Generalized gauss-chebyshev inequalities for unimodal distributions

Letg be an even function on ℝ which is nondecreasing in |x|. Letk be a positive constant. Sharp inequalities relatingP(|X|≥k) toEg(X) are obtained for random variablesX which are unimodal with mode 0, and for random variablesX which are unimodal with unspecified mode. The bounds in the mode 0 case generalize an inequality due to Gauss (1823), whereg(x)=x ². The bounds in the second case generalize inequalities of Vysochanskiĭ and Petunin (1980, 1983) and Dharmadhikari and Joag-dev (1985).     

Evaluating expectations of <Emphasis Type="Italic">L</Emphasis>-statistics by the Steffensen inequality

By combining the Moriguti and Steffensen inequalities, we obtain sharp upper bounds for the expectations of arbitrary linear combinations of order statistics from iid samples. The bounds are expressed in terms of expectations of the left truncated parent distribution and constants that depend only on the coefficients of the linear combination. We also present analogous results for dependent id samples. The bounds are especially useful for L-estimates of the scale parameter of the distribution.     

Inequalities compared

We know that the partial means m_rof a sequence of i.i.d. standardized random variables tend to 0 with probability 1. If we want P{m_k≥εfor some k ≥r}≤δ for given positive ε and δ, how large should we take r? Several (strong) inequalities for the distribution of partial sums providing an answer to this question can be found in the literature (Hájek -Rényi Robbins , Khan ). Furthermore there exist wellknown (weak) inequalities (Bienaymé -Chebyshev , Bernstein , Okamoto ) that give us values of rfor which P{m_r≥ε}≤δ. We compare these inequalities and illustrate them with numerical results for a fixed choice ofε and δ. After a general survey and introduction in section 1, the normal and the binomial distribution are considered in more detail in the sections 2 and 3, while in section 4 it is shown that the strong inequality essentially due to Robbins can give an inferior result for particular distributions.     

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.     

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.     

Games of chance with multiple objectives

We consider games of chance between two players: Player M can win only by amassing point totals in several categories before player N scores a prescribed total number n of points. Let M have k objectives, with m _i points required in category i and probability q _i of scoring a point in that category. We resolve certain special cases: (a) For all m _i equal, the probabilities of M winning are ordered by majorization of the vectors (q ₁,...,q _k ). (b) For all q _i equal, the probabilities of M winning are ordered by majorization of the vectors (m ₁,...,m _k ). (c) For all m _i equal and all q _i equal, the probability of M winning approaches 0 as n → ∞ or as k → ∞. The results, which follow from inequalities of majorization and Schur convexity, are in accord with intuition.      

Gumbel’s Identity,Binomial Moments,and Bonferroni Sums

Gumbel’s Identity equates the Bonferroni sum with the k ‐ th binomial moment of the number of events M_n which occur, out of n arbitrary events. We provide a unified treatment of familiar probability bounds on a union of events by Bonferroni, Galambos–Rényi, Dawson–Sankoff, and Chung–Erdös, as well as less familiar bounds by Fréchet and Gumbel, all of which are expressed in terms of Bonferroni sums, by showing that all these arise as bounds in a more general setting in terms of binomial moments of a general non‐negative integer‐valued random variable. Use of Gumbel’s Identity then gives the inequalities in familiar Bonferroni sum form. This approach simplifies existing proofs. It also allows generalization of the results of Fréchet and Gumbel to give bounds on the probability that at least t of n events occur for any A further consequence of the approach is an improvement of a recent bound of Petrov which itself generalizes the Chung–Erdös bound.     

A probability inequality for ranges and its application to maximum range test procedures

Summary It is proved that for any fixed argument the sequence (P _k) of the distribution functions of the ranges ofk i.i.d. univariate random variables is log-concave if the random variables have a log-concave density. If the support of the distribution is an infinite interval and the density is monotonous then the theorem holds also with “log-convex” instead of “log-concave”. The resulting inequalities can be used by a quick algorithm for closed maximum range test procedures for all pairwise comparisons (Royen 1988, 1989a, 1989b). Under the above assumptions the application of this algorithm can be extended e.g. to pairwise comparisons of variances.     

Inference on endogenously censored regression models using conditional moment inequalities

Under a quantile restriction, randomly censored regression models can be written in terms of conditional moment inequalities. We study the identified features of these moment inequalities with respect to the regression parameters where we allow for covariate dependent censoring, endogenous censoring and endogenous regressors. These inequalities restrict the parameters to a set. We show regular point identification can be achieved under a set of interpretable sufficient conditions. We then provide a simple way to convert conditional moment inequalities into unconditional ones while preserving the informational content. Our method obviates the need for nonparametric estimation, which would require the selection of smoothing parameters and trimming procedures. Without the point identification conditions, our objective function can be used to do inference on the partially identified parameter. Maintaining the point identification conditions, we propose a quantile minimum distance estimator which converges at the parametric rate to the parameter vector of interest, and has an asymptotically normal distribution. A small scale simulation study and an application using drug relapse data demonstrate satisfactory finite sample performance.     

Scharfe untere und obere Schranken für die Wahrscheinlichkeit der Realisation vonk untern Ereignissen

Summary LetA ₁,...,A _n be events in a probability space (,A,W). We denote byL _k the event, that at leastk events among then eventsA ₁,...A _n occur, and byK _k the event, that exactlyk events occur. If only the inequalities _i W(A _i ) _i ,i=1,...,n, are known, we calculate sharp lower and upper bounds forW(L _k ) andW(K _k ). These bounds only depend onn, k and _i , _i ,i=1,...,n. They are relevant, when treating combined tests or confidence procedures.     

