Concepts de dépendance et ordres stochastiques pour des lois bidimensionnelles

Yanagimoto and Okamoto (1969) introduced a stochastic ordering that generalizes a concept of monotone regression dependence introduced by Lehmann (1966). In this paper, we define and examine the properties of three new orderings which imply that of Yanagimoto and Okamoto. One of these orderings is seen to extend Shaked's (1977) notion of DTP(0, 1), and another includes Lehmann's concept of positive likelihood-ratio dependence as a special case. The proposed orderings are also compared with the TP₂ positive-dependence ordering defined by Kimeldorf and Sampson (1987).     

Extremes of Determinants and Optimality of Canonical Variables

Extremes of quadratic forms have been presented by several authors (Okamoto, 1969; Rao, 1973; Seber, 1984). The obvious multivariate extension of the extreme of quadratic forms is the extreme of the determinants as well as the ratios of the determinants. In this paper we develop some supremums of the determinants and the ratios of the determinants. A new optimality and equations of canonical variables are obtained.     

On approximating the distribution of indefinite quadratic forms

This paper provides a simple methodology for approximating the distribution of indefinite quadratic forms in normal random variables. It is shown that the density function of a positive definite quadratic form can be approximated in terms of the product of a gamma density function and a polynomial. An extension which makes use of a generalized gamma density function is also considered. Such representations are based on the moments of a quadratic form, which can be determined from its cumulants by means of a recursive formula. After expressing an indefinite quadratic form as the difference of two positive definite quadratic forms, one can obtain an approximation to its density function by means of the transformation of variable technique. An explicit representation of the resulting density approximant is given in terms of a degenerate hypergeometric function. An easily implementable algorithm is provided. The proposed approximants produce very accurate percentiles over the entire range of the distribution. Several numerical examples illustrate the results. In particular, the methodology is applied to the Durbin–Watson statistic which is expressible as the ratio of two quadratic forms in normal random variables. Quadratic forms being ubiquitous in statistics, the approximating technique introduced herewith has numerous potential applications. Some relevant computational considerations are also discussed.     

Some New Bounds on the Probability of Extinction of a Galton–Watson Process with Numerical Comparisons

Some new upper and lower bounds for the extinction probability of a Galton–Watson process are presented. They are very easy to compute and can be used even if the offspring distribution has infinite variance. These new bounds are numerically compared to previously discussed bounds. Some definite guidelines are given concerning when these new bounds are preferable. Some open problems are also discussed.     

A Note on the Upper Bound to Variance of the Sample Extreme from a Finite Population

We construct a new upper bound for the variance of the sample minimum and maximum in the case of a simple random sample drawn without replacement. The bound is optimal in the form provided. Similar bounds are shown for the other order statistics.     

Comment

Bonferroni inequalities often provide tight upper and lower bounds for the probability of a union of events. The bounds are especially useful when this probability is difficult to compute exactly. There are situations, however, in which the Bonferroni approach gives very poor results. An example is given in which the upper and lower Bonferroni bounds are far from the probability they seek to approximate and successive bounds do not converge. Even an improved first upper Bonferroni bound may not be close to the probability of the union of events.     

Some further aspects of the löwner-ordering antitonicity of the moore-penrose inverse

A survey is given of known proofs of the antitonicity of the inverse matrix function for positive definite matrices w.r.t. the Lowner partial ordering, and of the corresponding result for the Moore-Penrose inverse of nonnegative definite matrices [the theorem of Milliken and Akdeniz (1977)]. A short new proof of the latter result is obtained by employing an extremal representation of a nonnegative definite quadratic form. Another proof of this result involving Schur complements is also given, and is seen to be extendable to the case of symmetric (not necessarily nonnegative definite) matrices. A geometrical interpretation of Milliken and Akdeniz's theorem is presented. As an application, the relationship between the concepts of greater (maximum) concentration and smaller (minimum) dispersion is considered for a pair (class) of vector-valued statistics with possibly degenerate distributions.     

Probability inequalities for continuous unimodal random variables with finite support

A paramecer-free Bernstein-type upper bound is derived for the probability that the sum S of n i.i.d, unimodal random variables with finite support, X₁ ,X₂,…,X_n, exceeds its mean E(S) by the positive value nt. The bound for P{S - nμ ≥ nt} depends on the range of the summands, the sample size n, the positive number t, and the type of unimodality assumed for X_i. A two-sided Gauss-type probability inequality for sums of strongly unimodal random variables is also given. The new bounds are contrasted to Hoeffding's inequality for bounded random variables and to the Bienayme-Chebyshev inequality. Finally, the new inequalities are applied to a classic probability inequality example first published by Savage (1961).     

Bounds on expected generalized order statistics

We establish the upper nonpositive and all the lower bounds on the expectations of generalized order statistics based on a given distribution function with the finite mean and central absolute moment of a fixed order. We also describe the distributions for which the bounds are attained. The methods of deriving the lower nonpositive (upper nonnegative) and lower nonnegative (upper nonpositive) bounds are totally different. The first one, the greatest convex minorant method is the combination of the Moriguti and well-known Hölder inequalities and the latter one is based on the maximization of some norm on the properly chosen convex set. The paper completes the results of Cramer et al. [Evaluations of expected generalized order statistics in various scale units. Appl Math. 2002;29:285–295].     

