Bounds for moments through a general orthogonal expansion in a pre-hilbert space-I

Bounds are obtained for the product moments of an arbitrary finite number of ordered random variables. These bounds are obtained with the help of a representation of an arbitrary function in terms of a complete orthonormal system in a pre-Hilbert space of square integrable functions defined in a k-dimensional unit cube.     

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     

Post hoc false positive control for structured hypotheses

In a high-dimensional multiple testing framework, we present new confidence bounds on the false positives contained in subsets S of selected null hypotheses. These bounds are post hoc in the sense that the coverage probability holds simultaneously over all S, possibly chosen depending on the data. This article focuses on the common case of structured null hypotheses, for example, along a tree, a hierarchy, or geometrically (spatially or temporally). Following recent advances in post hoc inference, we build confidence bounds for some prespecified forest-structured subsets and deduce a bound for any subset S by interpolation. The proposed bounds are shown to improve substantially previous ones when the signal is locally structured. Our findings are supported both by theoretical results and numerical experiments. Moreover, our bounds can be obtained by an algorithm (with complexity bilinear in the sizes of the reference hierarchy and of the selected subset) that is implemented in the open-source R package sansSouci available from https://github.com/pneuvial/sanssouci , making our approach operational.     

Lower Bounds for the Efficiency of Designs with Respect to the D-Criterion when the Observations are Correlated

In the general linear model consider the experimental design problem for the Gauß-Markov estimator or least squares estimator when the observations are correlated. We prove new formulas for the efficiency of an exact design with respect to the D-criterion. For models with intercept term, for example, these formulas are useful to derive better lower bounds for the efficiency than the bounds recently given for an arbitrary linear model. These bounds are applied in examples to symmetrical regular circulants as covariance matrices. A byproduct of the investigations is some insight as to what kinds of designs might retain their optimality or high efficiency (for the uncorrelated homoscedastic case) under correlated observations.     

On the Parameter Estimation of the Asymmetric Multivariate Laplace Distribution

This article examines a family of three-parameter multivariate Laplace distributions ML _p (a, μ, Σ) which is closed under constant shifts. Parameter vectors a and μ are called shift and shape parameter, respectively, positive definite p × p-matrix Σ is a scale parameter. The first three moments are derived and used for estimating the parameters. The behavior of the obtained estimates is explored in a simulation experiment.     

Sharp bounds on the expectations of L-statistics expressed in the Gini mean difference units

We describe a method of calculating sharp lower and upper bounds on the expectations of arbitrary, properly centered L-statistics expressed in the Gini mean difference units of the original i.i.d. observations. Precise values of bounds are derived for the single-order statistics, their differences, and some examples of L-estimators. We also present the families of discrete distributions which attain the bounds, possibly in the limit.     

Moments of Order Statistics—From Even to Odd Sample Sizes

Abstract

A method is demonstrated to compute the complete set of first moments of order statistics for an arbitrary distribution, given only the first moments of the maximal order statistics either for all even sample sizes, or for all odd samples sizes.     

Kshirsagar-Type Lower Bounds for Mean Squared Error of Prediction

Let Y be an observable random vector and Z be an unobserved random variable with joint density f(y, z | θ), where θ is an unknown parameter vector. Considering the problem of predicting Z based on Y, we derive Kshirsagar type lower bounds for the mean squared error of any predictor of Z. These bounds do not require the regularity conditions of Bhattacharyya bounds and hence are more widely applicable. Moreover, the new bounds are shown to be sharper than the corresponding Bhattacharyya bounds. The conditions for attaining the new lower bounds are useful for easy derivation of best unbiased predictors, which we illustrate with some examples.     

R2 Bounds for Predictive Models: What Univariate Properties Tell us About Multivariate Predictability

ABSTRACT

A long-standing puzzle in macroeconomic forecasting has been that a wide variety of multivariate models have struggled to out-predict univariate models consistently. We seek an explanation for this puzzle in terms of population properties. We derive bounds for the predictive R² of the true, but unknown, multivariate model from univariate ARMA parameters alone. These bounds can be quite tight, implying little forecasting gain even if we knew the true multivariate model. We illustrate using CPI inflation data. Supplementary materials for this article are available online.     

Bootstrapping in a high dimensional but very low-sample size problem

This article is concerned with testing multiple hypotheses, one for each of a large number of small data sets. Such data are sometimes referred to as high-dimensional, low-sample size data. Our model assumes that each observation within a randomly selected small data set follows a mixture of C shifted and rescaled versions of an arbitrary density f. A novel kernel density estimation scheme, in conjunction with clustering methods, is applied to estimate f. Bayes information criterion and a new criterion weighted mean of within-cluster variances are used to estimate C, which is the number of mixture components or clusters. These results are applied to the multiple testing problem. The null sampling distribution of each test statistic is determined by f, and hence a bootstrap procedure that resamples from an estimate of f is used to approximate this null distribution.     

Hypothesis Testing and Confidence Regions for the Mean Sojourn Time of an M/M/1 Queueing System

This article discusses testing hypotheses and confidence regions with correct levels for the mean sojourn time of an M/M/1 queueing system. The uniformly most powerful unbiased tests for three usual hypothesis testing problems are obtained and the corresponding p values are provided. Based on the duality between hypothesis tests and confidence sets, the uniformly most accurate confidence bounds are derived. A confidence interval with correct level is proposed.     

