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.     

Robust bayesian analysis given bounds on the probability of a set

Suppose that just the lower and the upper bounds on the probability of a measurable subset K in the parameter space ω are a priori known. Instead of eliciting a unique prior probability measure, consider the class Γ of all the probability measures compatible with such bounds. Under mild regularity conditions about the likelihood function, both prior and posterior bounds on the expected value of any function of the unknown parameter ω are computed, as the prior measure varies in Γ. Such bounds are analysed according to the robust Bayesian viewpoint. Furthermore, lower and upper bounds on the Bayes factor are corisidered. Finally, the local sensitivity analysis is performed, considering the class Γ as a aeighbourhood of an elicited prior     

Bounds for the Expected Time to Extinction and the Probability of Extinction in the Galton-Watson Process

Upper bounds for the expected time to extinction in the Galton-Watson process are obtained. We also found upper and lower bounds for the probability of extinction of this process. These bounds improve some bounds previously obtained by other authors.     

Lower confidence bounds for percentiles of weibull and birnbaum-saunders distributions

Approximate lower confidence bounds on percentiles of the Weibull and the Birnbaum-Saunders distributions are investigated. Asymptotic lower confidence bounds based on Bonferroni's inequality and the Fisher information are discussed, and parametric bootstrap methods to provide better bounds are considered. Since the standard percentile bootstrap method typically does not perform well for confidence bounds on quantiles, several other bootstrap procedures are studied via extensive computer simulations. Results of the simulations indicate that the bootstrap methods generally give sharper lower bounds than the Bonferroni bounds but with coverages still near the nominal confidence level. Two illustrative examples are also presented, one for tensile strength of carbon micro-composite specimens and the other for cycles-to-failure data.     

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 estimates of multivariate Gaussian probability

ABSTRACT

Some lower and upper bounds of multivariate Gaussian probability are given based on the univariate Mills’ ratio. These bounds are sharper than known ones on the multivariate Mills’ ratio in many case.     

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.     

Multiple comparisons with a control for exponential location parameters under heteroscedasticity

In this paper, a new design-oriented two-stage two-sided simultaneous confidence intervals, for comparing several exponential populations with control population in terms of location parameters under heteroscedasticity, are proposed. If there is a prior information that the location parameter of k exponential populations are not less than the location parameter of control population, one-sided simultaneous confidence intervals provide more inferential sensitivity than two-sided simultaneous confidence intervals. But the two-sided simultaneous confidence intervals have advantages over the one-sided simultaneous confidence intervals as they provide both lower and upper bounds for the parameters of interest. The proposed design-oriented two-stage two-sided simultaneous confidence intervals provide the benefits of both the two-stage one-sided and two-sided simultaneous confidence intervals. When the additional sample at the second stage may not be available due to the experimental budget shortage or other factors in an experiment, one-stage two-sided confidence intervals are proposed, which combine the advantages of one-stage one-sided and two-sided simultaneous confidence intervals. The critical constants are obtained using the techniques given in Lam [9,10]. These critical constant are compared with the critical constants obtained by Bonferroni inequality techniques and found that critical constant obtained by Lam [9,10] are less conservative than critical constants computed from the Bonferroni inequality technique. Implementation of the proposed simultaneous confidence intervals is demonstrated by a numerical example.     

The multivariate Markov and multiple Chebyshev inequalities

Abstract

A sharp probability inequality named the multivariate Markov inequality is derived for the intersection of the survival functions for non-negative random variables as an extension of the Markov inequality for a single variable. The corresponding result in Chebyshev’s inequality is also obtained as a special case of the multivariate Markov inequality, which is called the multiple Chebyshev inequality to distinguish from the multivariate Chebyshev inequality for a quadratic form of standardized uncorrelated variables. Further, the results are extended to the inequalities for the union of the survival functions and those with lower bounds.     

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.     

Measurements of separation among probability densities or random variables

Distance between two probability densities or two random variables is a well established concept in statistics. The present paper considers generalizations of distances to separation measurements for three or more elements in a function space. Geometric intuition and examples from hypothesis testing suggest lower and upper bounds for such measurements in terms of pairwise distances; but also in L_p spaces some useful non-pairwise separation measurements always lie within these bounds. Examples of such separation measurements are the Bayes probability of correct classification among several arbitrary distributions, and the expected range among several random variables.     