Upper non positive bounds on expectations of generalized order statistics from DD and DDA populations

We present the upper non positive bounds on the expectations of gOSs centered about the sample mean, which are based on the parent distributions with decreasing density and decreasing density on average distributions. Such bounds can be obtained only for particular cases of gOSs and they are expressed in units generated by the central absolute moments of a fixed order. The attainability conditions are also described. The method of deriving presented bounds is based on the maximization of appropriate norms over properly chosen convex sets. The paper complements the results of Bieniek [J. Statist. Plann. Inference, 2008; 138:971–981].     

Rates of convergence of autocorrelation estimates for periodically correlated autoregressive Hilbertian processes

Autoregressive Hilbertian (ARH) processes are of great importance in the analysis of functional time series data and estimation of the autocorrelation operators attracts the attention of various researchers. In this paper, we study estimators of the autocorrelation operators of periodically correlated autoregressive Hilbertian processes of order one (PCARH(1)), which is an extension of ARH(1) processes. The estimation method is based on the spectral decomposition of the covariance operator and considers two main cases: known and unknown eigenvectors. We show the consistency in the mean integrated quadratic sense of the estimators of the autocorrelation operators and present upper bounds for the corresponding rates.     

Cohpasisons of improved bonferroni and sidak/slepian bounds with applications to normal markov processes

The recent literature contains theorems improving on both the standard Bonferroni inequality (Hoover (1990)) and the Sidak/Slepian inequalities (Glaz and Johnson (1984)), The application of these improved theorems to upper bounds for non coverage of simultaneous confidence intervals on multivariate normal variables is explored. The improved Bonferroni upper bounds always hold, while improved Sidak/Slepian bounds only apply to special cases. It is shown that improved Sidak/Slepian bounds will always hold for Normal Markov Processes, a commonly occuring and easily identifiable class of multivariate normal variables. The improved Sidak/Slepian upper bound, if it applies, is proven to be superior to the computationally equivalent improved Bonferroni bound. This improvement, however, is not great when both methods are used to determine upper bounds for Type I error in the range of .01 to .10.     

Inequalities for bivariate distribution with x ≤ y and marginals given

Upper and lower bounds on the joint bivariate distribution are found when the marginals are given and under the additional condition that X ≤ Y with probability one. The upper bound is the same as for the unrestricted bivariate distribution with marginals given, For the lower bound a simple inequality is derived which is exact, that is, achievable, in many cases including normal and exponential marginals.     

CONVERGENT SERIES EXPRESSIONS FOR INVERSE MOMENTS OF QUADRATIC FORMS IN NORMAL VARIABLES

Using relatively recent results from multivariate distribution theory, a direct approach to evaluating the inverse moments of a quadratic form in normal variables is proposed. Convergent infinite series expressions involving the invariant polynomials of matrix argument are obtained. The solution also depends upon a positive scalar which is arbitrarily chosen. For the solution to converge an upper bound upon this scalar is derived.     

Some bounds related to Harris family of distributions

As a lifetime distribution, Harris family of distributions are applied to the lifetime of a series system with random number of components. In this paper, properties of various ageing classes of mixtures of Harris family of distributions, where the tilt parameter of a Harris distribution is taken as a random variable, are studied. We obtain an upper bound for maximum error in evaluating its reliability function. Two bounds are also presented for survival function and expectation of the mixed Harris family. We also provide some interesting bounds for its residual survival function. Our results generalize several previous findings in this connection. Some illustrative examples are also provided.     

An upper bound for binary constant weight codes

An upper bound for binary constant weight codes is presented, and several examples which achieve this upper bound are given. Based on this upper bound, some properties of equidistant constant weight codes are obtained. A new proof for Grey–Rankin bound in coding theory is also presented in this paper.     

A note on bounding monte carlo variances michael lavine

A density bounded class P of probability distributions on a space χ is the set of all probability distributions corresponding to probability densities bounded below by a given subprob-ability density and bounded above by a given superprobability density. Density bounded classes arise in robust Bayesian analysis (Lavine 1991) and also in Monte Carlo integration (Fishman Granovsky and Rubin 1989). Finding upper and lower bounds on the variance over all p? P allows one to bound the Monte Carlo variance. Fishman Granovsky and Rubin (1989) find bounds on the variance over all p ? P and also find the densities in P achieving those bounds in the case where χ is discrete; that is, where P is actually a set of probability mass functions. This article generalizes their result by showing how to bound the variance and find the densities achieving the bounds when χ is continuous.     

Lower bounds on expectations of positive L-statistics from without-replacement models

We consider samples drawn without replacement from finite populations. We establish optimal lower non-negative and upper non-positive bounds on the expectations of linear combinations of order statistics centered about the population mean in units generated by the population central absolute moments of various orders. We also specify the general results for important examples of sample extremes, Gini mean differences and sample range. The paper completes the results of Papadatos and Rychlik [2004. Bounds on expectations of L-statistics from without replacement samples. J. Statist. Plann. Inference 124, 317–336], where sharp negative lower and positive upper bounds on the expectations of the combinations were presented for the without-replacement samples.     

The minimum of the expected value of the product of three random variables in the Fréchet class

Summary In this paper the minimum of the expected value of the product of three random variables is studied as their joint distribution function varies in the Fréchet class associated to the three given marginal distribution functions. The general problem is studied for three positive valued random variables and a lower bound for the minimum is provided. The case of three uniformly distributed random variables in [0, 1] is analyzed in more detail and an upper bound for the minimum is given. The Author conjectures that the distribution correspondent to the upper bound is a solution of the problem. Paper written with the contribution of MURST (funds 40%).