p + 1, ..., n, where p ≥ 1 and q = n − p ≥ 1. Denote by D _i:n (p,q) = X _i:n −X _i-1:n the ith spacing of the order statistics X _1:n ≤ X _2:n ≤ ... ≤ X _n:n , i = 1, ..., n, where X _0:n ≡ 0. The purpose of this paper is to investigate multivariate likelihood ratio orderings between spacings D _i:n (p,q), generalizing univariate comparison results in Wen et al.(J Multivariate Anal 98:743–756, 2007). We also point out that such multivariate likelihood ratio orderings do not hold for order statistics instead of spacings. Supported by National Natural Science Foundation of China, the Program for New Century Excellent Talents in University (No.: NCET-04-0569), and by the Knowledge Innovation Program of the Chinese Academy of Sciences (No.: KJCX3-SYW-S02).     

Bounds for the mean residual life function of a <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system

In the reliability studies, k-out-of-n systems play an important role. In this paper, we consider sharp bounds for the mean residual life function of a k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F, measured in location and scale units of the residual life random variable X _t  = (X−t|X > t). We characterize the probability distributions for which the bounds are attained. We also evaluate the so obtained bounds numerically for various choices of k and n.     

Largest excess of boundary crossings for martingales

Summary LetX ₁,X ₂, ... form a sequence of martingale differences and denote byZ(a, α) = sup n (S _n −an ^α)⁺ the largest excess forS _n =X ₁ + ... +X _n crossing the boundaryan ^α. We give a sufficient condition for the finiteness ofEZ(a, α)^β which is formulated in terms of bounds forE(X _i ^{+ p} andE(|X _i |γ|X ₁, ...,X _i-1), whereα, β, γ, p are suitably related. This general result is then applied to the case of independent random variables.     

Locally minimax test of independence in elliptically symmetrical distributions with additional observations

Summary LetX=(X _ij )=(X ₁, ...,X _n )’,X’ _i =(X _i1, ...,X _ip )’,i=1,2, ...,n be a matrix having a multivariate elliptical distribution depending on a convex functionq with parameters, 0,σ. Let ϱ²=ϱ ₂ ^-2 be the squared multiple correlation coefficient between the first and the remainingp ₂+p ₃=p−1 components of eachX _i . We have considered here the problem of testingH ₀:ϱ²=0 against the alternativesH ₁:ϱ ₁ ^-2 =0, ϱ ₂ ^-2 >0 on the basis ofX andn ₁ additional observationsY ₁ (n ₁×1) on the first component,n ₂ observationsY ₂(n ₂×p ₂) on the followingp ₂ components andn ₃ additional observationsY ₃(n ₃×p ₃) on the lastp ₃ components and we have derived here the locally minimax test ofH ₀ againstH ₁ when ϱ ₂ ^-2 →0 for a givenq. This test, in general, depends on the choice ofq of the familyQ of elliptically symmetrical distributions and it is not optimality robust forQ.     

Strong consistency of the MLE under random censoring

LetX ₁, ...,X _n be an i.i.d. sample from some parametric family {_θ :θ (Θ} of densities. In the random censorship model one observesZ _i =min (X _i ,Y _i ) andδ _i =1_{ x _i ≤Y _i}, whereY _i is a censoring variable being independent ofX _i . In this paper we investigate the strong consistency ofθ _n maximizing the modified likelihood function based on (Z _i ,δ _i , 1≤i≤n. The main result constitutes an extension of Wald’s theorem for complete data to censored data. Work partially supported by the “Deutsche Forschungsgemeinschaft”.     

Non-positive upper bounds on expectations of small order statistics from DDA and DFRA populations

Rychlik [Appl Math (Warsaw) 29:15–32, 2002] presented positive sharp upper bounds on the expectations of order statistics with sufficiently large ranks, based on i.i.d. samples from the decreasing density and failure rate populations (DDA and DFRA, for short). They were expressed in terms of the population mean and standard deviation. Here we provide respective non-positive upper tight evaluations for expected small order statistics centered about the population mean, measured in various scale units.     

On the power function of the exact test for ther×c contingency table

Summary LetN=[n _ij ] (i=1, …,r;j=1, …,c) be the matrix of observed frequencies in anr×c contingency table fromr possibly different multinomial populations with respective probabilitiesp _i =(p _i1, …,p _ic ).Freeman andHalton have proposed an exact conditional test for the hypothesisH ₀ :p _i =(p ₁, …p _c ) of the exact test is derived. Numerical values forβ(p) were previously computed for the special case:r=3,c=2 [Bennett andNakamura, 1964].     