Lower bounds for expected sample size of sequential procedures for the multinomial selection problems

In this article, lower bounds for expected sample size of sequential selection procedures are constructed for the problem of selecting the most probable event of k-variate multinomial distribution. The study is based on Volodin’s universal lower bounds for expected sample size of statistical inference procedures. The obtained lower bounds are used to estimate the efficiency of some selection procedures in terms of their expected sample sizes.     

Sharp bounds on the expectations of second record values from symmetric populations

Combining the greatest convex minorant approximation (Moriguti, S. (1953). A modification of Schwarz's inequality with applications to distributions. Ann. Math. Statist., 24, 107–113.) with the Hölder inequality, we establish sharp bounds on the expectations of the second record statistics from symmetric populations. We also determine the distributions for which the bounds are attained. The optimal bounds are numerically evaluated and compared with other classical rough ones.     

Bounds for Expectations of Differences of Generalized Order Statistics Based on General and Life Distributions

Let X _? ^(r), r ≥ 1, denote generalized order statistics based on an arbitrary distribution function F with finite pth absolute moment for some 1 ≤ p ≤ ∞. We present sharp upper bounds on E(X _? ^(s) ? X _? ^(r)), 1 ≤ r < s, for F being either general or life distribution. The bounds are expressed in various scale units generated by pth central absolute or raw moments of F, respectively. The distributions achieving the bounds are specified.     

L1-distance and classification problem by Bayesian method

In this article we suggest a definition for the notion of L¹-distance that combines probability density functions and prior probabilities. We also obtain the upper and lower bounds for this distance as well as its relation to other measures. Besides, the relationship between the proposed distance and quantities involved in classification problem by Bayesian method will be established. In practice, calculations are performed by Matlab procedures. As an illustration for applications of the obtained results, the article gives here an estimation for the ability to repay bank debt of some companies in Can Tho City, Vietnam.     

Testing for Common Features

This article introduces a class of statistical tests for the hypothesis that some feature that is present in each of several variables is common to them. Features are data properties such as serial correlation, trends, seasonality, heteroscedasticity, autoregressive conditional hetero-scedasticity, and excess kurtosis. A feature is detected by a hypothesis test taking no feature as the null, and a common feature is detected by a test that finds linear combinations of variables with no feature. Often, an exact asymptotic critical value can be obtained that is simply a test of overidentifying restrictions in an instrumental variable regression. This article tests for a common international business cycle.     

Prediction intervals for ordinary and dual generalized order statistics from two independent sequences

ABSTRACT

Based on the observed dual generalized order statistics drawn from an arbitrary unknown distribution, nonparametric two-sided prediction intervals as well as prediction upper and lower bounds for an ordinary and a dual generalized order statistic from another iid sequence with the same distribution are developed. The prediction intervals for dual generalized order statistics based on the observed ordinary generalized order statistics are also developed. The coverage probabilities of these prediction intervals are exact and free of the parent distribution, F. Finally, numerical computations and real examples of the coverage probabilities are presented for choosing the appropriate limits of the prediction.     

Methods for calculating stationary distribution in linear models of time series

The article deals with methods for computing the stationary marginal distribution in linear models of time series. Two approaches are described. First, an algorithm based on approximation of solution of the corresponding integral equation is briefly reviewed. Then, we study the limit behaviour of the partial sums c ₁ η₁+c ₂ η₂+···+c _n η _n where η _i are i.i.d. random variables and c _i real constants. We generalize procedure of Haiman (1998) [Haiman, G., 1998, Upper and lower bounds for the tail of the invariant distribution of some AR(1) processes. Asymptotic Methods in Probability and Statistics, 45, 723–730.] to an arbitrary causal linear process and relax the assumptions of his result significantly. This is achieved by investigating the properties of convolution of densities.     

The discrepancy of P-values and posterior probability: Two-sided hypothesis of variance of normal distribution with unknown mean

This article is concerned with the comparison of P-value and Bayesian measure in point null hypothesis for the variance of Normal distribution with unknown mean. First, using fixed prior for test parameter, the posterior probability is obtained and compared with the P-value when an appropriate prior is used for the mean parameter. In the second, lower bounds of the posterior probability of H₀ under a reasonable class of prior are compared with the P-value. It has been shown that even in the presence of nuisance parameters, these two approaches can lead to different results in the statistical inference.     

Lower bounds of the wrap-around -discrepancy and relationships between MLHD and uniform design with a large size

The wrap-around (WD) L₂-discrepancy has been commonly used in experimental designs. In this paper, some lower bounds of the WD L₂-discrepancy for asymmetrical U-type designs are given and the expectation and variance of midpoint Latin hypercube designs (LHD) are also obtained. Relationships between midpoint LHD and uniform designs for symmetrical and asymmetrical cases are discussed in the sense of comparing the lower bound and the expectation of squared wrap-around L₂-discrepancy of U-type designs. Some comparisons between simple random sampling and the lower bounds of U-type designs are given.