Estimation of multivariate conditional-tail-expectation using Kendall's process

This paper deals with the problem of estimating the multivariate version of the Conditional-Tail-Expectation, proposed by Di Bernardino et al. [(2013), ‘Plug-in Estimation of Level Sets in a Non-Compact Setting with Applications in Multivariable Risk Theory’, ESAIM: Probability and Statistics, (17), 236–256]. We propose a new nonparametric estimator for this multivariate risk-measure, which is essentially based on Kendall's process [Genest and Rivest, (1993), ‘Statistical Inference Procedures for Bivariate Archimedean Copulas’, Journal of American Statistical Association, 88(423), 1034–1043]. Using the central limit theorem for Kendall's process, proved by Barbe et al. [(1996), ‘On Kendall's Process’, Journal of Multivariate Analysis, 58(2), 197–229], we provide a functional central limit theorem for our estimator. We illustrate the practical properties of our nonparametric estimator on simulations and on two real test cases. We also propose a comparison study with the level sets-based estimator introduced in Di Bernardino et al. [(2013), ‘Plug-In Estimation of Level Sets in A Non-Compact Setting with Applications in Multivariable Risk Theory’, ESAIM: Probability and Statistics, (17), 236–256] and with (semi-)parametric approaches.     

Mann–Whitney test with empirical likelihood methods for pretest–posttest studies

Pretest–posttest studies are an important and popular method for assessing the effectiveness of a treatment or an intervention in many scientific fields. While the treatment effect, measured as the difference between the two mean responses, is of primary interest, testing the difference of the two distribution functions for the treatment and the control groups is also an important problem. The Mann–Whitney test has been a standard tool for testing the difference of distribution functions with two independent samples. We develop empirical likelihood-based (EL) methods for the Mann–Whitney test to incorporate the two unique features of pretest–posttest studies: (i) the availability of baseline information for both groups; and (ii) the structure of the data with missing by design. Our proposed methods combine the standard Mann–Whitney test with the EL method of Huang, Qin and Follmann [(2008), ‘Empirical Likelihood-Based Estimation of the Treatment Effect in a Pretest–Posttest Study’, Journal of the American Statistical Association, 103(483), 1270–1280], the imputation-based empirical likelihood method of Chen, Wu and Thompson [(2015), ‘An Imputation-Based Empirical Likelihood Approach to Pretest–Posttest Studies’, The Canadian Journal of Statistics accepted for publication], and the jackknife empirical likelihood method of Jing, Yuan and Zhou [(2009), ‘Jackknife Empirical Likelihood’, Journal of the American Statistical Association, 104, 1224–1232]. Theoretical results are presented and finite sample performances of proposed methods are evaluated through simulation studies.     

On the estimation of extropy

Recently, Lad, Sanfilippo, and Agro [(2015), ‘Extropy: Complementary Dual of Entropy’, Statistical Science, 30, 40–58.] showed the measure of entropy has a complementary dual, which is termed extropy. The present article introduces some estimators of the extropy of a continuous random variable. Properties of the proposed estimators are stated, and comparisons are made with Qiu and Jia’s estimators [(2018a), ‘Extropy Estimators with Applications in Testing uniformity’, Journal of Nonparametric Statistics, 30, 182–196]. The results indicate that the proposed estimators have a smaller mean squared error than competing estimators. A real example is presented and analysed.     

A likelihood ratio like permutation test for one way designs

We consider permutation tests based on a likelihood ratio like statistic for the one way or k sample design used in an example in Kolassa and Robinson [(2011), ‘Saddlepoint Approximations for Likelihood Ratio Like Statistics with Applications to Permutation Tests’, Annals of Statistics, 39, 3357–3368]. We give explicitly the region in which the statistic exists, obtaining results which permit calculation of the statistic on the boundary of this region. Numerical examples are given to illustrate improvement in the power of the tests compared to the classical statistics for long-tailed error distributions and no loss of power for normal error distributions.     

