Probability Inequalities Research Articles

Probability Inequalities for Sums of Independent Unbounded Random Variables

The tail probability inequalities for the sum of independent unbounded random variables on a probability space (Ω, T, P) were studied and a new method was proposed to treat the sum of independent unbounded random variables by truncating the original probability space (Ω, T, P). The probability exponential inequalities for sums of independent unbounded random variables were given. As applications of the results, some interesting examples were given. The examples show that the method proposed in the paper and the results of the paper are quite useful in the study of the large sample properties of the sums of independent unbounded random variables.

Some liminf results on increments of the primitives of Brownian motion

Let {W(t); t≥0} be a standard Brownian motion. For a positive integer m, define a Gaussian process $$X_m \left( t \right) = \frac{1}{{m!}}\int_0^t { \left( {t - s} \right)^m dW\left( s \right)} $$ . In this paper the liminf behavior of the increments of this process is discussed by establishing some probability inequalities. Some previous results are extended and improved.

Approximations for hybrids of empirical and partial sums processes

Abstract We obtain probability inequalities and almost sure rates for the approximations of the hybrids of empirical and partial sums processes. The proofs are based on Poisson approximations for empirical processes and Gaussian approximations for partial sums. Some consequences of the approximations are also discussed.

Some inequalities for the kernel density estimator under random censorship*

In this paper, the kernel density estimator proposed by Blum and Susarla (1980) is investigated. Three inequalities, a L1 error inequality, a pth, p>=2, order absolute moment inequality and a probability inequality, are established, respectively.

Variance upper bounds and a probability inequality for discrete α-unimodality

Variance upper bounds for discrete α-unimodal distributions defined on a finite support are established. These bounds depend on the support and the unimodality index α. They increase as the unimodality index α increases. More information about the underly

Simultaneous confidence intervals in the analysis of orthogonal saturated designs

Abstract We present the first known method of constructing exact simultaneous confidence intervals for the analysis of orthogonal, saturated factorial designs. Given m independent, normally distributed, unbiased estimators of treatment contrasts, if there is an independent chi-squared estimator of error variance, then simultaneous confidence intervals based on the Studentized maximum modulus distribution are exact under all parameter configurations. In this paper, an analogous method is developed for the case of an orthogonal saturated design, for which the treatment contrasts are independently estimable but there is no independent estimator of error variance. Lacking an independent estimator of the error variance, the smallest sums of squares of effect estimators are pooled. The simultaneous confidence intervals are based on a probability inequality, for which the simultaneous confidence coefficient is achieved in the null case.

Moment and probability inequalities for the bivariate product-limit estimator

Let S( s, t) be a bivariate survival function. Let S ̂ n(s,t) be the bivariate product limit estimator of S( s, t) proposed by Campbell and Földes (1982, In: Gnedenko, B.V., Puri, M.C., Vincze, I. (Eds.), Nonparametric Statistical Inference. North-Holland, Amsterdam, pp. 103–122). In this paper, absolute moment and probability inequalities for S ̂ n(s,t) are derived, respectively.

Stochastic Processes and Related Topics. In Memory of Stamatis Cambanis 1943-1995

