Nonparametric estimation of copula-based measures of multivariate association from contingency tables

Nonparametric estimation of copula-based measures of multivariate association in a continuous random vector X=(X₁, …, X_d) is usually based on complete continuous data. In many practical applications, however, these types of data are not readily available; instead aggregated ordinal observations are given, for example, ordinal ratings based on a latent continuous scale. This article introduces a purely nonparametric and data-driven estimator of the unknown copula density and the corresponding copula based on multivariate contingency tables. Estimators for multivariate Spearman's rho and Kendall's tau are based thereon. The properties of these estimators in samples of medium and large size are evaluated in a simulation study. An increasing bias can be observed along with an increasing degree of association between the components. As it is to be expected, the bias is severely influenced by the amount of information available. Additionally, the influence of sample size is only marginal. We further give an empirical illustration based on daily returns of five German stocks.     

Some robust tests of independence in symmetrical multivariate distributions

Let X =(x)^ij=(11¹, …, X,)_T, i = l, …n, be an n X random matrix having multivariate symmetrical distributions with parameters μ, Σ. The p-variate normal with mean μ and covariance matrix is a member of this family. Let be the squared multiple correlation coefficient between the first and the succeeding p₁ components, and let p² = + be the squared multiple correlation coefficient between the first and the remaining p₁ + p₂ =p – 1 components of the p-variate normal vector. We shall consider here three testing problems for multivariate symmetrical distributions. They are (A) to test p² =0 against; (B) to test against =0, 0; (C) to test against p² =0, We have shown here that for problem (A) the uniformly most powerful invariant (UMPI) and locally minimax test for the multivariate normal is UMPI and is locally minimax as p² 0 for multivariate symmetrical distributions. For problem (B) the UMPI and locally minimax test is UMPI and locally minimax as for multivariate symmetrical distributions. For problem (C) the locally best invariant (LBI) and locally minimax test for the multivariate normal is also LBI and is locally minimax as for multivariate symmetrical distributions.     

Ordered Alternatives and the 1 1/2-Tail Test

Under proper conditions, two independent tests of the null hypothesis of homogeneity of means are provided by a set of sample averages. One test, with tail probability P ₁, relates to the variation between the sample averages, while the other, with tail probability P ₂, relates to the concordance of the rankings of the sample averages with the anticipated rankings under an alternative hypothesis. The quantity G = P ₁ P ₂ is considered as the combined test statistic and, except for the discreteness in the null distribution of P ₂, would correspond to the Fisher statistic for combining probabilities. Illustration is made, for the case of four means, on how to get critical values of G or critical values of P ₁ for each possible value of P ₂, taking discreteness into account. Alternative measures of concordance considered are Spearman's ρ and Kendall's τ. The concept results, in the case of two averages, in assigning two-thirds of the test size to the concordant tail, one-third to the discordant tail.     

McDonald's ωt,Cronbach's α, and Generalized θ for Composite Reliability of Common Factors Structures

In the common factor model for subtest scores, several reliability coefficients, including Cronbach's α, have been found to be biased. In this article, we introduce a new coefficient, θ_G, or Generalized θ, which is a generalized version of Armor's θ coefficient and is equal to the true reliability when the dimensions are orthogonal and the measures are parallel. We assessed the McDonald's ω_t, α, and θ_G in terms of mean bias, efficiency, and precision using a Monte Carlo simulation. θ_G outperformed ω_t when the factors were orthogonal or nearly orthogonal with low correlations between them.     

Regression-based orderings and measures of stochastic depence

General axiomatic approach to the so-called global dependence of a random variable xon a random vector Y= Y _t,Y _n) is proposed. natural orderings and measures of global dependence are discussed and examplified by some real and function-valued measures of dependence. orderings and measures to be introduced are referred o as regression-based as they depend only on the distributions of EX|Y X.     

Testing and dating structural changes in copula-based dependence measures

This paper is concerned with testing and dating structural breaks in the dependence structure of multivariate time series. We consider a cumulative sum (CUSUM) type test for constant copula-based dependence measures, such as Spearman''s rank correlation and quantile dependencies. The asymptotic null distribution is not known in closed form and critical values are estimated by an i.i.d. bootstrap procedure. We analyze size and power properties in a simulation study under different dependence measure settings, such as skewed and fat-tailed distributions. To date breakpoints and to decide whether two estimated break locations belong to the same break event, we propose a pivot confidence interval procedure. Finally, we apply the test to the historical data of 10 large financial firms during the last financial crisis from 2002 to mid-2013.     

Multivariate quality control

This paper includes both the motivation for multivariate quality control, and a discussion of some ot rhe techniques currently available. The emphasis focuses primarily on control charts and includes the T² -chart, the use of principal components anm some recent developments, multivariate analogs of CUSUM cnarts and the use of the Andrews procedure. Some of the problems associated with multivariate acceptance sampling are presented, and the paper concludes with some recommendauons for future researcn and development.     

