Testing homogeneity of several covariance matrices and multi-sample sphericity for high-dimensional data under non-normality

A test for homogeneity of g ? 2 covariance matrices is presented when the dimension, p, may exceed the sample size, n_i, i = 1, …, g, and the populations may not be normal. Under some mild assumptions on covariance matrices, the asymptotic distribution of the test is shown to be normal when n_i, p → ∞. Under the null hypothesis, the test is extended for common covariance matrix to be of a specified structure, including sphericity. Theory of U-statistics is employed in constructing the tests and deriving their limits. Simulations are used to show the accuracy of tests.     

Note on the bias in the estimation of the serial correlation coefficient of AR(1) processes

We derive approximating formulas for the mean and the variance of an autocorrelation estimator which are of practical use over the entire range of the autocorrelation coefficient ρ. The least-squares estimator ∑ ^{n −1} _{i =1}ε _i ε _{i +1} / ∑ ^{n −1} _{i =1}ε² _i is studied for a stationary AR(1) process with known mean. We use the second order Taylor expansion of a ratio, and employ the arithmetic-geometric series instead of replacing partial Cesàro sums. In case of the mean we derive Marriott and Pope's (1954) formula, with (n− 1)⁻¹ instead of (n)⁻¹, and an additional term α (n− 1)⁻². This new formula produces the expected decline to zero negative bias as ρ approaches unity. In case of the variance Bartlett's (1946) formula results, with (n− 1)⁻¹ instead of (n)⁻¹. The theoretical expressions are corroborated with a simulation experiment. A comparison shows that our formula for the mean is more accurate than the higher-order approximation of White (1961), for |ρ| > 0.88 and n≥ 20. In principal, the presented method can be used to derive approximating formulas for other estimators and processes. Received: November 30, 1999; revised version: July 3, 2000     

A Test of Independence Based on the Likelihood of Cut-Points

We define a test statistic C _n based on the sum of the likelihood ratio statistics for testing independence in the 2 × 2 tables defined at n sample cut-points (X _i , Y _i ). The asymptotic distribution of C _n , given the cut-points, is sum of dependent χ² variables with one degree of freedom. We use the bootstrap to obtain the distribution of C _n . We compare the performance of several tests of bivariate independence, including Pearson, Spearman, and Kendall correlations, Blum-Kiefer-Rosenblatt statistic, and C _n under several copulas and given marginal distributions.     

Adjusted Confidence Bands in Nonparametric Regression

Suppose we have {(x _i , y _i )} i = 1, 2,…, n, a sequence of independent observations. We wish to find approximate 1 ? α simultaneous confidence bands for the regression curve. Many previous confidence bands in the literature have practical difficulties. In this article, the local linear smoother is used to estimate the regression curve. The bias of the estimator is considered. Different methods of constructing confidence bands are discussed. Finally, a possible method incorporating logistic regression in an innovative way is proposed to construct the bands for random designs. Simulations are used to study the performance or properties of the methods. The procedure for constructing confidence bands is entirely data-driven. The advantage of the proposed method is that it is simple to use and can be applied to random designs. It can be considered as a practically useful and efficient method.     

THE CALCULATION OF CONFIDENCE REGIONS FOR EIGENVECTORS

An explicit algorithm is given for constructing a confidence region on the appropriate unit sphere for an eigenvector, given a large sample. It is assumed the eigenvector corresponds to the largest eigenvalue of Exx^T, a matrix with distinct eigenvalues, and that the estimation uses n^-1S?x_ix_ix^T. While the theory is generally applicable, the writer has in mind the special case where ∥x∥= 1 i.e. directional data.     

On constructing T 2 control charts for retrospective examination

In this paper we consider the issue of constructing retrospective T ² control chart limits so as to control the overall probability of a false alarm at a specified value. We describe an exact method for constructing the control limits for retrospective examination. We then consider Bonferroni-adjustments to Alt's control limit and to the standard x ² control limit as alternatives to the exact limit since it is computationally cumbersome to find the exact limit. We present the results of some simulation experiments that are carried out to compare the performance of these control limits. The results indicate that the Bonferroni-adjusted Alt's control limit performs better that the Bonferroni-adjusted x ² control limit. Furthermore, it appears that the Bonferroni-adjusted Alt's control limit is more than adequate for controlling the overall false alarm probability at a specified value.     

Upper tolerance limit for the largest observation from censored samples

This paper deals with obtaining an upper tolerance limit for a largest observation X_(n) in an ordered sample of size n from a continuous distribution where the first m observations X₍₁₎ < X₍₂₎ < … < X_(m), l ≤ m < n, have been observed. A criterion of “goodness” of tolerance limit is developed, and a method is given to obtain the best tolerance limit. This method is applied to exponential and Pareto distributions.     

On the bounded regret of empirical bayes estimators

Let (θ₁,x₁),…,(θ_n,x_n) be independent and identically distributed random vectors with E(xθ) = θ and Var(x|θ) = a + bθ + cθ². Let t_i be the linear Bayes estimator of θ_i and θ~_i be the linear empirical Bayes estimator of θ_i as proposed in Robbins (1983). When Ex and Var x are unknown to the statistician. The regret of using θ~_i instead of t_i because of ignorance of the mean and the variance is r_i = E(θ_i ? θ_i)² ?E(t_i ?θ_i)². Under appropriate conditions cumulative regret R_n = r₁+…r_n is shown to have a finite limit even when n tends to infinity. The limit can be explicitly computed in terms of a,b,c and the first four moments of x.     

