The echelon Markov and Chebyshev inequalities

Abstract

A multivariate version of the sharp Markov inequality is derived, when associated probabilities are extended to segments of the supports of non-negative random variables, where the probabilities take echelon forms. It is shown that when some positive lower bounds of these probabilities are available, the multivariate Markov inequality without the echelon forms is improved. The corresponding results for Chebyshev’s inequality are also obtained.     

PROBABILITY APPROXIMATIONS FOR DIVISIBLE DISCRETE DISTRIBUTIONS

If an integer-valued random variable can be represented as a sum of independent random variables, then powerful tools exist to derive approximations to its distribution. We apply this idea to examples in some of which it is not clear how to give a physical interpretation to the independent sum-mands. We consider bounds on the accuracy of single term approximations, Edgeworth expansions and saddlepoint approximations for both individual probabilities and cumulative probabilities.     

Some Improvements on Markov's Theorem with Extensions

ABSTRACT

Markov's theorem for an upper bound of the probability related to a nonnegative random variable has been improved using additional information in almost the nontrivial entire range of the variable. In the improvement, Cantelli's inequality is applied to the square root of the original variable, whose expectation is finite when that of the original variable is finite. The improvement has been extended to lower bounds and monotonic transformations of the original variable. The improvements are used in Chebyshev's inequality and its multivariate version.     

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.     

Poisson approximation and dna sequence matching

The Poisson distribution is commonly used to model the number of occurrences of independent rare events. However, many instances arise where dependence exists, for example, in counting the length of long head runs in coin tossing, or matches between two DNA sequences. The Chen-Stein method of Poisson approximation yields bounds on the error incurred when approximating the number of occurrences of possibly dependent events by a Poisson random variable of the same mean. In addition to the problems related to the motivating examples from molecular biology involving runs and matches, the method may be applied to questions as varied as calculating probabilities involving extremes of sequences of random variables and approximating the probability of general birthday coincidences.     

Half-normal approximation for statistics of symmetric simple random walk

In 2013, Döbler used Stein’s method to obtain the uniform bounds in half-normal approximation for three statistics of a symmetric simple random walk; the maximum value, the number of returns to the origin and the number of sign changes up to a given time n. In this paper, we give the non-uniform bounds for these statistics by using Stein’s method and the concentration inequality approach.     

The variance upper bound for a mixed random variable

In this note, we derive upper bounds on the variance of a mixed random variable. Our results are an extension of previous results for unimodal and symmetric random variables. The novelty of our findings is that this mixed random variable does not necessarily need to be symmetric and is multimodal. We also characterize the cases when these bounds are optimal.     

Applications of a probability inequality to ranking and selection and other related problems

A probability inequality of conditionally independent and identically distributed (i.i.d.) random variables obtained recently by the author is applied to ranking and selection problems. It is shown that under both the indifference-zone and the subset formulations, the probability of a correct selection (PCS) is a cumulative probability of conditionally i.i.d, random variables. Therefore bounds on both the PCS and the sample size required can be obtained from that probability inequality. Applications of that inequality to other multiple decision problems are also considered. It is illustrated that general results concerning conditionally i.i.d. random variables are applicable to many problems in multiple decision theory.     

Complete Convergence for Weighted Sums and Arrays of Rowwise Extended Negatively Dependent Random Variables

In this article, we study the complete convergence for weighted sums of extended negatively dependent random variables and row sums of arrays of rowwise extended negatively dependent random variables. We apply two methods to prove the results: the first of is based on exponential bounds and second is based on the generalization of the classical moment inequality for extended negatively dependent random variables.     

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).     

Bayesian prediction with a random sample size for the burr lifetime distribution

This paper is concerned with the problem of deriving Bayesian prediction bounds for the Burr distribution when the sample size is a random variable. Prediction bounds for both the future observations (the case of two-sample prediction) and the remaining observations in the same sample (the case of one-sample prediction) will be derived. The analysis will depend mainly on assuming that the size of the sample is a random variable having the Poisson distribution. Finally, numerical examples are given to illustrate the results.     

Further properties of a random graph with duplications and deletions

We deal with a random graph model where at each step, a vertex is chosen uniformly at random, and it is either duplicated or its edges are deleted. Duplication has a given probability. We analyze the limit distribution of the degree of a fixed vertex and derive a.s. asymptotic bounds for the maximal degree. The model shows a phase transition phenomenon with respect to the probabilities of duplication and deletion.     

