A Generalization of Hotelling's T 2

ABSTRACT

We derive an asymptotic version of Hotelling's T ² for the multivariate proper dispersion models of Jøtrgensen and Lauritzen (2000 Jørgensen , B. , Lauritzen , S. L. ( 2000 ). Multivariate dispersion models . J. Multivariate Anal. 74 : 267 – 281 . [CSA] [CROSSREF]  [Google Scholar]), the main tool being the saddlepoint approximation. Multivariate dispersion models are suitable for positive, directional, compositional, and other non normal data. We illustrate the results by a multivariate gamma model.     

A Similarity Measure Based on Species Proportions

ABSTRACT

There are several indices for measuring the similarity of two populations, including the ratio of the number of shared species to the number of distinct species (Jaccard's index) and the conditional probability of observing a shared species (Smith et al., 1996 Smith , W. , Solow , A. R. , Preston , P. E. ( 1996 ). An estimator of species overlap using a modified beta-binomial model. Biometrics 52 : 1472 – 1477 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). However, these indices only take into account the number of species and species proportions of shared species. In this article, we propose a new similarity index which includes the species proportions of both the shared and non shared species in each population, and also propose a Nonparametric Maximum Likelihood Estimator (NPMLE) for this index. Bootstrap and delta methods are used to evaluate the standard errors of the NPMLE. Based on a loss function, we also compare a class of nonparametric estimators for the proposed index in various situations.     

ROBUST TESTS FOR THE SIGNIFICANCE OF ORTHANT RESTRICTED MEAN VECTOR

Mudholkar and Srivastava [1] Mudholkar, G. S. and Srivastava, D. K. A class of robust stepwise alternatives to Hotelling's T²tests. Submitted to the Journal of Applied Statistics 1999 [Google Scholar]adapted Mudholkar and Subbaiah's [2] Mudholkar, G. S. and Subbaiah, P. 1980. Testing significance of a mean vector–a possible alternative to Hotelling's T². Ann. Inst. Statist. Math., 32(A): 43–52.  [Google Scholar]modified stepwise procedure, using the trimmed means in place of the means and appropriate studentization, to construct robust tests for the significance of a mean vector. They concluded that the robust alternatives provide excellent type I error control, and a substantial gain in power over Hotelling's T ²test in case of heavy tailed populations without significant loss of power when the population is normal. In this paper we adapt the modified stepwise approach to construct simple tests for the significance of the orthant constrained mean vector of a p-variate normal population with unknown covariance matrix, and also for constructing robust tests without assuming normality. The simple normal theory tests have exact type I error, whereas the robust tests provide a reasonably type I error control and substantial power advantage over Perlman's [3] Perlman, M. D. 1969. One-sided testing problems in multivariate analysis. Annals of Mathematical Statistics, 40: 549–567. [Crossref] , [Google Scholar]likelihood ratio test.     

On goodness-of-fit tests for Aalen's additive model based on left-truncated right-censored or doubly censored data

ABSTRACT

Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) proposed goodness-of-fit tests for Aalen's additive risk model. In this article, we demonstrate that the approach of Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) can be applied to left-truncated right-censored (LTRC) data and doubly censored data. A simulation study is conducted to investigate the performance of the proposed tests. The proposed tests are illustrated using heart transplant data.     

The structural Sharpe model under t-distributions

In this paper we consider Sharpe's single-index model or Sharpe's model, by assuming that the returns obtained follow a multivariate t elliptical distribution. Also, given that the returns of the market are not observable, the statistical analysis was made in the context of an errors-in-variables model. In order to analyze the sensibility to possible outliers and/or atypical returns of the maximum likelihood estimators the local influence method [10 Cook, R. D. 1986. Assessment of local influence. J. Roy. Statist. Soc. B, 48: 133–169.  [Google Scholar]] was implemented. The results are illustrated by using a set of shares of companies belonging to the Chilean Stock Market. The main conclusion is that the t model with small degrees of freedom is able to incorporate possible outliers and influential returns in the data.     

