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.     

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).     

Linearity of regression for non-adjacent order statistics

Let X ₁,X ₂,…,X _n be a random sample from a continuous distribution with the corresponding order statistics X _1:n≤X _2:n≤…≤X _n:n. All the distributions for which E(X _{k+r: n}|X _k:n)=a X _k:n+b are identified, which solves the problem stated in Ferguson (1967). Received February 1998     

On characterizing discrete distributions via conditional expectations of weak record values

Let (W _n ,n ≥ 0) denote the sequence of weak records from a distribution with support S = { α₀,α₁,...,α _N }. In this paper, we consider regression functions of the form ψ _n (x) = E(h(W _n ) |W _n+1 = x), where h(·) is some strictly increasing function. We show that a single function ψ _n (·) determines F uniquely up to F(α₀). Then we derive an inversion formula which enables us to obtain F from knowledge of ψ _n (·), ψ _n-1(·), h(·) and F(α₀).     

r-te Momente von Punktprozessen

Zusammenfassung {X (t): tR ⁺} sei ein Punktprozeß,H (x) eine konvexe nicht-negative Funktion. Mit Hilfe der bedingten Wahrscheinlichkeitenp _n (t) für genaun Ereignisse (Punkte) im Zeitpunkt punktt unter der Bedingung, daß im Zeitpunktt mindestens ein Ereignis eintritt, wird eine Beziehung formuliert, die für die Existenz des ErwartungswertesE (H (X (t ₀))) notwendig ist. Hat der Punktprozeß unabhängige Zuwächse, und erfüllt die FunktionH (x) einige weitere Bedingungen, so ist die angegebene Beziehung auch hinreichend für die Existenz dieses Erwartungswertes. Für Punktprozesse mit unabhängigen Zuwächsen ergibt sich als unmittelbare Anwendung dieser Aussagen eine notwendige und hinreichende Bedingung für die Existenz vonE X (t ₀) ^r für reellesr1.

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Summary Let {X (t): tR ⁺} be a point process andH (x) a convex non-negative function. Using the conditional probabilitiesp _n (t) thatn events (points) occur at timet given that at least one event occurs att a condition is formulated which is necessary for the existence ofE (H (X (t ₀))). This condition is sufficient, too, if the point process has independent increments and the functionH (x) fulfils some further conditions. Using these statements one gets a necessary and sufficient condition for the existence ofE X (t ₀) ^r for realr1.

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Herrn ProfessorWeissinger zum 65. Geburtstag am 12. Mai 1978 gewidmet     

Slow convergence of the number of near-maxima for Burr XII distributions

A near-maximum is an observation which falls within a distance a of the maximum observation in an independent and identically distributed sample of size n. Subject to some conditions on the tail thickness of the population distribution, the number K _n (a) of near-maxima is known to converge in probability to one or infinity, or in distribution to a shifted geometric law. In this paper we show that for all Burr XII distributions K _n (a) converges almost surely to unity, but this convergence property may not become clear under certain cases even for very large n. We explore the reason of such slow convergence by studying a distributional continuity between Burr XII and Weibull distributions. We have also given a theoretical explanation of slow convergence of K _n (a) for the Burr XII distributions by showing that the rate of convergence in terms of P{K _n (a) > 1} tending to zero changes very little with the sample size n. Illustrations of the limiting behaviour K _n (a) for the Burr XII and the Weibull distributions are given by simulations and real data. The study also raises an important issue that although the Burr XII provides overall better fit to a given data set than the Weibull distribution, cautions should be taken for the extrapolation of the upper tail behaviour in the case of slow convergence.      

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.     

Convergence of a branching type recursion with non-stationary immigration

The asymptotic distribution of a branching type recursion with non-stationary immigration is investigated. The recursion is given by , where (X _l ) is a random sequence, (L _n ⁻¹⁽¹⁾ ) are iid copies ofL _n−1,K is a random number andK, (L _n ⁻¹⁽¹⁾ ), {(X _l ),Y _n } are independent. This recursion has been studied intensively in the literature in the case thatX _l ≥0,K is nonrandom andY _n =0 ∀n. Cramer, Rüschendorf (1996b) treat the above recursion without immigration with starting conditionL ₀=1, and easy to handle cases of the recursion with stationary immigration (i.e. the distribution ofY _n does not depend on the timen). In this paper a general limit theorem will be deduced under natural conditions including square-integrability of the involved random variables. The treatment of the recursion is based on a contraction method. The conditions of the limit theorem are built upon the knowledge of the first two moments ofL _n . In case of stationary immigration a detailed analysis of the first two moments ofL _n leads one to consider 15 different cases. These cases are illustrated graphically and provide a straight forward means to check the conditions and to determine the operator whose unique fixed point is the limit distribution of the normalizedL _n .     

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.     