UPPER BOUNDS FOR THE HARMONIC MEAN, WITH AN APPLICATION TO EXPERIMENTAL DESIGN

For positive-valued random variables, the paper provides a sequence of upper bounds for the harmonic mean, the ith of these bounds being exact if and only if the random variable is essentially i-valued. Sufficient conditions for the convergence of the bounds to the harmonic mean are given. The bounds have a number of applications, particularly in experimental design where they may be used to check how close a given design is to A-optimality     

On A method for selecting a distributional model

A method for selecting a distributional model for a random variable, given a random sample of observations of it, is studied for various cases. The problems considered include those of choosing between the Weibull and lognormal distributions, between the lognormal and gamma distributions, and between the gamma and Weibull distributions, as well as choosing one of the three. Simulation studies were performed to estimate probabilities of correct selection for the method when it is applied to these problems     

Shortening the distance between Edgeworth and Berry–Esseen in the classical case

We obtain near optimal Berry–Esseen bounds for standardized sums of independent identically distributed random variables. This is achieved by distinguishing the lattice and the non-lattice cases, as one-term Edgeworth expansions do. The main tool is an easy inequality involving the usual second modulus of continuity, in substitution of Esseen's smoothing inequality. An illustrative example concerning the exponential distribution is also considered.     

General characterization theorems via the mean absolute deviation

The Chernoff–Borovkov–Utev inequality resulted owing to earlier inequalities established by Chernoff (1981) and Borovkov and Utev (1983), respectively, giving bounds for the variance of functions of normal r.v.’s and leading to characterizations of normality. Subsequently, several analytic properties of variance bounds and other relevant results were established by others. Defining the mean absolute deviation (about a median) as E|X−med(X)| where med(X) is a median of the distribution of the random variable X, Freimer and Mudholkar (1991) gave a bound for the mean absolute deviation of a certain real-valued function of an absolutely continuous random variable (w.r.t. Lebesgue measure) and Korwar (1991) presented an analogue of this in the discrete case; these authors, also, characterized the Laplace and a mixture of two Waring distributions via the respective bounds.We extend these latter results theorems to the case where the distributions are not necessarily purely discrete or absolutely continuous, via the approach of Alharbi and Shanbhag (1996). The results in Freimer and Mudholkar (1991) and Korwar (1991) are now corollaries to our findings. Also, following Alharbi and Shanbhag (1996), we relate these results to the modified version of Cox’s representation for a survival function in terms of the hazard measure, given in Kotz and Shanbhag (1980). (The original version of the representation mentioned had appeared in Cox (1972).)     

Markov's Inequality and Chebyshev's Inequality for Tail Probabilities: A Sharper Image

Markov's inequality gives an upper bound on the probability that a nonnegative random variable takes large values. For example, if the random variable is the lifetime of a person or a machine, Markov's inequality says that the probability that an individual survives more than three times the average lifetime in the population of such individuals cannot exceed one-third. Here we give a simple, intuitive geometric interpretation and derivation of Markov's inequality. These results lead to inequalities sharper than Markov's when information about conditional expectations is available, as in reliability theory, demography, and actuarial mathematics. We use these results to sharpen Chebyshev's tail inequality also.     

Sharp bounds for L-statistics from dependent samples of random length

Several distribution-free bounds on expected values of L-statistics based on the sample of possibly dependent and nonidentically distributed random variables are given in the case when the sample size is a random variable, possibly dependent on the observations, with values in the set {1,2,…}. Some bounds extend the results of Papadatos (2001a) to the case of random sample size. The others provide new evaluations even if the sample size is nonrandom. Some applications of the presented bounds are also indicated.     

Why a time effect often has a limited impact on capture‐recapture estimates in closed populations

The author is concerned with log‐linear estimators of the size N of a population in a capture‐recapture experiment featuring heterogeneity in the individual capture probabilities and a time effect. He also considers models where the first capture influences the probability of subsequent captures. He derives several results from a new inequality associated with a dispersive ordering for discrete random variables. He shows that in a log‐linear model with inter‐individual heterogeneity, the estimator N is an increasing function of the heterogeneity parameter. He also shows that the inclusion of a time effect in the capture probabilities decreases N in models without heterogeneity. He further argues that a model featuring heterogeneity can accommodate a time effect through a small change in the heterogeneity parameter. He demonstrates these results using an inequality for the estimators of the heterogeneity parameters and illustrates them in a Monte Carlo experiment     

On Stein's method, smoothing estimates in total variation distance and mixture distributions

In many settings it is useful to have bounds on the total variation distance between some random variable Z and its shifted version Z+1. For example, such quantities are often needed when applying Stein's method for probability approximation. This note considers one way in which such bounds can be derived, in cases where Z is either the equilibrium distribution of some birth-death process or the mixture of such a distribution. Applications of these bounds are given to translated Poisson and compound Poisson approximations for Poisson mixtures and the Pólya distribution.