Scheffés mixed model for multivariate repeated measures:a relative efficiency evaluation

Scheffé’s mixed model, generalized for application to multivariate repeated measures, is known as the multivariate mixed model (MMM). The primary advantages the MMM are (1) the minimum sample size required to conduct an analysis is smaller than for competing procedures and (2) for certain covariance structures, the MMM analysis is more powerful than its competitors. The primary disadvantage is that the MMM makes a very restrictive covariance assumption; namely multivariate sphericity. This paper shows, first, that even minor departures from multivariate sphericity inflate the size of MMM based tests. Accordingly, MMM analyses, as computed in release 4.0 of SPSS MANOVA (SPSS Inc., 1990), can not be recommended unless it is known that multivariate sphericity is satisfied. Second, it is shown that a new Box-type (Box, 1954) Δ-corrected MMM test adequately controls test size unless departure from multivariate sphericity is severe or the covariance matrix departs substantially from a multiplicative-Kronecker structure. Third, power functions of adjusted MMM tests for selected covariance and noncentrality structures are compared to those of doubly multivariate methods that do not require multivariate sphericity. Based on relative efficiency evaluations, the adjusted MMM analyses described in this paper can be recommended only when sample sizes are very small or there is reason to believe that multivariate sphericity is nearly satisfied. Neither the e-adjusted analysis suggested in the SPSS MANOVA output (release 4.0) nor the adjusted analysis suggested by Boik (1988) can be recommended at all.     

The Ranges of Limiting Values of Some Partial Correlations under Conditional Independence

Pearson's partial correlation, Kendall's partial tau, and a partial correlation based on Spearman's rho need not be consistent estimators of zero under conditional independence. The ranges of possible limiting values of these correlations are computed under multivariate normality and lognormality. Students should exercise caution when interpreting these partial correlations as a measure of conditional independence.     

Selection of Variables in Multivariate Regression Models for Large Dimensions

The Akaike information criterion, AIC, and Mallows’ C _p statistic have been proposed for selecting a smaller number of regressors in the multivariate regression models with fully unknown covariance matrix. All of these criteria are, however, based on the implicit assumption that the sample size is substantially larger than the dimension of the covariance matrix. To obtain a stable estimator of the covariance matrix, it is required that the dimension of the covariance matrix is much smaller than the sample size. When the dimension is close to the sample size, it is necessary to use ridge-type estimators for the covariance matrix. In this article, we use a ridge-type estimators for the covariance matrix and obtain the modified AIC and modified C _p statistic under the asymptotic theory that both the sample size and the dimension go to infinity. It is numerically shown that these modified procedures perform very well in the sense of selecting the true model in large dimensional cases.     

Some complex variable transformations and exact power comparisons of two-sided tests of equality of two Hermitian covariance matrices

Power studies of tests of equality of covariance matrices of two p-variate complex normal populations σ₁ = σ₂ against two-sided alternatives have been made based on the following five criteria: (1) Roy's largest root, (2) Hotelling's trace, (4) Wilks' criterion and (5) Roy's largest and smallest roots. Some theorems on transformations and Jacobians in the two-sample complex Gaussian case have been proved in order to obtain a general theorem for establishing the local unbiasedness conditions connecting the two critical values for tests (1)–(5). Extensive unbiased power tabulations have been made for p=2, for various values of n₁, n₂, λ₁ and λ₂ where n₁ is the df of the SP matrix from the ith sample and λ₁ is the ith latent root of σ₁σ^-1₂ (i=1, 2). Equal tail areas approach has also been used further to compute powers of tests (1)–(4) for p=2 for studying the bias and facilitating comparisons with powers in the unbiased case. The inferences have been found similar to those in the real case. (Chu and Pillai, Ann. Inst. Statist. Math. 31.     

A note on L1 consistent estimation

Let (??, ??) be a space with a σ-field, M = {P_s; s o} a family of probability measures on A, Θ arbitrary, X₁,…,X_n independently and identically distributed P random variables. Metrize Θ with the L₁ distance between measures, and assume identifiability. Minimum-distance estimators are constructed that relate rates of convergence with Vapnik-Cervonenkis exponents when M is “regular”. An alternative construction of estimates is offered via Kolmogorov's chain argument.     

