Robust methods of estimation of regression coefficients 1

Recently, cumulative residual entropy (CRE) has been found to be a new measure of information that parallels Shannon's entropy (see Rao et al. [Cumulative residual entropy: A new measure of information, IEEE Trans. Inform. Theory. 50(6) (2004), pp. 1220–1228] and Asadi and Zohrevand [On the dynamic cumulative residual entropy, J. Stat. Plann. Inference 137 (2007), pp. 1931–1941]). Motivated by this finding, in this paper, we introduce a generalized measure of it, namely cumulative residual Renyi's entropy, and study its properties. We also examine it in relation to some applied problems such as weighted and equilibrium models. Finally, we extend this measure into the bivariate set-up and prove certain characterizing relationships to identify different bivariate lifetime models.     

Hayter and Tsui's test with double sampling for the vector mean of multivariate normal population

Abstract

In this paper, we introduce a version of Hayter and Tsui's statistical test with double sampling for the vector mean of a population under multivariate normal assumption. A study showed that this new test was more or as efficient than the well-known Hotelling's T² with double sampling. Some nice features of Hayter and Tsui's test are its simplicity of implementation and its capability of identifying the errant variables when the null hypothesis is rejected. Taking that into consideration, a new control chart called HTDS is also introduced as a tool to monitor multivariate process vector mean when using double sampling.     

Bounds on Bivariate Distribution Functions with Given Margins and Known Values at Several Points

Let H(x, y) be a continuous bivariate distribution function with known marginal distribution functions F(x) and G(y). Suppose the values of H are given at several points, H(x _i , y _i ) = θ _i , i = 1, 2,…, n. We first discuss conditions for the existence of a distribution satisfying these conditions, and present a procedure for checking if such a distribution exists. We then consider finding lower and upper bounds for such distributions. These bounds may be used to establish bounds on the values of Spearman's ρ and Kendall's τ. For n = 2, we present necessary and sufficient conditions for existence of such a distribution function and derive best-possible upper and lower bounds for H(x, y). As shown by a counter-example, these bounds need not be proper distribution functions, and we find conditions for these bounds to be (proper) distribution functions. We also present some results for the general case, where the values of H(x, y) are known at more than two points. In view of the simplification in notation, our results are presented in terms of copulas, but they may easily be expressed in terms of distribution functions.     

A NEW TEST FOR EQUALITY OF VARIANCES FOR k NORMAL POPULATIONS

ABSTRACT

A simple test based on Gini's mean difference is proposed to test the hypothesis of equality of population variances. Using 2000 replicated samples and empirical distributions, we show that the test compares favourably with Bartlett's and Levene's test for the normal population. Also, it is more powerful than Bartlett's and Levene's tests for some alternative hypotheses for some non-normal distributions and more robust than the other two tests for large sample sizes under some alternative hypotheses. We also give an approximate distribution to the test statistic to enable one to calculate the nominal levels and P-values.     

Inferences Based on a Skipped Correlation Coefficient

The most popular method for trying to detect an association between two random variables is to test H ₀ ?:?ρ=0, the hypothesis that Pearson's correlation is equal to zero. It is well known, however, that Pearson's correlation is not robust, roughly meaning that small changes in any distribution, including any bivariate normal distribution as a special case, can alter its value. Moreover, the usual estimate of ρ, r, is sensitive to only a few outliers which can mask a true association. A simple alternative to testing H ₀ ?:?ρ =0 is to switch to a measure of association that guards against outliers among the marginal distributions such as Kendall's tau, Spearman's rho, a Winsorized correlation, or a so-called percentage bend correlation. But it is known that these methods fail to take into account the overall structure of the data. Many measures of association that do take into account the overall structure of the data have been proposed, but it seems that nothing is known about how they might be used to detect dependence. One such measure of association is selected, which is designed so that under bivariate normality, its estimator gives a reasonably accurate estimate of ρ. Then methods for testing the hypothesis of a zero correlation are studied.     

