Minimax estimators andΓ-minimax estimators for a bounded normal mean under the lossl p (θ, d)=|θ-d|p

Summary Let the random variableX be normal distributed with known varianceσ ²>0. It is supposed that the unknown meanθ is an element of a bounded intervalΘ. The problem of estimatingθ under the loss functionl _p (θ, d)=|θ-d| ^p p≥2 is considered. In case the length of the intervalθ is sufficiently small the minimax estimator and theΓ(β, τ)-minimax estimator, whereΓ(β, τ) represents special vague prior information, are given.     

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     

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.     

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 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).     

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

Suffizienz bei verschiebungsinvarianten Schätzproblemen

Zusammenfassung Es wird gezeigt, daß beim Schätzen eines die Verteilung einer ZufallsgrößeX (mit Dichte) charakterisierenden Lageparameters verschiebungsinvariante FunktionenZ ₁=a ₁(X ₁,...,X _n ),...,Z _m =a _m (X ₁,...,X _n ) dern unabhängigen WiederholungenX ₁,...,X _n vonX genau dann suffizient sind, wenn für jede konvexe Schadensfunktion ein gleichmäßig bestes, nur vonZ ₁,...,Z _m abhängendes verschiebungsinvariantes Schätzverfahren existiert. Weiter wird bewiesen, daßX genau dann normalverteilt ist, wenn zu jeder konvexen Schadensfunktion ein existiert derart, daß ein gleichmäßig bestes verschiebungsinvariantes Schätzverfahren ist.

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Summary LetX ₁,...,X _n be independent random variables with density functionf(x–) and unknown location parameter R ¹; furthermore leta _i (x ₁,...,x _n ),i=1,..., m, be functions which are invariant with respect to translations. ThenZ _i =a _i (X ₁,...,X _n ),i=1,...,m, are sufficient iff for every convex loss functions (.) there exists a functionh(z ₁,...,z _m ) such thath(Z ₁,...,Z _m ) is a best invariant estimate for the location parameter . Furthermore we show thatX ₁,...,X _n is a sample from a normal distribution if for every convex loss functions (.) there exists a constant such that is a best invariant estimate for .

     

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.     

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.     

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”.     

Inference ony>y o by using a correlated variable: A three decision approach

Summary In an extension of the two decision approach [Bauer, Scheiber andWohlzogen, 1975] a Bayes solution is aimed at for the three decisiony>y _o,yy _o or no classification on the basic of the measurement of a positively correlated random variableX, which can be measured more easily and/or with smaller expense. Assuming a bivariate normal distribution forX andY optimal decision regions for the measuredx are derived in the case of constant or exponentially increasing losses.

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Zusammenfassung In Erweiterung des Zwei-Entscheidungsproblems [Bauer, Scheiber undWohlzogen, 1975] wird eine Bayes-Lösung für die drei Entscheidungeny>y ₀,yy ₀ oder keine Zuordnung aufgrund der Messung einer mitY positiv korrelierten, einfacher und/oder billiger zugänglichen ZufallsvariablenX angestrebt. Optimale Entscheidungsbereiche für die Messungenx werden bei Voraussetzung einer bivariaten Normalverteilung fürX undY unter der Annahme konstanter oder exponentiell wachsender Verluste bestimmt.

     

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.     

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.     

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.     

A method to obtain new copulas from a given one

Given a strictly increasing continuous function φ from [0, 1] to [0, 1] and its pseudo-inverse φ^[−1], conditions that φ must satisfy for C_φ(x₁, . . . ,x_n)=φ^[−1](C(φ(x₁), . . . ,φ(x_n))) to be a copula for any copula C are studied. Some basic properties of the copulas obtained in this way are analyzed and several examples of generator functions φ that can be used to construct copulas C_φ are presented. In this manner, a method to obtain from a given copula C a variety of new copulas is provided. This method generalizes that used to construct Archimedean copulas in which the original copula C is the product copula, and it is related with mixtures     

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.      

On the upper and lower semicontinuity of the Aumann integral

Let (T,τ,μ) be a finite measure space, X be a Banach space, P be a metric space and let L₁(μ,X) denote the space of equivalence classes of X-valued Bochner integrable functions on (T,τ,μ). We show that if φ:T×P→2^X is a set-valued function such that for each fixed pεP, φ(·,p) has a measurable graph and for each fixed tεT, φ(t,·) is either upper or lower semicontinuous then the Aumann integral of φ, i.e.,∫_Tφ(t,p)dμ(t)= {∫_Tx(t)dμ(t):xεS_φ(p)}, where S_φ(p)= {yεL₁(μ,X):y(t)εφ(t,p)μ−a.e.}, is either upper or lower semicontinuous in the variable p as well. Our results generalize those of Aumann (1965, 1976) who has considered the above problem for X=Rⁿ, and they have useful applications in general equilibrium and game theory.