Minimax estimation of a cumulative distribution function by converting to a parametric problem

Let X = (X ₁,...,X _n ) be a sample from an unknown cumulative distribution function F defined on the real line . The problem of estimating the cumulative distribution function F is considered using a decision theoretic approach. No assumptions are imposed on the unknown function F. A general method of finding a minimax estimator d(t;X) of F under the loss function of a general form is presented. The method of solution is based on converting the nonparametric problem of searching for minimax estimators of a distribution function to the parametric problem of searching for minimax estimators of the probability of success for a binomial distribution. The solution uses also the completeness property of the class of monotone decision procedures in a monotone decision problem. Some special cases of the underlying problem are considered in the situation when the loss function in the nonparametric problem is defined by a weighted squared, LINEX or a weighted absolute error.     

The internal rate of return of fuzzy cash flow

An internal rate of return (IRR) of an investment of financing project with cash flow (a₀, a₁, a₂,..., a_n) is usually defined as a rate of interest such that $$a_0 + a_{\text{1}} {\text{(1}} + r{\text{)}}^{ - 1} + ... + a_n (1 + r)^{ - n} = 0$$ . If the cash flow has one sign change then the previous equation has a unique solution τ>?1. Generally the IRR does not extend to fuzzy cash flows, as it can be seen with examples (see [2]). In this paper we show that under suitable hypotheses a unique fuzzy IRR exists for a fuzzy cash flow.     

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 .

     

Distribution of run statistics in partially exchangeable processes

Let { X_i} _{i 3 1}{\{ X_{i}\} _{i\geq 1}} be an infinite sequence of recurrent partially exchangeable binary random variables. We study the exact distributions of two run statistics (total number of success runs and the longest success run) in { X_i} _{i 3 1}{\{ X_{i}\} _{i\geq1}} . Since a flexible class of models for binary sequences can be obtained using the concept of partial exchangeability, as a special case of our results one can obtain the distribution of runs in ordinary Markov chains, exchangeable and independent sequences. The results also enable us to study the distribution of runs in particular urn models.     

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).     

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”.     

Funktionen homogener Markoffscher Ketten als homogene bzw. inhomogene Markoffsche Ketten

Summary In this paper is considered a functionf of a finite-state, homogeneous Markov chain {X _n, n=1,2,...}. Sufficient and necessary conditions are derived that {X _n). n=1,2,...} is a homogeneous or inhomogeneous Markov chain. Furthermore the case is considered that {f (X _n), n=1,2,...} is Markovian whatever the initial dist     

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     

On the power of nonparametric changepoint-tests

We consider a sequenceX _1n,..., X_nn, n N, of independent random elements. Suppose there exists a [0, 1) such thatX _1n,...,X _(n),n have the distribution v₁ andX _[n]+1.n ,...,X _nn have the distribution v₂v₁. We construct consistent level- tests forH ₀:=0 versusH ₁:(0, 1), which are based on certainU-statistic type processes. A detailed investigation of the power function is also provided.     

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

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 .     

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.     

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}.     

Regression analysis and dependence

In this paper we study the relationship between regression analysis and a multivariate dependency measure. If the general regression model Y=f() holds for some function f, where 1i₁< i₂<···i_m k, and X₁,...,X_k is a set of possible explanatory random variables for Y. Then there exists a dependency relation between the random variable Y and the random vector (). Using the dependency statistic defined below, we can detect such dependency even if the function f is not linear. We present several examples with real and simulated data to illustrate this assertion. We also present a way to select the appropriate subset among the random variables X₁,X₂,...,X_k, which better explain Y.     

Contributions to nonparametric generalized failure rate function estimation

For a specified distribution functionG with densityg, and unknown distribution functionF with densityf, the generalized failure rate function (x)=f(x)/gG ^–1 F(x) may be estimated by replacingf andF byf _n and , wheref _n is an empirical density function based on a sample of sizen from the distribution functionF, and . Under regularity conditions we show and, under additional restrictions whereC is a subset ofR and _n. Moreover, asymptotic normality is derived and the Berry-Esséen type bound is shown to be related to a theorem which concerns the sum of i.i.d. random variables. The order boundO(n^–1/2+c _n ^1/2 ) is established under mild conditions, wherec _n is a sequence of positive constants related tof _n and tending to 0 asn.Research was supported in part by the Army, Navy and Air Force under Office of Naval Research contract No. N00014-76-C-0608. AMS 1970 subject classifications. Primary 62G05. Secondary 60F15.     

Rangvariable bei Vorliegen von Bindungen

Summary The family of distributions of the random sampleX ₁,...,X _N not being specified solution of different statistical problems is obtained by methods based on ranks, because these methods are distribution-free under the assumption ofQ:=P(X _i =X _j )=0. Frequently, in case ofQ>0, the use of midranks is proposed, though only few is known about their distribution depending on the (usually unknown) distribution of the random sample. Here general formulas of variance and covariance of midranks are derived, after the importance of the termQ determining variance and covariance has been briefly pointed out.     

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.