Test for the equality of several correlation coefficients

We derive two C(α) statistics and the likelihood-ratio statistic for testing the equality of several correlation coefficients, from k ≥ 2 independent random samples from bivariate normal populations. The asymptotic relationship of the C(α) tests, the likelihood-ratio test, and a statistic based on the normality assumption of Fisher's Z-transform of the sample correlation coefficient is established. A comparative performance study, in terms of size and power, is then conducted by Monte Carlo simulations. The likelihood-ratio statistic is often too liberal, and the statistic based on Fisher's Z-transform is conservative. The performance of the two C(α) statistics is identical. They maintain significance level well and have almost the same power as the other statistics when empirically calculated critical values of the same size are used. The C(α) statistic based on a noniterative estimate of the common correlation coefficient (based on Fisher's Z-transform) is recommended.     

Gluing Copulas

We present a new way of constructing n-copulas, by scaling and gluing finitely many n-copulas. Gluing for bivariate copulas produces a copula that coincides with the independence copula on some grid of horizontal and vertical sections. Examples illustrate how gluing can be applied to build complicated copulas from simple ones. Finally, we investigate the analytical as well as statistical properties of the copulas obtained by gluing, in particular, the behavior of Spearman's ρ and Kendall's τ.     

Testing for Zero Intercept in Multivariate Normal Regression Using the Univariate Multiple-Regression F Test

A likelihood ratio test is derived for comparing the performance potential of a subset of a population of financial assets to the performance potential of the entire population. The test is shown to be equivalent to a test for zero intercept in a multivariate normal regression model. Rao's F approximation to Wilks' Lamda is shown to be equivalent in this case to the conventional F test used to test the significance of a subset of regressors in a univariate multiple-regression model. The test is illustrated using a sample of returns from ten stocks from the New York Stock Exchange.     

Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles

Let (X, Y) be a bivariate random vector whose distribution function H(x, y) belongs to the class of bivariate extreme-value distributions. If F₁ and F₂ are the marginals of X and Y, then H(x, y) = C{F₁(x),F₂(y)}, where C is a bivariate extreme-value dependence function. This paper gives the joint distribution of the random variables Z = {log F₁(X)}/{log F₁(X)F₂(Y)} and W = C{F₁{(X),F₂(Y)}. Using this distribution, an algorithm to generate random variables having bivariate extreme-value distribution is présentés. Furthermore, it is shown that for any bivariate extreme-value dependence function C, the distribution of the random variable W = C{F₁(X),F₂(Y)} belongs to a monoparametric family of distributions. This property is used to derive goodness-of-fit statistics to determine whether a copula belongs to an extreme-value family.     

Sample Sizes in the Interval Estimation of the Correlation Coefficient

An expression is derived for the maximum length of the interval estimator of the correlation coefficient, p, under bivariate normal assumptions. The prespecification of this minimum attainable precision and the confidence level results in an expression for the sample size required. An approximate expression for the sample size is proposed and is numerically shown to be as good as or better than that based on the Fisher's Z transformation.     

Probabilistic tolerance design for a subsystem under burr distribution using taguchi's loss functions

Taguchi (1986) has derived tolerances for subcomponents, subsystems, parts and materials in which the relationship between a higher-level (Y) and a lower-level (X) quality characteristic is assumed to be deterministic and linear, namely, Y=_α+βX, without an error term. Tsai (1990) developed a probabilistic tolerance design for a subsystem in which a bivariate normal distribution between the above two quality characteristics as well as Taguchi's quadratic loss function were considered together to develop a closed form solution of the tolerance design for a subsystem. The Burr family is very rich for fitting sample data, and has positive domain. A bivariate Burr distribution can describe a nonlinear relationship between two quality characteristics, hence, it is adopted instead of a bivariate normal distribution and the simple solutions of three probabilistic tolerance desings for a subsystem are obtained for three cases of “nominal-is-best”, “smaller-is-berrer”, and “larger-is-beter” quality characteristics, by using Taguchi’ los functions, respectively.     