Designing and construction of tightened-normal-tightened variables sampling scheme

The tightened-normal-tightened (TNT) attributes sampling scheme was devised by Calvin (1977). In this paper, a TNT Scheme with variables sampling plan as the reference plan, designated as TNTVSS (n_σ; k_T, k_N) is introduced, where n_σ is the sample size under the reference plan, and k_T and k_N are the acceptance constants corresponding to tightened and normal plans respectively. The behaviour of OC curves of the TNTVSS (n_σ; k_T, k_N) is studied. The efficiency of TNTVSS (n_σ; k_T, k_N) with respect to smaller sample sizes has been established over the attributes scheme. The TNTVSS is matched with the TNT (n; c_N, c_T) of Vijayaraghavan and Soundararajan (1996), for the specified points on the OC curves, namely (p₁, α) and (p₂, β) and it is shown that the sample size of the variables scheme is much smaller than that of the attributes scheme. The TNT scheme with an unknown σ variables plan as the reference plan is also introduced along with the procedure of selection of the parameters. The method of designing the scheme based on the given AQL (Acceptable Quality level), α (producer's risk), LQL (Limiting Quality Level) and β (consumer's risk) is indicated. Among the class of TNTVSS which exists, for a given (p₁,α) and (p₂, β), a scheme, which will have a more steeper OC curve than that of any other scheme, is identified and given.     

Remarks on multivariate analogues of kendall's tau

The aim of the paper is to discuss a decision theoretical interpretation of multivariate analogues of Kendall's tau.     

Some competitors of the mood test for the two-sample scale problem

Let X_l,…,X_n (Y_l,…,Y_m) be a random sample from an absolutely continuous distribution with distribution function F(G).A class of distribution-free tests based on U-statistics is proposed for testing the equality of F and G against the alternative that X's are more dispersed then Y's. Let 2 ? C ? n and 2 ? d ? m be two fixed integers. Let ?_c,d(X_il,…,X_ic ; Y_jl,…,X_jd)=1(-1)when max as well as min of {X_il,…,X_ic ; Y_jl,…,Y_jd } are some X_i's (Y_j's)and zero oterwise. Let S_c,d be the U-statistic corresponding to ?_c,d.In case of equal sample sizes, S22 is equivalent to Mood's Statistic.Large values of S_c,d are significant and these tests are quite efficient     

A note on Stein's lemma for multivariate elliptical distributions

When two random variables are bivariate normally distributed Stein's original lemma allows to conveniently express the covariance of the first variable with a function of the second. Landsman and Neslehova (2008) extend this seminal result to the family of multivariate elliptical distributions. In this paper we use the technique of conditioning to provide a more elegant proof for their result. In doing so, we also present a new proof for the classical linear regression result that holds for the elliptical family.     

THE CONTROL CHART FOR INDIVIDUAL OBSERVATIONS FROM A MULTIVARIATE NON-NORMAL DISTRIBUTION

The Hotelling's T²statistic has been used in constructing a multivariate control chart for individual observations. In Phase II operations, the distribution of the T²statistic is related to the F distribution provided the underlying population is multivariate normal. Thus, the upper control limit (UCL) is proportional to a percentile of the F distribution. However, if the process data show sufficient evidence of a marked departure from multivariate normality, the UCL based on the F distribution may be very inaccurate. In such situations, it will usually be helpful to determine the UCL based on the percentile of the estimated distribution for T². In this paper, we use a kernel smoothing technique to estimate the distribution of the T²statistic as well as of the UCL of the T²chart, when the process data are taken from a multivariate non-normal distribution. Through simulations, we examine the sample size requirement and the in-control average run length of the T²control chart for sample observations taken from a multivariate exponential distribution. The paper focuses on the Phase II situation with individual observations.     

Algorithms

Abstract

The main reason for the limited use of multivariate discrete models is the difficulty in calculating the required probabilities. The task is usually undertaken via recursive relationships which become quite computationally demanding for high dimensions and large values. The present paper discusses efficient algorithms that make use of the recurrence relationships in a manner that reduces the computational effort and thus allow for easy and cheap calculation of the probabilities. The most common multivariate discrete distribution, the multivariate Poisson distribution is treated. Real data problems are provided to motivate the use of the proposed strategies. Extensions of our results are discussed. It is shown that probabilities, for a large family of multivariate distributions, can be computed efficiently via our algorithms.     

Multivariate analysis with an autoregressive covariance model

This paper addresses the problem of testing the multivariate linear hypothesis when the errors follow an antedependence model (Gabriel, 1961, 1962). Antedependence can be formulated as a nonstationary autoregressive model of general order. Three test statistics are derived that provide analogs to three commonly used MANOVA statistics: Wilks' Lambda, the Lawley-Hotelling Trace, and Pillai's Trace. Formulas are given for each of these statistics that show how they can be obtained From any statistical computing package that calculates the usual MANOVA statistics. These antedependent statistics would be appropriate in analyzing certain multivariate data sets in which repeated measurements are taken on the same subjects over a period of time.     

On the analysis of multivariate repeated measures designs

The relationship between the mixed-model analysis and multivariate approach to a repeated measures design with multiple responses is presented. It is shown that by taking the trace of the appropriate submatrix of the hypothesis (error) sums of squares and crossproducts (SSCP) matrix obtained from the multivariate approach, one can get the hypothesis (error) SSCP matrix for the mixed-model analysis. Thus, when analyzing data from a multivariate repeated measures design, it is advantageous to use the multivariate approach because the result of the mixed-model analysis can also be obtained without additional computation.