Improved local polynomial estimation in time series regression

We propose a modification of local polynomial estimation which improves the efficiency of the conventional method when the observation errors are correlated. The procedure is based on a pre-transformation of the data as a generalization of the pre-whitening procedure introduced by Xiao et al. [(2003), ‘More Efficient Local Polynomial Estimation in Nonparametric Regression with Autocorrelated Errors’, Journal of the American Statistical Association, 98, 980–992]. While these authors assumed a linear process representation for the error process, we avoid any structural assumption. We further allow the regressors and the errors to be dependent. More importantly, we show that the inclusion of both leading and lagged variables in the approximation of the error terms outperforms the best approximation based on lagged variables only. Establishing its asymptotic distribution, we show that the proposed estimator is more efficient than the standard local polynomial estimator. As a by-product we prove a suitable version of a central limit theorem which allows us to improve the asymptotic normality result for local polynomial estimators by Masry and Fan [(1997), ‘Local Polynomial Estimation of Regression Functions for Mixing Processes’, Scandinavian Journal of Statistics, 24, 165–179]. A simulation study confirms the efficiency of our estimator on finite samples. An application to climate data also shows that our new method leads to an estimator with decreased variability.     

On shrinking minimax convergence in nonparametric statistics

‘?…?if we are prepared to assume that the unknown density has k derivatives, then?…?the optimal mean integrated squared error is of order n^{?2 k/(2 k+1)}?…?’ The citation is from Silverman [(1986), Density Estimation for Statistics and Data Analysis, London: Chapman &; Hall] and its assertion is based on a classical minimax lower bound which is the pillar of the modern nonparametric statistics. This paper proposes a new minimax methodology that implies a faster decreasing minimax lower bound that is attainable by a data-driven estimator, and the same estimator is also minimax under the classical approach. The recommendation is to test performance of estimators via the new and classical minimax approaches.     

Cytometry inference through adaptive atomic deconvolution

In this paper, we consider a statistical estimation problem known as atomic deconvolution. Introduced in reliability, this model has a direct application when considering biological data produced by flow cytometers. From a statistical point of view, we aim at inferring the percentage of cells expressing the selected molecule and the probability distribution function associated with its fluorescence emission. We propose here an adaptive estimation procedure based on a previous deconvolution procedure introduced by Es, Gugushvili, and Spreij [(2008), ‘Deconvolution for an atomic distribution’, Electronic Journal of Statistics, 2, 265–297] and Gugushvili, Es, and Spreij [(2011), ‘Deconvolution for an atomic distribution: rates of convergence’, Journal of Nonparametric Statistics, 23, 1003–1029]. For both estimating the mixing parameter and the mixing density automatically, we use the Lepskii method based on the optimal choice of a bandwidth using a bias-variance decomposition. We then derive some convergence rates that are shown to be minimax optimal (up to some log terms) in Sobolev classes. Finally, we apply our algorithm on the simulated and real biological data.     

Data-driven smooth tests for the extreme value distribution

In this paper we present data-driven smooth tests for the extreme value distribution. These tests are based on a general idea of construction of data-driven smooth tests for composite hypotheses introduced by Inglot, T., Kallenberg, W. C. M. and Ledwina, T. [(1997). Data-driven smooth tests for composite hypotheses. Ann. Statist., 25, 1222–1250] and its modification for location-scale family proposed in Janic-Wróblewska, A. [(2004). Data-driven smooth test for a location-scale family. Statistics, in press]. Results of power simulations show that the newly introduced test performs very well for a wide range of alternatives and is competitive with other commonly used tests for the extreme value distribution.     