Change-Point Detection With Non-Parametric Regression

We consider the regression model y_i = ?(x_i ) + ε in which the function ? or its pth derivative ?^(p) may have a discontinuity at some unknown point τ. By fitting local polynomials from the left and right, we test the null that ?^(p) is continuous against the alternative that ?^(p)(τ?) ≠ ?^(p)(τ+). We obtain Darling-Erdös type limit theorems for the test statistics under the null hypothesis of no change, as well as their limits in probability under the alternative. Consistency of the related change-point estimators is also established.     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

On the Failure Probability of Used Coherent Systems

In analyzing the lifetime properties of a coherent system, the concept of “signature” is a useful tool. Let T be the lifetime of a coherent system having n iid components. The signature of the system is a probability vector s=(s₁, s₂, …, s_n), such that s_i=P(T=X_i:n), where, X_i:n, i=1, 2, …, n denote the ordered lifetimes of the components. In this note, we assume that the system is working at time t>0. We consider the conditional signature of the system as a vector in which the ith element is defined as p_i(t)=P(T=X_i:n|T>t) and investigate its properties as a function of time.     

Improved bootstrap confidence intervals in certain toxicological experiments

The bootstrap, the jackknife, and classical methods are compared through their confidence intervals for the proportion of affected fetuses in a common type of animal experiment. Specifically, suppose that for the ith of M pregnant animals, there are x _i affected fetuses out of n _i total in the litter. The conditional distribution of x _i given n _i is sometimes modeled as binomial (n _i p _i ), where p _i is a realization from some unknown continuous density. The p _i are not observable and it is of interest in some toxicological experiments to find confidence intervals for E(p). Theory suggests that the proposed parametric bootstrap should produce higher order agreement between the nominal and actual coverage than that exhibited by the usual nonparametric bootstrap. Some simulation results provide additional evidence of this superiority of the modified parametric bootstrap over the jack-knife and classical approaches. The proposed resampling is flexible enough to handle a more general model allowing correlation between p _i and n _i .     

ON ASYMPTOTIC OPTIMALITY OF SOME TESTS FOR EXPONENTIAL DISTRIBUTIONS

The probability density function (pdf) of a two parameter exponential distribution is given by f(x; p, s?) =s?^-1 exp {-(x - ρ)/s?} for x≥ρ and 0 elsewhere, where 0 < ρ < ∞ and 0 < s?∞. Suppose we have k independent random samples where the ith sample is drawn from the ith population having the pdf f(x; ρ_i, s?_i), 0 < ρ_i < ∞, 0 < s?_i < s?_i < and f(x; ρ, s?) is as given above. Let X_i1 < X_i2 <… < X_iri denote the first r_i order statistics in a random sample of size n_i, drawn from the ith population with pdf f(x; ρ_i, s?_i), i = 1, 2,…, k. In this paper we show that the well known tests of hypotheses about the parameters ρ_i, s?_i, i = 1, 2,…, k based on the above observations are asymptotically optimal in the sense of Bahadur efficiency. Our results are similar to those for normal distributions.     

Estimation of the sum of differences distribution

Consider the distribution of Z_i d_iwhere the d.d_i?s are 1=1 lldifferences independently, identically and symmetrically distributed with mean zero. The problem is to determine properties of the sdd given the distribution of the d._i?fs and the sample size n. The standardized moments as a function of the moments of the d._i!s are developed. A variance reduction technique for estimating the quantiles of the sdd using Monte Carlo methods is developed based on using the randomization sample consisting of the 2ⁿ values of Z _i+d. rather than the single observation i=l lZ d. corresponding to each sample d_id_n. The randomization sample is shown to produce unbiased and consistent estimators.     

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.     

Pooling operators with the marginalization property

In this paper, we consider the problem of combining a number of opinions which have been expressed as probability measures P₁, …, P_n, over some space. It is shown that a pooling formula which has the marginalization property of McConway (1981) must be of the form T = Σⁿ_i=₁W_i P_i + (1 - Σⁿ_i =₁W_i)Q, where Q is an arbitrary measure and W₁, …, W_n ϵ [—1,1] are weights such that| Σ_J Σ _j w_j | ≤ 1 for every subset J of {1, …, n}. If, in addition, T is required to preserve the independence of arbitrary events A and B whenever these events are independent under each P_i, then either T = P_i for some 1 ≤ i ≤ n or T = Q, in which case Q takes values in {0, l}.     

Convergence properties for weighted sums of NSD random variables

Abstract

Let {X_n, n ? 1} be a sequence of negatively superadditive dependent (NSD, in short) random variables and {b_ni, 1 ? i ? n, n ? 1} be an array of real numbers. In this article, we study the strong law of large numbers for the weighted sums ∑ⁿ_{i = 1}b_niX_i without identical distribution. We present some sufficient conditions to prove the strong law of large numbers. As an application, the Marcinkiewicz-Zygmund strong law of large numbers for NSD random variables is obtained. In addition, the complete convergence for the weighted sums of NSD random variables is established. Our results generalize and improve some corresponding ones for independent random variables and negatively associated random variables.     

Randomly weighted sums and their maxima with heavy-tailed increments and dependence structure

Consider the randomly weighted sums S_m(θ) = ∑^m_{i = 1}θ_iX_i, 1 ? m ? n, and their maxima M_n(θ) = max?_{1 ? m ? n}S_m(θ), where X_i, 1 ? i ? n, are real-valued and dependent according to a wide type of dependence structure, and θ_i, 1 ? i ? n, are non negative and arbitrarily dependent, but independent of X_i, 1 ? i ? n. Under some mild conditions on the right tails of the weights θ_i, 1 ? i ? n, we establish some asymptotic equivalence formulas for the tail probabilities of S_n(θ) and M_n(θ) in the case where X_i, 1 ? i ? n, are dominatedly varying, long-tailed and subexponential distributions, respectively.