BOUNDS ON THE BIAS OF WINSORIZED MEANS

This paper provides sharp upper bounds on the bias of the Winsorized mean as an estimator of the population mean in a nonparametric setting. The resulting bounds are numerically evaluated. Further, the probability distributions attaining the bounds are determined.     

Non‐parametric Estimation of the Number of Zeros in Truncated Count Distributions

We present some lower bounds for the probability of zero for the class of count distributions having a log‐convex probability generating function, which includes compound and mixed‐Poisson distributions. These lower bounds allow the construction of new non‐parametric estimators of the number of unobserved zeros, which are useful for capture‐recapture models, or in areas like epidemiology and literary style analysis. Some of these bounds also lead to the well‐known Chao's and Turing's estimators. Several examples of application are analysed and discussed.     

On characterizations and variance bounds of discrete α-unimodality

Characterizations of α-unimodality for integer-valued random variables about a specific mode are established in terms of their probability mass functions, distribution functions and characteristic functions. Using these characterizations variance lower bounds in terms of α and the mode are derived. For α=1 all these results are reduced to ordinary unimodality. The new variance lower bounds for discrete unimodality is sharper than its continuous counterpart. An upper bound for the variance of discrete unimodal distribution defined on a finite support is discussed.     

Confidence and prediction intervals based on interpolated records

In several statistical problems, nonparametric confidence intervals for population quantiles can be constructed and their coverage probabilities can be computed exactly, but cannot in general be rendered equal to a pre-determined level. The same difficulty arises for coverage probabilities of nonparametric prediction intervals for future observations. One solution to this difficulty is to interpolate between intervals which have the closest coverage probability from above and below to the pre-determined level. In this paper, confidence intervals for population quantiles are constructed based on interpolated upper and lower records. Subsequently, prediction intervals are obtained for future upper records based on interpolated upper records. Additionally, we derive upper bounds for the coverage error of these confidence and prediction intervals. Finally, our results are applied to some real data sets. Also, a comparison via a simulation study is done with similar classical intervals obtained before.     

Distribution-free prediction intervals for the future current record statistics

Prediction of records plays an important role in many applications, such as, meteorology, hydrology, industrial stress testing and athletic events. In this paper, based on the observed current records of an iid sequence sample drawn from an arbitrary unknown distribution, we develop distribution-free prediction intervals as well as prediction upper and lower bounds for current records from another iid sequence. We also present sharp upper bounds for the expected lengths of the so obtained prediction intervals. Numerical computations of the coverage probabilities are presented for choosing the appropriate limits of the prediction intervals.      

On lower confidence bounds for pcs in truncated location parameter models

We are concerned with deriving lower confidence bounds for the probability of a correct selection in truncated location-parameter models. Two cases are considered according to whether the scale parameter is known or unknown. For each case, a lower confidence bound for the difference between the best and the second best is obtained. These lower confidence bounds are used to construct lower confidence bounds for the probability of a correct selection. The results are then applied to the problem of seleting the best exponential populationhaving the largest truncated location-parameter. Useful tables are provided for implementing the proposed methods.     

Reporting Bayes factors or probabilities to decision makers of unknown loss functions

When competing interests seek to influence a decision maker, a scientist must report a posterior probability or a Bayes factor among those consistent with the evidence. The disinterested scientist seeks to report the value that is least controversial in the sense that it is best protected from being discredited by one of the competing interests. If the loss function of the decision maker is not known but can be assumed to satisfy two invariance conditions, then the least controversial value is a weighted generalized mean of the upper and lower bounds of the interval.     

Bounds on Sample Variation Measures Based on Majorization

Using majorization theory, upper and lower bounds are derived for different measures of variation as progressively more items of information are available about the sample data. As a convenient starting point, bounds are first established for a one-parameter family of variation measures, which is a generalized mean difference measure of which Gini's mean difference, the standard deviation, and the range are particular cases. While, as pointed out, some of the derived bounds are well known, others do not appear to have been published and are tighter than established bounds. Some 40 different bounds are derived, besides any number of bounds given for the generalized family of variation measures. A number of interesting inequalities are also derived on the basis of some of the bounds. While the bounds have been developed in terms of real-valued sample data generally, the paper concludes with a brief discussion of the bounds for categorical data when the sample data consists of frequencies (counts).