On approximating the distributions ofFriedman's 75-175-175-1and related statistics

Error bounds depending explicitly on parameters of the problem are given for large-sample approximations to the central and noncentral distributions ofFriedman's [1937]x _r ² statistic,Steel's [1959] statistics for comparingr treatments with a control, andNemenyi's [1963] statistics for pair-wise comparisons among all treatments. These bounds are of ordero(N ^–1/2); an improved bound of ordero (N ^–r/(r+1)) is given forFriedman's statistic in the central case. Applications yield bounds on the actual type 1 and type 2 error probabilities in terms of their normal-theory approximations.Supported in part by ONR Contract N00014-72A-0136-003 from the U.S. Office of Naval Research.     

A fixed sample size selection procedure for negative binomial populations

Summary A fixed sample size procedure for selecting the ‘best’ ofk negative binomial populations is developed. Selection is made in such a way that the probability of correct selection is at leastP* whenever the distance between the probabilities of success is at leastδ*. The exponentr is assumed to be known and the same for all populations. Extensive computer calculations* were employed to obtain the exact least favorable configuration. The smallest sample sizes needed to meet specifications (P*,δ*) are tabulated forr=1 (1)5;δ*=0.05 (0.05) 0.55 andP*=0.75, 0.80, 0.90, 0.95, 0.98, 0.99 involvingk=3 (1) 6, 8, 10 populations. All the computations were carried out on the Alabama Supercomputer. Part of this work was completed when the authors were at the Department of Statistics, Oklahoma State University, Stillwater, OK 74078.     

Upper and lower bounds on infinite time ruin probabilities in case of constraints on claim size distributions

Probabilities of ruin (or non-ruin) are solutions of differential or integro-differential equations. Solving these kinds of equations analytically and/or numerically causes a lot of mathematical difficulties. In addition there exists a practical problem of determining, estimating or guessing the distribution of the risk. A realistic way to deal with this problem consists in deriving upper and lower bounds for the ruin probability in case of incomplete information on the distribution F. The present contribution is inspired by and generalises a result of G. Taylor who uses the concept of ordering of risks to order ruin probabilities. We show how some of the results obtained by F. De Vylder for deriving sharp bounds on the stop-loss premium in case of incomplete information can be applied to the evaluation of practical bounds on infinite time ruin probabilities.     

Probability Integrals of the Multivariate t Distribution

Results on probability integrals of multivariate t distributions are reviewed. The results discussed include: Dunnett and Sobel's probability integrals, Gupta and Sobel's probability integrals, John's probability integrals, Amos and Bulgren's probability integrals, Steffens' non‐central probabilities, Dutt's probability integrals, Amos' probability integral, Fujikoshi's probability integrals, probabilities of cone, probabilities of convex polyhedra, probabilities of linear inequalities, maximum probability content, and Monte Carlo evaluation.     

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     

Simultaneous estimation of hazard rates of several exponential populations

Suppose independent random samples are drawn from k (2) populations with a common location parameter and unequal scale parameters. We consider the problem of estimating simultaneously the hazard rates of these populations. The analogues of the maximum likelihood (ML), uniformly minimum variance unbiased (UMVU) and the best scale equivariant (BSE) estimators for the one population case are improved using Rao‐Blackwellization. The improved version of the BSE estimator is shown to be the best among these estimators. Finally, a class of estimators that dominates this improved estimator is obtained using the differential inequality approach.     

Single-stage election procedures for large electorates

The Condorcet efficiency of single-stage election procedures is considered under the assumption of impartial culture for large electorates. The most efficient ranked voting rule is either Borda rule or a truncated scoring rule. A decision rule is established to determine the number of candidates, k, that individuals should be required to vote for, whether or not ranking should be required, and the scoring rule that should be used if ranking is required. This decision depends upon the number of candidates available and the probabilities that individuals will vote if they must rank k candidates or simply report k candidates.     

Probabilities of election outcomes with two parameters: The relative impact of unifying and polarizing candidates

Consider an election on three candidates for n voters with complete and transitive preference rankings on the candidates. Let k (r) denote the minimum total number of last (middle) position rankings for each of the three candidates. If k is close to zero, some candidate is seldom disliked and is a unifying candidate. If r is close to zero, some candidate is always either liked or disliked and is a polarizing candidate. A procedure is developed to obtain representations for conditional probabilities of election outcomes, when parameters like k or r are specified. Representations are obtained for the conditional probability that a pairwise majority rule winner, or PMRW, exists, given k and given r. Results show significant differences in the impact that unifying and polarizing candidates have on the probability that a PMRW exists. Received: 19 February 2004, Accepted: 29 September 2005 JEL Classification: D7 A preliminary version of this paper was presented at the Bosphorus Workshop on Economic Design and at the Abdus Salam International Center for Theoretical Physics. Comments from participants at both presentations were very helpful in developing the final version of the paper.     

Response Surface Regressions for Critical Value Bounds and Approximate p-values in Equilibrium Correction Models1

We consider the popular ‘bounds test’ for the existence of a level relationship in conditional equilibrium correction models. By estimating response surface models based on about 95 billion simulated F-statistics and 57 billion t-statistics, we improve upon and substantially extend the set of available critical values, covering the full range of possible sample sizes and lag orders, and allowing for any number of long-run forcing variables. By computing approximate P-values, we find that the bounds test can be easily oversized by more than 5 percentage points in small samples when using asymptotic critical values.     

Revelations of a gambler

Knowing that a decision maker maximizes expected utility with respect to some (unknown) utility U and some (unknown) probability P, what can one tell about P by observing his decisions? We discuss this revealed preference question primarily in the simple case of a two-element (H and T) state space, and show that the possible revelations of P_T/P_H are precisely those of the form P_T/P_Hε∪^K_k=1 (γ_k,δ_k), for some algebraic numbers γ_k,δ_k.