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).     

Distributions determined by conditioning on a pair of order statistics

LetX ₁,X ₂, ...,X _n (n≥3) be a random sample on a random variableX with distribution functionF having a unique continuous inverseF ⁻¹ over (a,b), −∞≤a<b≤∞ the support ofF. LetX _1:n <X _2:n <...<X _n:n be the corresponding order statistics. Letg be a nonconstant continuous function over (a,b). Then for some functionG over (a, b) and for some positive integersr ands, 1<r+1<s≤n

On maximum likelihood prediction based on Type II doubly censored exponential data

In this paper, the maximum likelihood predictor (MLP) of the kth ordered observation, t _k, in a sample of size n from a two-parameter exponential distribution as well as the predictive maximum likelihood estimators (PMLE's) of the location and scale parameters, θ and β, based on the observed values t _r, …, t _s (1≤r≤s<k≤n), are obtained in closed forms, contrary to the belief they cannot be so expressed. When θ is known, however, the PMLE of β and MLP of t _k do not admit explicit expressions. It is shown here that they exist and are unique; sharp lower and upper bounds are also provided. The derived predictors and estimators are reasonable and also have good asymptotic properties. As applications, the total duration time in a life test and the failure time of a k-out-of-n system may be predicted. Finally, an illustrative example is included. Received: August 1999     

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.     

A consistent test for multivariate normality based on the empirical characteristic function

LetX ₁,X ₂, …,X _n be independent identically distributed random vectors in IR ^d ,d ⩾ 1, with sample mean and sample covariance matrixS _n. We present a practicable and consistent test for the composite hypothesisH _d: the law ofX ₁ is a non-degenerate normal distribution, based on a weighted integral of the squared modulus of the difference between the empirical characteristic function of the residualsS _n ^−1/2 (X _j − ) and its pointwise limit exp (−1/2|t|²) underH _d. The limiting null distribution of the test statistic is obtained, and a table with critical values for various choices ofn andd based on extensive simulations is supplied.     

Two-sample rank tests with truncated populations

LetX ₁,…,X _m andY ₁,…,Y _n be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S ₍₁₎,…,S _(n)), whereS _(j)=# (X _i ’s≤Y _(j)) andY _(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test. We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests based onS ₍₁₎ andS _(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c).     

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”.     

Characterization of the multivariate normal distribution by conditional normal distributions

Summary LetX andY be two random vectors with values in ℝ ^k and ℝ∝, respectively. IfZ=(X ^T,Y ^T) ^T is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vectorZ=(X ₁ ^T , …,X _t ^T ) ^T is splitted intot≥3 partsX ₁, …,X _t.     

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     

Multivariate L1 mean

The center of a univariate data set {x ₁,…,x _n} can be defined as the point μ that minimizes the norm of the vector of distances y′=(|x ₁−μ|,…,|x _n−μ|). As the median and the mean are the minimizers of respectively the L ₁- and the L ₂-norm of y, they are two alternatives to describe the center of a univariate data set. The center μ of a multivariate data set {x ₁,…,x _n} can also be defined as minimizer of the norm of a vector of distances. In multivariate situations however, there are several kinds of distances. In this note, we consider the vector of L ₁-distances y′₁=(∥x ₁- μ∥₁,…,∥x _n- μ∥₁) and the vector of L ₂-distances y′₂=(∥x ₁- μ∥₂,…,∥x _n-μ∥₂). We define the L ₁-median and the L ₁-mean as the minimizers of respectively the L ₁- and the L ₂-norm of y ₁; and then the L ₂-median and the L ₂-mean as the minimizers of respectively the L ₁- and the L ₂-norm of y ₂. In doing so, we obtain four alternatives to describe the center of a multivariate data set. While three of them have been already investigated in the statistical literature, the L ₁-mean appears to be a new concept. Received January 1999     

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.     

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.     

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].     

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.     

Tests of significance for the average square deviation from target in a finite lot

Consider lots of discrete items 1, 2, …,N with quality characteristicsx ₁,x ₂, …,x _N . Leta be a target value for item quality. Lot quality is identified with the average square deviation from target per item in the lot (lot average square deviation from target). Under economic considerations this is an appropriate lot quality indicator if the loss respectively the profit incurred from an item is a quadratic function ofx _i −a. The present paper investigates tests of significance on the lot average square deviationz under the following assumptions: The lot is a subsequence of a process of production, storage, transport; the random quality characteristics of items resulting from this process are i.i.d. with normal distributionN(μ, σ ²); the target valuea coincides with the process meanμ.     

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.     

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).     

Characterizations of distributions via linearity of regression of generalized order statistics

Let X (r, n, m, k), 1 r n, denote generalized order statistics based on an absolutely continuous distribution function F. We characterize all distribution functions F for which the following linearity of regression holds E(X(r+l,n,m,k) | X(r,n,m,k))=aX(r,n,m,k)+b.We show that only exponential, Pareto and power distributions satisfy this equation. Using this result one can obtain characterizations of exponential, Pareto and power distributions in terms of sequential order statistics, Pfeifers records and progressive type II censored order statistics. Received July 2001/Revised August 2002     

One-Sample Bayesian Predictive Interval of Future Ordered Observations for the Pareto Distribution

Nigm et al. (2003, statistics 37: 527–536) proposed Bayesian method to obtain predictive interval of future ordered observation Y _(j) (r < j≤ n ) based on the right type II censored samples Y ₍₁₎ < Y ₍₂₎ < ... < Y _(r) from the Pareto distribution. If some of Y ₍₁₎ < ... < Y _(r-1) are missing or false due to artificial negligence of typist or recorder, then Nigm et al.’s method may not be an appropriate choice. Moreover, the conditional probability density function (p.d.f.) of the ordered observation Y _(j) (r < j ≤ n ) given Y ₍₁₎ <Y ₍₂₎ < ... < Y _(r) is equivalent to the conditional p.d.f. of Y _(j) (r < j ≤ n ) given Y _(r). Therefore, we propose another Bayesian method to obtain predictive interval of future ordered observations based on the only ordered observation Y _(r), then compares the length of the predictive intervals when using the method of Nigm et al. (2003, statistics 37: 527–536) and our proposed method. Numerical examples are provided to illustrate these results.