Reverse submartingale property arising from exchangeable random variables

In this work, for an exchangeable sequence of random variables {X_i, i̿}, and two nondecreasing sequences of positive integers {h_n, ǹ} and {k_n, ǹ}, where h_n+k_nhn, Qǹ, we prove that {R_n,hn,kn/n, ǹ} forms a reverse submartingale sequence, where R_{n,h_n,k_n}={\displaystyle {1\over k_n}} ~^{k_n-1}_{j=0} X_{n-j,n}-{\displaystyle {1\over h_n}} ~^{h_n}_{j=1} X_{j,n}$R_{n,h_n,k_n}={\displaystyle {1\over k_n}} ~^{k_n-1}_{j=0} X_{n-j,n}-{\displaystyle {1\over h_n}} ~^{h_n}_{j=1} X_{j,n}, and X_1,nhX_2,nh…hX_n,n are the order statistics based on {X₁,…,X_n}.     

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.

Rate of convergence in the central limit theorem and in the strong law of large numbers for von mises statistics

This paper provides the rate of convergence in the central limit theorem and in the strong law of large numbers forvon Mises statistics , based on i.i.d. random variablesX ₁ ,..., X _N .The proofs rely on a decomposition ofvon Mises statistics into a linear combination ofU-statistics and then use (generalized) results on the convergence rates forU-statistics obtained byGrams/Serfling [1973] andCallaert/Janssen [1978].     

On characterizations of distributions by regression of adjacent generalized order statistics

Let , r ≥ 1, denote generalized order statistics, with arbitrary parameters , based on distribution function F. In this paper we characterize continuous distributions F by the regression of adjacent generalized order statistics, i.e. where are continuous and increasing functions and ψ is strictly increasing. Further we investigate in detail the case when ψ(x) = x and g is a linear function of the form g(x) = cx + d for some .     

Runs in an ordered sequence of random variables

Let be independent and identically distributed random variables with continuous distribution function. Denote by the corresponding order statistics. In the present paper, the concept of -neighbourhood runs, which is an extension of the usual run concept to the continuous case, is developed for the sequence of ordered random variables      

A remark on robustness of linear best estimates

Let s₀ = ||x- h₀ (x)||₀ , s(B₁ ) = ||x- h₁ (x)||₀ , s₁ (B₁ ) = ||x- h₁ (x)||₁ , \begin{gathered} \sigma _0 = \parallel \xi - \eta _0 (\xi )\parallel _0 , \hfill \\ \sigma (B_1 ) = \parallel \xi - \eta _1 (\xi )\parallel _0 , \hfill \\ \sigma _1 (B_1 ) = \parallel \xi - \eta _1 (\xi )\parallel _1 , \hfill \\ \end{gathered}     

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

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

Multiple comparisons with the best,with economic applications

In this paper we discuss a statistical method called multiple comparisons with the best, or MCB. Suppose that we have N populations, and population i has parameter value θ_i. Let $\theta _{(N)}={\rm max}_{i=1,\ldots ,N}\theta _{i}$\nopagenumbers\end , the parameter value for the ‘best’ population. Then MCB constructs joint confidence intervals for the differences $[\theta _{(N)}‐\theta _{1},\theta _{(N)}‐\theta _{2},\ldots ,\theta _{(N)}‐\theta _{N}]$\nopagenumbers\end . It is not assumed that it is known which population is best, and part of the problem is to say whether any population is so identified, at the given confidence level. This paper is meant to introduce MCB to economists. We discuss possible uses of MCB in economics. The application that we treat in most detail is the construction of confidence intervals for inefficiency measures from stochastic frontier models with panel data. We also consider an application to the analysis of labour market wage gaps. Copyright © 2000 John Wiley & Sons, Ltd.     

A note on minimum variance

Summary Minimizing is discussed under the unbiasedness condition: and the condition (A):f _i (x) (i=1, ..., p) are linearly independent , and .     

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.     

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.     

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”.     

Über die Anzahl von Zufallspunkten mit typ-gleichem nächsten Nachbarn und einen multivariaten Zwei-Stichproben-Test

Summary For independents-variate samplesX ₁, ...,X _m i.i.d.f. (.),Y ₁, ...,Y _n i.i.d. g. (.), where the densitiesf (.),g (.) are assumed to be continuous on their respective sets of positivity, consider the numberT _m,n of pointsZ of the pooled sample (which are either of typeX or of typeY) such that the nearest neighbor ofZ is of the same type asZ. We show that, as , independently of (.). An omnibus test for the two sample problem f(.)g(.) orf(.)g(.)? may be obtained by rejecting the hypothesisf(.)g(.) for large values ofT _m,n.     

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.     

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.