The Design of Scoring Schemes for Surveys Using the Block Total Response Method

ABSTRACT

In this article we investigate the design of scoring schemes for surveys using the block total response method. This method was first proposed by Raghavarao and Federer (1979 Raghavarao , D. , Federer , W. T. ( 1979 ). Block total response as an alternative to the randomized response method in surveys . J. Roy. Statist. Soc. 41 ( 1 ): 40 – 45 . [CSA] [CROSSREF]  [Google Scholar]) to provide accurate estimates of the base rates of sensitive characteristics using balanced incomplete block designs. The scoring scheme used in Raghavarao and Federer (1979 Raghavarao , D. , Federer , W. T. ( 1979 ). Block total response as an alternative to the randomized response method in surveys . J. Roy. Statist. Soc. 41 ( 1 ): 40 – 45 . [CSA] [CROSSREF]  [Google Scholar]) did not guarantee anonymity of answers and so the possibility of improving on this basic scoring scheme is considered in this article.     

Stratification: Measuring and Inference

ABSTRACT

Stratification of distribution functions is an important issue in the area of income distributions. Two distribution functions form a perfect stratification if they occupy disjoint ranges on the horizontal axis. Otherwise, there is overlapping. A measure which quantifies the amount of stratification is introduced by Yitzhaki (1994 Yitzhaki , S. ( 1994 ). Economic distance and overlapping of distributions . J. Econometrics 61 : 147 – 159 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), but no procedure for drawing inference is suggested. We develop a consistent estimator of the degree of overlapping and offer a nonparametric procedure for inference. Its limiting distribution, properly standardized, is normal. The asymptotic variance can be estimated using the jackknife method, and simulations show that the suggested procedure works well for sample sizes of 50 (100 for some cases).     

Income inequality versus utility inequality

Bentham (1789 Bentham, J. (1789). Utilitarianism, reproduced by Nabu Public Domain Reprints. Available at: http://www.publicdomainreprints.org. [Google Scholar]) introduced utility as the pursuit of happiness, with happiness defined in his philosophical view as existing if “pleasure” predominated over “pain.” This paper is the first to derive and compare the relative frequency distributions of income and utility in Bentham's classical sense. A utility-based Gini coefficient is formulated from a utility distribution derived from the better known income distribution. A utility-based Lorenz curve is defined as the accumulated sum of pain and pleasure, as Bentham defined them. A utility-based polarization index is created as the ratio of accumulated pain to accumulated pleasure. If raw income data are unavailable, an income distribution can still be estimated from a summary inequality statistic such as the Gini coefficient using the maximum entropy method. In this study, a measure of true income inequality, utility inequality, and the “pain” suffered by the poorest group are quantified and estimated using National Tax Service data from Korea.     

On Testing Sample Selection Bias Under the Multicollinearity Problem

ABSTRACT

This paper reviews and extends the literature on the finite sample behavior of tests for sample selection bias. Monte Carlo results show that, when the “multicollinearity problem” identified by Nawata (1993 Nawata , K. ( 1993 ). A note on the estimation of models with sample-selection biases . Economics Letters 42 : 15 – 24 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) is severe, (i) the t-test based on the Heckman–Greene variance estimator can be unreliable, (ii) the Likelihood Ratio test remains powerful, and (iii) nonnormality can be interpreted as severe sample selection bias by Maximum Likelihood methods, leading to negative Wald statistics. We also confirm previous findings (Leung and Yu, 1996 Leung , S. F. , Yu , S. ( 1996 ). On the choice between sample selection and two-part models . Journal of Econometrics 72 : 197 – 229 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) that the standard regression-based t-test (Heckman, 1979 Heckman , J. J. ( 1979 ). Sample selection bias as a specification error . Econometrica 47 : 153 – 161 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) and the asymptotically efficient Lagrange Multiplier test (Melino, 1982 Melino , A. ( 1982 ). Testing for sample selection bias . Review of Economic Studies 49 : 151 – 153 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), are robust to nonnormality but have very little power.     