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.     

Effects of the generalized Box–Cox transformation on Type I error rate and power of Hotelling's T 2

Most multivariate statistical techniques rely on the assumption of multivariate normality. The effects of nonnormality on multivariate tests are assumed to be negligible when variance–covariance matrices and sample sizes are equal. Therefore, in practice, investigators usually do not attempt to assess multivariate normality. In this simulation study, the effects of skewed and leptokurtic multivariate data on the Type I error and power of Hotelling's T ² were examined by manipulating distribution, sample size, and variance–covariance matrix. The empirical Type I error rate and power of Hotelling's T ² were calculated before and after the application of generalized Box–Cox transformation. The findings demonstrated that even when variance–covariance matrices and sample sizes are equal, small to moderate changes in power still can be observed.     

An Adjustment to the Bartlett's Test for Small Sample Size

The Bartlett's test (1937) for equality of variances is based on the χ² distribution approximation. This approximation deteriorates either when the sample size is small (particularly < 4) or when the population number is large. According to a simulation investigation, we find a similar varying trend for the mean differences between empirical distributions of Bartlett's statistics and their χ² approximations. By using the mean differences to represent the distribution departures, a simple adjustment approach on the Bartlett's statistic is proposed on the basis of equal mean principle. The performance before and after adjustment is extensively investigated under equal and unequal sample sizes, with number of populations varying from 3 to 100. Compared with the traditional Bartlett's statistic, the adjusted statistic is distributed more closely to χ² distribution, for homogeneity samples from normal populations. The type I error is well controlled and the power is a little higher after adjustment. In conclusion, the adjustment has good control on the type I error and higher power, and thus is recommended for small samples and large population number when underlying distribution is normal.     

A new revised version of McNemar's test for paired binary data

By considering separately B and C, the frequencies of individuals who consistently gave positive or negative answers in before and after responses, a new revised version of McNemar's test is derived. It improves upon Lu's revised formula, which considers B and C together. When both B and C are 0, the new revised version produces the same results as McNemar's test. When one of B and C is 0, the new revised test produces the same results as Lu's version. Compared to Lu's version, the new revised test is a more complete revision of McNemar's test.     

Transformations for Testing the Fit of the Inverse-Gaussian Distribution

Abstract

Use of the MVUE for the inverse-Gaussian distribution has been recently proposed by Nguyen and Dinh [Nguyen, T. T., Dinh, K. T. (2003). Exact EDF goodnes-of-fit tests for inverse Gaussian distributions. Comm. Statist. (Simulation and Computation) 32(2):505–516] where a sequential application based on Rosenblatt's transformation [Rosenblatt, M. (1952). Remarks on a multivariate transformation. Ann. Math. Statist. 23:470–472] led the authors to solve the composite goodness-of-fit problem by solving the surrogate simple goodness-of-fit problem, of testing uniformity of the independent transformed variables. In this note, we observe first that the proposal is not new since it was proposed in a rather general setting in O'Reilly and Quesenberry [O'Reilly, F., Quesenberry, C. P. (1973). The conditional probability integral transformation and applications to obtain composite chi-square goodness-of-fit tests. Ann. Statist. I:74–83]. It is shown on the other hand that the results in the paper of Nguyen and Dinh (2003) are incorrect in their Sec. 4, specially the Monte Carlo figures reported. Power simulations are provided here comparing these corrected results with two previously reported goodness-of-fit tests for the inverse-Gaussian; the modified Kolmogorov–Smirnov test in Edgeman et al. [Edgeman, R. L., Scott, R. C., Pavur, R. J. (1988). A modified Kolmogorov-Smirnov test for inverse Gaussian distribution with unknown parameters. Comm. Statist. 17(B): 1203–1212] and the A ² based method in O'Reilly and Rueda [O'Reilly, F., Rueda, R. (1992). Goodness of fit for the inverse Gaussian distribution. T Can. J. Statist. 20(4):387–397]. The results show clearly that there is a large loss of power in the method explored in Nguyen and Dinh (2003) due to an implicit exogenous randomization.     