Multivariate gamma distributions with one-factorial accompanying correlation matrices and applications to the distribution of the multivariate range

Summary A new representation for the characteristic function of the joint distribution of the Mahalanobis distances betweenk independentN(μ, Σ)-distributed points is given. Especially fork=3 the corresponding distribution function is obtained as a special case of multivariate gamma distributions whose accompanying normal distribution has a positive semidefinite correlation matrix with correlationsϱ _ij=−a _i a _j. These gamma distribution functions are given here by one-dimensional parameter integrals. With some further trivariate gamma distributions third order Bonferroni inequalities are derived for the upper tails of the distribution function of the multivariate range ofk independentN(μ, I)-distributed points. From these inequalities very accurate (conservative) approximations to upperα-level bounds can also be computed for studentized multivariate ranges.     

Increasing failure rate and decreasing reversed hazard rate properties of the minimum and maximum of multivariate distributions with log-concave densities

For a multivariate random vector X = (X ₁,...,X _n ) with a log-concave density function, it is shown that the minimum min{X ₁,...,X _n } has an increasing failure rate, and the maximum max{X ₁,...,X _n } has a decreasing reversed hazard rate. As an immediate consequence, the result of Gupta and Gupta (in Metrika 53:39–49, 2001) on the multivariate normal distribution is obtained. One error in Gupta and Gupta method is also pointed out.      

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).     

Vergleich zweier Operationscharakteristiken

Summary We compare the OC-curvesL _n.c (p) (1) andL _n.c ^* (p) (2). The first is founded on the binomial distribution, the latter relates to the Poisson distribution and is often used as approximation. These OC-curves occur in Statistical Quality Control as probabilities for the acception of a lot as approximations for such probabilities; they are regarded as functions of the fraction defectivep. It is shown that the two OC-curves have exactly one intersection point between 0 and 1, if the acceptance numberc is 1 and the sample sizen is >c+1.Forp between 0 and the intersection pointp _s we have thenL _n.c.(p)>L _n.c ^* (p); from p _s <p1 followsL _n.c(p)n.c ^* (p).An interval is given which coversp _s and with an example it is shown how one might use the results of this paper for the construction of sampling plans.     

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 note on strategies for bandit problems with infinitely many arms

A bandit problem consisting of a sequence of n choices (n) from a number of infinitely many Bernoulli arms is considered. The parameters of Bernoulli arms are independent and identically distributed random variables from a common distribution F on the interval [0,1] and F is continuous with F(0)=0 and F(1)=1. The goal is to investigate the asymptotic expected failure rates of k-failure strategies, and obtain a lower bound for the expected failure proportion over all strategies presented in Berry et al. (1997). We show that the asymptotic expected failure rates of k-failure strategies when 0<b1 and a lower bound can be evaluated if the limit of the ratio F(1)–F(t) versus (1–t)^b exists as t1^– for some b>0.     

Evaluation of P(Xt>Yt ) when both Xt and Yt are residual lifetimes of two systems

Let the random variables X and Y denote the lifetimes of two systems. In reliability theory to compare between the lifetimes of X and Y there are several approaches. Among the most popular methods of comparing the lifetimes are to compare the survival functions, the failure rates and the mean residual lifetime functions of X and Y. Assume that both systems are operating at time t > 0. Then the residual lifetimes of them are X_t=X?t | X>t and Y_t=Y?t | Y>t, respectively. In this paper, we introduce, by taking into account the age of systems, a time‐dependent criterion to compare the residual lifetimes of them. In other words, we concentrate on function R(t ):=P(X_t>Y_t) which enables one to obtain, at time t, the probability that the residual lifetime X_t is greater than the residual lifetime Y_t. It is mentioned, in Brown and Rutemiller (IEEE Transactions on Reliability, 22 , 1973) that the probability of type R(t) is important for designing as long‐lived a product as possible. Several properties of R(t) and its connection with well‐known reliability measures are investigated. The estimation of R(t) based on samples from X and Y is also discussed.     

Erneuerungsprozesse mit Wartungsstrategien

Summary The following renewal process is considrred: given intervals (kt ₁,(k+1)t ₁],k=0, 1, 2, ..., 0<t ₁<, there will be with probabilityp, 0p1, a renewal in each interval at a time selected by random. The costs for each of this renewals are a units, while the costs of the other renewals areb units each. The renewal function and the cost function are derivided and their asymptotic behavior is discussed.     

The problem of optimum stratification and allocation withq=n/N>0

Summary Dalenius/Gurney [1951] published necessary conditions for the stratum boundaries, so that with Neyman's optimal allocation of the sample sizen the variance of the sample mean will become a minimum. They introduced in the variance of the sample mean for the sample sizesn _h the opti mal values according to Neyman and differentiated this variance with respect to the stratum boundaries. Because Neyman's allocation formula yields only feasible solutions forn _h N _h , the conditions ofDalenius result in wrong, i.e. nonfeasible solutions, if one of the restrictionsn _h N _h (h=1 (1) L) is violated.By the example of a logarithmic normal distribution with =0, =1,5 forL=2 the behaviour of the Dalenius-Neyman-minimum and that of the feasible minimum will be shown in dependence on the sampling fractionq=n/N and a critical valueq _c will be given. For valuesq>q _c the Dalenius-Neyman-minimum is no longer feasible.For the same logarithmic normal distribution andL=2 (1) 10 this critical sampling fractionq _c will be given (section 5).For different values of andq the optimal stratum boundaries and sampling fractions are listed in section 6 forL=2;3;4.