Stamatis Cambanis - a glimpse of his life and work, G. Kallianpur, R. Leadbetter. Publications of Stamatis Cambanis: spectral representation and structure of stable self-similar processes, K. Burnecki et al three elementary proofs of Central Limit theorem with applications to random sums, T. Cacoullos et al almost everywhere convergence and SLLN under rearrangements, S. Chobanyan, V. Mandrekar sufficient conditions for existence of conditional moments of stable random variables, R. Cioczek-Georges, M.S. Taqqu how heavy are tails of a stationary HARCH(k) process? - a study of moments, P. Embrechts et al use of stochastic comparisons in communication networks, A. Ephremide on conditional variance-covariance of stable random vectors II, S.B. Fotopoulos interacting particle approximation for fractal Burgers equation, T. Funaki, W.A. Woyczynski optimal transformations for prediction in continuous-time stochastic processes, B. Gidas, A. Murua algebraic methods toward higher-order probability inequalities, K.I. Gross, D.St.P. Richards comparison and deviation from a representation formula, C. Houdre components of strong Markov property, O. Kallenberg Russian options, G. Kallianpur cycle representation of Markov processes - an application to rotational partitions, S. Kalpazidou on extreme values in stationary random fields, M.R. Leadbetter, H. Rootzen norming operators for operator-shelf-similar processes, M. Maejima multivariate probability density and regression functions estimation of continuous-time stationary processes from discrete-time data, e. Masry tracing path of Wright-Fisher process with one-way mutation in case of a large deviation, F. Papangelou a distribution inequality for martingales with bounded symmetric differences, L.D. Pitt moment comparison of multilineal forms in stable and semistable random variables with application to semistable multiple integrals, B.S. Rajput et al global dependency measure for sets of random elements - the Italian problem and some consequences, J. Stoyanov.

On tail probabilities of Kolmogorov–Smirnov statistics based on uniform mixing processes

Abstract Sen (1974, Weak convergence of multidimensional empirical processes for stationary φ-mixing processes. Ann. Probab. 2, 147–154, Theorem 3.2) established the tail probability inequalities for Kolmogorov–Smirnov statistics for sequences of φ-mixing random variables. In the present note, Sen's result is improved considerably.

A Course in Mathematical Statistics

Concepts of Set Theory Some Probabilistic Concepts and Results On Random variables and Their Distributions Distribution Functions, Probability Densities, and their Relationships Moments of Random Variables-Some Moment and Probability Inequalities Characteristic Functions, Moment Generating Functions, and Related Theorems Stochastic Independence with Some Applications Basic Limit Theorems Transformations of Random Variables and Random Vectors Order Statistics and Related Theorems Sufficiency and Related Theorems Point Estimation Testing Hypotheses Sequential Procedures Confidence Regions-Tolerance Intervals The General Linear Hypothesis Analysis of Variance The Multivariate Normal Distribution Quadratic Forms Nonparametric Inference Appendices Table of Selected Discrete and Continuous Distributions and Some of their Characteristics Some Notation and Abbreviations

Probability inequalities in multivariate distributions

For a bivariate binary response model y, = 1 (xj βj+j > 0), j=1,2, we propose to estimate nonpararnetrically the quadrant correlation E{sgn(u1) *sgn(u2)} between the two error terms ul and u2 without specifjing the error term distribution. The quadrant correlation accounts for the relationship between yl and y2 that is not explained by xl and x2, and can be used in testing for the specification of endogenous dummy variable models. The quadrant correlation is further generalized into orthant dependence allowing unknown regression functions, unknown error term distribution and arbitrary forms of heteroskedasticity. A simulation study is provided, followed by a brief application to a real data set.

Two-sided Inequalities for Laplace Transforms

Two-sided inequalities for Laplace transforms of nonnegative random variables and random vectors with nonnegative elements are obtained in terms of moments (i.e., in terms of derivatives of Laplace transforms at 0). The proofs are based on the connections between Laplace transforms, generating functions, and probabilities of the unions of events; inequalities for such probabilities are also used.

Sub-Bernoulli Functions, Moment Inequalities and Strong Laws for Nonnegative and symmetrized U-Statistics

This paper concerns moment and tail probability inequalities and the strong law of large numbers for $U$-statistics with nonnegative or symmetrized kernels and their multisample and decoupled versions. Sub-Bernoulli functions are used to obtain the moment and tail probability inequalities, which are then used to obtain necessary and sufficient conditions for the almost sure convergence to zero of normalized $U$-statistics with nonnegative or completely symmetrized kernels, without further regularity conditions on the kernel or the distribution of the population, for normalizing constants satisfying a simple condition. Moments of $U$-statistics are bounded from above and below by that of maxima of certain kernels, up to scaling constants. The multisample and decoupled versions of these results are also considered.

How Big Are the Increments of a Two-Parameter Gaussian Process?