Strong uniform convergence for the estimator of the regression function under ϕ-mixing conditions

Suppose the observations (X _i,Y _i), i=1,…, n, are ϕ-mixing. The strong uniform convergence and convergence rate for the estimator of the regression function was studied by serveral authors, e.g. G. Collomb (1984), L. Gy?rfi et al. (1989). But the optimal convergence rates are not reached unless the Y _i are bounded or the E exp (a|Y _i|) are bounded for some a>0. Compared with the i.i.d. case the convergence of the Nadaraya-Watson estimator under ϕ-mixing variables needs strong moment conditions. In this paper we study the strong uniform convergence and convergence rate for the improved kernel estimator of the regression function which has been suggested by Cheng P. (1983). Compared with Theorem A in Y. P. Mack and B. Silverman (1982) or Theorem 3.3.1 in L. Gy?rfi et al. (1989), we prove the convergence for this kind of estimators under weaker moment conditions. The optimal convergence rate for the improved kernel estimator is attained under almost the same conditions of Theorem 3.3.2 in L. Gy?rfi et al. (1989). Received: September 1999     

The average number of events per time unit and its estimation

Let ζ _t be the number of events which will be observed in the time interval [0;t] and define as the average number of events per time unit if this limit exists. In the case of i.i.d. waiting-times between the events,E[ζ _t ] is the renewal function and it follows from well-known results of renewal theory thatA exists and is equal to 1/τ, if τ>0 is the expectation of the waiting-times. This holds true also when τ = ∞.A may be estimate by ζ _t /t or where is the mean of the firstn waiting-timesX ₁,X ₂, ...,X _n . Both estimators converage with probability 1 to 1/τ if theX _i are i.i.d.; but the expectation of may be infinite for alln and also if it is finite, is in general a positively biased estimator ofA. For a stationary renewal process, ζ _t /t is unbiased for eacht; if theX _i are i.i.d. with densityf(x), then ζ _t /t has this property only iff(x) is of the exponential type and only for this type the numbers of events in consecutive time intervals [0,t], [t, 2t], ... are i.i.d. random variables for arbitraryt > 0.     

Asymptotic results and tests for the choice of approximative models in nonlinear two-phases regression models, heteroscedatic case

This paper is devoted to the study of the least squares estimator of f for the classical, fixed design, nonlinear model X (t _i)=f(t _i)+ε(t _i), i=1,2,…,n, where the (ε(t _i))_i=1,…,n are independent second order r.v.. The estimation of f is based upon a given parametric form. In Brodeau (1993) this subject has been studied in the homoscedastic case. This time we assume that the ε(t _i) have non constant and unknown variances σ²(t _i). Our main goal is to develop two statistical tests, one for testing that f belongs to a given class of functions possibly discontinuous in their first derivative, and another for comparing two such classes. The fundamental tool is an approximation of the elements of these classes by more regular functions, which leads to asymptotic properties of estimators based on the least squares estimator of the unknown parameters. We point out that Neubauer and Zwanzig (1995) have obtained interesting results for connected subjects by using the same technique of approximation. Received: February 1996     

Minimax estimators for a bounded location parameter

Summary Recently, Bischoff and Fieger (1992) considered the classical problem of estimating a bounded normal mean when the loss is thep-th power of the error. They proved that forp>-2 a two point prior is least favourable in case the parameter interval is small enough. In the present paper it is shown that this result remains valid forp>1. Moreover, the normal family is generalized to location parameter families. Finally, it is proved that no two point prior is least favourable for absolute error loss, i.e., forp=1.     

A note on the maximum deviation of the scale-contaminated normal to the best normal distribution

In this paper we consider the case of the scale-contaminated normal (mixture of two normals with equal mean components but different component variances: (1−p)N(μ,σ²)+pN(μ,τ²) with σ and τ being non-negative and 0≤p≤1). Here is the scale error and p denotes the amount with which this error occurs. It's maximum deviation to the best normal distribution is studied and shown to be montone increasing with increasing scale error. A closed-form expression is derived for the proportion which maximizes the maximum deviation of the mixture of normals to the best normal distribution. Implications to power studies of tests for normality are pointed out. Received May 2001     

A study on the mean past lifetime of the components of (n − k + 1)-out-of-n system at the system level

In the present paper, we consider a (n − k + 1)-out-of-n system with identical components where it is assumed that the lifetimes of the components are independent and have a common distribution function F. We assume that the system fails at time t or sometime before t, t > 0. Under these conditions, we are interested in the study of the mean time elapsed since the failure of the components. We call this as the mean past lifetime (MPL) of the components at the system level. Several properties of the MPL are studied. It is proved that the relation between the proposed MPL and the underlying distribution is one-to-one. We have shown that when the components of the system have decreasing reversed hazard then the MPL of the system is increasing with respect to time. Some examples are also provided.     