Calculating the mvue of the fraction nonconforming for the bivariate normal

The minimum variance unbiased estimator of the proportion lying outside an m-dimensional rectangle for multivariate normal populations was derived by Baillie (1987a, b). The estimator is a natural extension of a univariate estimator widely used in acceptance sampling. Computation of the multivariate estimator is nontrivial; one must integrate a multivariate density over the intersection of an m-dimensional ellipsoid and an m-dimensional rectangle. We propose an algorithm for the bivariate case which involves a one-dimensional numerical integration and calls to routines for either an incomplete beta function or a Student's t cumulative distribution function     

Concordance coefficients to measure the agreement among several sets of ranks

In this paper, two measures of agreement among several sets of ranks, Kendall's concordance coefficient and top-down concordance coefficient, are reviewed. In order to illustrate the utility of these measures, two examples, in the fields of health and sports, are presented. A Monte Carlo simulation study was carried out to compare the performance of Kendall's and top-down concordance coefficients in detecting several types and magnitudes of agreements. The data generation scheme was developed in order to induce an agreement with different intensities among m (m>2) sets of ranks in non-directional and directional rank agreement scenarios. The performance of each coefficient was estimated by the proportion of rejected null hypotheses, assessed at 5% significance level, when testing whether the underlying population concordance coefficient is sufficiently greater than zero. For the directional rank agreement scenario, the top-down concordance coefficient allowed to achieve a percentage of significant concordances that was higher than the one achieved by Kendall's concordance coefficient. Mainly, when the degree of agreement was small, the results of the simulation study pointed to the advantage of using a weighted rank concordance, namely the top-down concordance coefficient, simultaneously with Kendall's concordance coefficient, enabling the detection of agreement (in a top-down sense) in situations not detected by Kendall's concordance coefficient.     

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     

Extensions of Stein's Lemma for the Skew-Normal Distribution

When two random variables have a bivariate normal distribution, Stein's lemma (Stein, 1973 Stein , C. M. ( 1973 ). Estimation of the mean of a multivariate normal distribution . Proc. Prague Symp. Asymptotic Statist. 345 – 381 . [Google Scholar] 1981 Stein , C. M. ( 1981 ). Estimation of the mean of a multivariate normal distribution . Ann. Statist. 9 : 1135 – 1151 .[Crossref], [Web of Science ®] , [Google Scholar]), provides, under certain regularity conditions, an expression for the covariance of the first variable with a function of the second. An extension of the lemma due to Liu (1994 Liu , J. S. ( 1994 ). Siegel's formula via Stein's identities . Statist. Probab. Lett. 21 : 247 – 251 .[Crossref], [Web of Science ®] , [Google Scholar]) as well as to Stein himself establishes an analogous result for a vector of variables which has a multivariate normal distribution. The extension leads in turn to a generalization of Siegel's (1993 Siegel , A. F. ( 1993 ). A surprising covariance involving the minimum of multivariate normal variables . J. Amer. Statist. Assoc. 88 : 77 – 80 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) formula for the covariance of an arbitrary element of a multivariate normal vector with its minimum element. This article describes extensions to Stein's lemma for the case when the vector of random variables has a multivariate skew-normal distribution. The corollaries to the main result include an extension to Siegel's formula. This article was motivated originally by the issue of portfolio selection in finance. Under multivariate normality, the implication of Stein's lemma is that all rational investors will select a portfolio which lies on Markowitz's mean-variance efficient frontier. A consequence of the extension to Stein's lemma is that under multivariate skew-normality, rational investors will select a portfolio which lies on a single mean-variance-skewness efficient hyper-surface.