Some limit theorems on the increments of a two-parameter Gaussian process are obtained via estimating large deviation probability inequalities on the suprema of the Gaussian process which is a generalization of a two-parameter Levy Brownian motion.

Probabilistic analysis on the splitting-shooting method for image transformations

This paper explores a new topic in image processing where accuracy of images even in details is crucial, and adopts a new methodology dealing with discrete topics by continuous mathematics and numerical approximation. The key idea is that a pixel of images at different levels can be quantified by a greyness value, which can then be regarded as the mean of an integral of continuous functions over a small pixel region, and evaluated by numerical integration approximately. Hence, new treatments of approximate integration and new discrete algorithms of images have been developed. This paper also integrates different mathematics disciplines: numerical analysis, geometry, probability and statistics to discrete images that can be applied to many areas in computer sciences: image processing, computer graphics, computer vision, geometric added designs, and pattern recognition. In this paper, new error analysis in terms of probability theory is explored for the popular splitting-shooting method (SSM) and the combination (CSIM) of the splitting-shooting-integrating methods proposed in [35–37], and convergence rates in probability of image greyness are proven to be O p( 1 N 1.5 ) higher than O( 1 N ) reported in [45] by strict error analysis. Moreover, a new partial refinement technique of pixel partition is also proposed in this paper, to achieve the convergence rate O p( 1 N 2 ) in probability for SSM. By the new study of this paper, the SSM can be applied to real images with 256 greyness levels. The numerical and graphical experiments are also provided to confirm the theoretical analysis made. Both the strict error bounds and the probabilistic error bounds with explicit constants are also derived for general α-norms as α ⩾ 1, and the countable probability inequalities for several sums of random variables are developed from probability theory, which are more suited to numerical computations.

Some probability inequalities for ordered $\rm MTP\sb 2$ random variables: a proof of the Simes conjecture

Some new probability inequalities involving the ordered components of an $MTP_2$ random vector are derived, which provide an analytical proof of an important conjecture in the field of multiple hypothesis testing. This conjecture has been mostly validated so far using simulation.

Some large sample results for a class of functionals of Kaplan-Meier estimator

In this paper, a class of functionals of Kaplan-Meier estimator is investigated. Counting process martingale methods are used to show the asymptotic normality, and we establish a mean square error inequality and a probability inequality of them without the assumption thatF, G are continuous, where,F, G are survival time distribution and censoring time distribution respectively.

Multiple Comparisons in the General Linear Model

Whereas multiple comparisons computations in a one-way model are well understood, multiple comparisons computations in a general linear model (GLM) are not. For models with the so-called “one-way structure,” no new technique is needed beyond proper substitution of terms. Examples of designs that guarantee a one-way structure include variance balanced designs and orthogonal designs. For models without a one-way structure, more sophisticated computational techniques are needed. Approximations based on the probabilistic inequalities of Bonferroni, Šidák, and Slepian are too conservative. Even the second-order Hunter—Worsley inequality is rather conservative. The so-called factor analytic approximation is quite accurate for multiple comparison with a control (MCC) and multiple comparison with the best (MCB), but conditions for it to be conservative are not known. This article describes a highly accurate, deterministic, conservative approximation that is applicable to a popular class of general linear models, and a fast, stochastic, conservative approximation that is generally applicable.

Directional monotonicity properties of the power functions of likelihood ratio tests for cone-restricted hypotheses of normal means

In some cone-restricted testing problems, it is known that the power function of the likelihood ratio test has a certain monotonicity property with respect to the direction of the alternatives. In this paper, we establish a unified result concerning such directional monotonicity by using an argument based on a cone ordering and a related probability inequality. From this result, we show that the power function is unimodal in direction when the cone is suitably symmetric.

A generalization of Ueno's inequality for n-step transition probabilities

We provide a generalization of Ueno's inequality for n-step transition probabilities of Markov chains in a general state space. Our result is relevant to the study of adaptive control problems and approximation problems in the theory of discrete-time Mark