A bound on the expected overshoot for some concave boundaries

LetX ₁,X ₂,… be i.i.d. with finite meanμ>0,S _n =X ₁+…+X _n . Forf(n)=n ^β ,c>0 we consider the stopping timesT _c =inf{n:S _n >c+f(n)} with overshootR _c =S _{T c} −(c+f(T _c )). For 0<β<1 we give a bound for sup _c≥0 ER _c in the spirit of Lorden’s well-known inequality forf=0.     

On the discrete mean past lifetime

Let T denote a positive discrete survival time and n a non-negative integer number. Properties of the mean past lifetime E(n  −  T|T  <  n) are provided. M. Asadi’s work is supported by the University of Isfahan Grant 850713.     

Asymptotic normality of M-estimators in a semiparametric model with longitudinal data

Suppose that the longitudinal observations (Y _ij , X _ij , t _ij ) for i = 1, . . . ,n; j = 1, . . . ,m _i are modeled by the semiparamtric model where β ₀ is a k × 1 vector of unknown parameters, g(·) is an unknown estimated function and e _ij are unobserved disturbances. This article consider M-type regressions which include mean, median and quantile regressions. The M-estimator of the slope parameter β ₀ is obtained through piecewise local polynomial approximation of the nonparametric component. The local M-estimator of g(·) is also obtained by replacing β ₀ in model with its M-estimator and using local linear approximation. The asymptotic distribution of the estimator of β ₀ is derived. The asymptotic distributions of the local M-estimators of g(·) at both interior and boundary points are also established. Various applications of our main results are given. The research is supported in part by National Natural Science Foundation of China (Grant No. 10671089).     

Multivariate multistage methodologies for simultaneous all pairwise comparisons

We consider the problem of constructing simultaneous fixed-width confidence intervals for all pairwise treatment differences μ₁−μ _J , in the presence ofk(≥2) independent populationsN _p (μ₁,Σ), 1≤i≠j≤k. Appropriate purely sequential, accelerated sequential and three-stage sampling strategies have been developed and variousfirst-order asymptotic properties are then derived when Σ _pxp is completely unknown, but positive definite (p.d.). In the two special cases when the largest component variance in Σ is a known multiple of one of the variances or Σ=σ² H where σ(>0) is unknown, butH _pxp is known and p.d., the original multistage sampling strategies are specialized. Under such special circumstances, associatedsecond-order characteristics are then developed. It is to be noted that our present formulation and the methodologies fill important voids in the context of multivariate multiple comparisons which is a challenging area that has not yet been fully explored. Moderate sample performances of the proposed techniques were very encouraging and detailed remarks on these were included in Mukhopadhyay and Aoshima (1997).     

Some properties of the minimum and the maximum of random variables with joint logconcave distributions

It is shown that if (X ₁, X ₂, . . . , X _n ) is a random vector with a logconcave (logconvex) joint reliability function, then X _P = min _i∈P X _i has increasing (decreasing) hazard rate. Analogously, it is shown that if (X ₁, X ₂, . . . , X _n ) has a logconcave (logconvex) joint distribution function, then X ^P  = max _i∈P X _i has decreasing (increasing) reversed hazard rate. If the random vector is absolutely continuous with a logconcave density function, then it has a logconcave reliability and distribution functions and hence we obtain a result given by Hu and Li (Metrika 65:325–330, 2007). It is also shown that if (X ₁, X ₂, . . . , X _n ) has an exchangeable logconcave density function then both X _P and X ^P have increasing likelihood ratio.     

A characterization of the exponential distribution by higher order gap

Summary SupposeX is a non-negative random variable with an absolutely continuous (with respect to Lebesgue measure) distribution functionF (x) and the corresponding probability density functionf(x). LetX ₁,X ₂,...,X _n be a random sample of sizen fromF andX _i,n is thei-th smallest order statistics. We define thej-th order gapg _i,j(n) asg _i,j(n)=X _i+j,n–X_i,n 1i<n, 1nn–i. In this paper a characterization of the exponential distribution is given by considering a distribution property ofg _i,j(n).     

Estimating a restricted normal mean

Let X ₁, X ₂, ..., X _n be a random sample from a normal distribution with unknown mean μ and known variance σ ². In many practical situations, μ is known a priori to be restricted to a bounded interval, say [−m, m] for some m > 0. The sample mean , then, becomes an inadmissible estimator for μ. It is also not minimax with respect to the squared error loss function. Minimax and other estimators for this problem have been studied by Casella and Strawderman (Ann Stat 9:870–878, 1981), Bickel (Ann Stat 9:1301–1309, 1981) and Gatsonis et al. (Stat Prob Lett 6:21–30, 1987) etc. In this paper, we obtain some new estimators for μ. The case when the variance σ ² is unknown is also studied and various estimators for μ are proposed. Risk performance of all estimators is numerically compared for both the cases when σ ² may be known and unknown.