Balakrishnan skew logistic distribution

Abstract

Based on Balakrishnan's (2002 Balakrishnan, N. (2002). Discussion on “Skew multivariate models related to hidden truncation and/or selective reporting” by B.C. Arnold and R.J. Beaver. Test 11: 37–39.[Web of Science ®] , [Google Scholar]) novel idea, a new skew logistic distribution is proposed. It contains Azzalini's skew logistic and standard logistic distributions as particular cases. Several mathematical properties of the proposed distribution (including characteristic function, cumulative distribution function, stochastic orderings, and simulation schemes) are derived. A real-data application is used to illustrate practical usefulness.     

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.     

Regularity in individual shopping trips: implications for duration models in marketing

Most models for purchase-timing behavior of households do not take into account that many households have regular and non-shopping days. We propose a statistical model for purchase timing that exploits information on the shopping days of households. The model is formulated in a counting process framework that counts the recurrent purchases for each household over (calendar) time. In our empirical application of yogurt and detergent purchases from the ERIM1 The data can be found at http://gsbww.uchicago.edu/research/mkt/Databases/ERIM/ERIM.html View all notes database, we show that calendar time effects and regular and non-shopping days are important features to include in models for purchase-timing behavior. We find, for instance, that for these product categories the probability of purchasing is 50–60% higher on Saturdays and 70% higher on regular shopping days. We highlight the managerial implications of these model features by simulating some promotional actions.     

The Maximum Probability 2 × c Contingency Tables and the Maximum Probability Points of the Multivariate Hypergeometric Distribution

Abstract

The problem of obtaining the maximum probability 2 × c contingency table with fixed marginal sums, R  = (R ₁, R ₂) and C  = (C ₁, … , C _c ), and row and column independence is equivalent to the problem of obtaining the maximum probability points (mode) of the multivariate hypergeometric distribution MH(R ₁; C ₁, … , C _c ). The most simple and general method for these problems is Joe's (Joe, H. (1988 Joe, H. 1988. Extreme probabilities for contingency tables under row and column independence with application to Fisher's exact test. Commun. Statist. Theory Meth., 17(11): 3677–3685. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Extreme probabilities for contingency tables under row and column independence with application to Fisher's exact test. Commun. Statist. Theory Meth. 17(11):3677–3685.) In this article we study a family of MH's in which a connection relationship is defined between its elements. Based on this family and on a characterization of the mode described in Requena and Martín (Requena, F., Martín, N. (2000 Requena, F. and Martín, N. 2000. Characterization of maximum probability points in the multivariate hypergeometric distribution. Statist. Probab. Lett., 50: 39–47.  [Google Scholar]). Characterization of maximum probability points in the multivariate hypergeometric distribution. Statist. Probab. Lett. 50:39–47.), we develop a new method for the above problems, which is completely general, non recursive, very simple in practice and more efficient than the Joe's method. Also, under weak conditions (which almost always hold), the proposed method provides a simple explicit solution to these problems. In addition, the well-known expression for the mode of a hypergeometric distribution is just a particular case of the method in this article.     

The Modified Pulak and Al-Sultan's Model for Determining the Optimum Process Parameters

Economic selection of process parameters has been an important topic in modern statistical process control. The optimum process parameters setting have a major effect on the expected profit/cost per item. There are some concerns on the problem of setting process parameters. Boucher and Jafari (1991 Boucher , T. O. , Jafari , M. A. ( 1991 ). The optimum target value for single filling operations with quality sampling plans . J. Qual. Technol. 23 : 44 – 47 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) first considered the attribute single sampling plan applied in the selection of process target. Pulak and Al-Sultan (1996 Pulak , M. F. S. , Al-Sultan , K. S. ( 1996 ). The optimum targeting for a single filling operation with rectifying inspection . Omega 24 : 727 – 733 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) extended Boucher and Jafari's model and presented the rectifying inspection plan for determining the optimum process mean. In this article, we further propose a modified Pulak and Al-Sultan model for determining the optimum process mean and standard deviation under the rectifying inspection plan with the average outgoing quality limit (AOQL) protection. Taguchi's (1986 Taguchi , G. ( 1986 ). Introduction to Quality Engineering . Asian Productivity Organization . [Google Scholar]) symmetric quadratic quality loss function is adopted for evaluating the product quality. By solving the modified model, we can obtain the optimum process parameters with the maximum expected profit per item and the specified quality level can be reached.     

Generalizing the Marshall's Inequality

In the article, we generalizes the Marshall's inequality in Marshall (1960 Marshall , A. W. ( 1960 ). A one-sided analog of Kolmogorov's inequality . Ann. Math. Statist. 31 : 483 – 487 .[Crossref] , [Google Scholar]) under the condition that the p-order moment exists by a new approach. Some properties of the inequality and further conclusions are given.     

Nonparametric Estimation and Hypothesis Testing on the Partial Area Under Receiver Operating Characteristic Curves

ABSTRACT

The area under a receiver operating characteristic curve is a useful index of the accuracy of a diagnostic test. When the diagnostic ability of a new biomarker is of interest only in a certain range of specificity, the partial area under the curve becomes desirable. In this article, we extend Bamber's (1975 Bamber , D. ( 1975 ). The area above the ordinal dominance graph and the area below the receiver operating characteristics graph. J. Math. Psychol. 12 : 387 – 415 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) results and show that the partial area under a receiver operating characteristic curve is the probability of a constrained stochastic ordering. We then construct a ‘weighted’ Mann-Whitney statistic as an estimator of the partial area and investigate its statistical properties. A testing procedure is also developed to compare partial area under two receiver operating characteristic curves. The methods are exemplified with data from biomarkers associated with coronary heart disease.     

Multivariate Aging with Archimedean Dependence Structures in High Dimensions

Bivariate aging notions for a vector X of lifetimes based on stochastic comparisons between X and X _t, where X _t is the multivariate residual lifetime after time t > 0, have been studied in Pellerey (2008 Pellerey , F. ( 2008 ). On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas . Kybernetika 44 : 795 – 806 .[Web of Science ®] , [Google Scholar]) under the assumption that the dependence structure in X is described by an Archimedean survival copula. Similar stochastic comparisons between X _t and X _t+s, for all t; s > 0, were considered in Mulero and Pellerey (2010 Mulero , J. , Pellerey , F. ( 2010 ). Bivariate aging properties under Archimedean dependence structures . Commun. Statist. Theor. Meth. 39 : 3108 – 3121 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). In this article, these results are generalized and extended to the multivariate case. Two illustrative examples are also provided.     

Nonparametric Bivariate Estimation with Left-Truncated and Right-Censored Data

ABSTRACT

In this article we consider estimating the bivariate survival function observations where one of the components is subject to left truncation and right censoring and the other is subject to right censoring only. Two types of nonparametric estimators are proposed. One is in the form of inverse-probability-weighted average (Satten and Datta, 2001 Satten , G. A. , Datta , S. ( 2001 ). The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average . Amer. Statist. 55 : 207 – 210 . [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and the other is a generalization of Dabrowska's 1988 Dabrowska , D. M. ( 1988 ). Kaplan–Meier estimate on the plane . Ann. Statist. 18 : 308 – 325 . [Google Scholar] estimator. The two are then compared based on their empirical performances.     

GMM in linear regression for longitudinal data with multiple covariates measured with error

Griliches and Hausman 5 Griliches, Z. and Hausman, J. A. 1986. Errors in variables in panel data. J. Econometrics, 32: 93–118. [Crossref], [Web of Science ®] [Google Scholar] and Wansbeek 11 Wansbeek, T. J. 2001. GMM estimation in panel data models with measurement error. J. Econometrics, 104: 259–268. [Crossref], [Web of Science ®] [Google Scholar] proposed using the generalized method of moments (GMM) to obtain consistent estimators in linear regression models for longitudinal data with measurement error in one covariate, without requiring additional validation or replicate data. For usefulness of this methodology, we must extend it to the more realistic situation where more than one covariate are measured with error. Such an extension is not straightforward, since measurement errors across different covariates may be correlated. By a careful construction of the measurement error correlation structure, we are able to extend Wansbeek's GMM and show that the extended Griliches and Hausman's GMM is equivalent to the extended Wansbeek's GMM. For illustration, we apply the extended GMM to data from two medical studies, and compare it with the naive method and the method assuming only one covariate having measurement error.