A new estimator of entropy and its application in testing normality

In this paper, we introduce a new estimator of entropy of a continuous random variable. We compare the proposed estimator with the existing estimators, namely, Vasicek [A test for normality based on sample entropy, J. Roy. Statist. Soc. Ser. B 38 (1976), pp. 54–59], van Es [Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Statist. 19 (1992), pp. 61–72], Correa [A new estimator of entropy, Commun. Statist. Theory and Methods 24 (1995), pp. 2439–2449] and Wieczorkowski-Grzegorewski [Entropy estimators improvements and comparisons, Commun. Statist. Simulation and Computation 28 (1999), pp. 541–567]. We next introduce a new test for normality. By simulation, the powers of the proposed test under various alternatives are compared with normality tests proposed by Vasicek (1976) and Esteban et al. [Monte Carlo comparison of four normality tests using different entropy estimates, Commun. Statist.–Simulation and Computation 30(4) (2001), pp. 761–785].     

Variable selection for additive model via cumulative ratios of empirical strengths total

We propose a data-driven method to select significant variables in additive model via spline estimation. The additive structure of the regression model is imposed to overcome the ‘curse of dimensionality’, while the spline estimators provide a good approximation to the additive components of the model. The additive components are ordered according to their empirical strengths, and the significant variables are chosen at the first crossing of a predetermined threshold by the CUmulative Ratios of Empirical Strengths Total of the components. Consistency of the proposed method is established when the number of variables are allowed to diverge with sample size, while extensive Monte-Carlo study demonstrates superior performance of the proposed method and its advantages over the BIC method of Huang and Yang [(2004), ‘Identification of Nonlinear: Additive Autoregressive Models’, Journal of the Royal Statistical Society Series B, 66, 463–477] in terms of speed and accuracy.     

On the entropy estimators

The paper introduces an estimator of the entropy of a continuous random variable. The estimator is obtained by modifying the estimator proposed by Ebrahimi et al. [Two measures of sample entropy, Statist. Probab. Lett. 20 (1994), pp. 225–234]. The consistency of the estimator is proved and comparisons are made with Vasicek's estimator [A test for normality based on sample entropy, J. R. Stat. Soc. Ser. B 38 (1976), pp. 54–59], van Es estimator [Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Statist. 19 (1992), pp. 61–72], Ebrahimi et al. estimator and Correa estimator [A new estimator of entropy, Comm. Statist. Theory Methods 24 (1995), pp. 2439–2449]. The results indicate that the proposed estimator has smaller mean-squared error than above estimators. A real example is presented and analysed.     

On the moment generating functions for integer valued random variables

Abstract

In this note, we give explicit expressions of moment generating functions for integer valued random variables in both univariate and multivariate cases, which extend the results obtained by Nadarajah and Mitov [Communications in Statistics–Theory and Methods, 32, 2003, 47–60] and more recently by Chakraborti, Jardim and Epprecht [The American Statistician, 2017], Kwong and Nadarajah [Communications in Statistics–Theory and Methods, 2017]. Some examples are also discussed.     

On improving convergence rate of Bernstein polynomial density estimator

This paper is concerned with the Bernstein estimator [Vitale, R.A. (1975), ‘A Bernstein Polynomial Approach to Density Function Estimation’, in Statistical Inference and Related Topics, ed. M.L. Puri, 2, New York: Academic Press, pp. 87–99] to estimate a density with support [0, 1]. One of the major contributions of this paper is an application of a multiplicative bias correction [Terrell, G.R., and Scott, D.W. (1980), ‘On Improving Convergence Rates for Nonnegative Kernel Density Estimators’, The Annals of Statistics, 8, 1160–1163], which was originally developed for the standard kernel estimator. Moreover, the renormalised multiplicative bias corrected Bernstein estimator is studied rigorously. The mean squared error (MSE) in the interior and mean integrated squared error of the resulting bias corrected Bernstein estimators as well as the additive bias corrected Bernstein estimator [Leblanc, A. (2010), ‘A Bias-reduced Approach to Density Estimation Using Bernstein Polynomials’, Journal of Nonparametric Statistics, 22, 459–475] are shown to be O(n^?8/9) when the underlying density has a fourth-order derivative, where n is the sample size. The condition under which the MSE near the boundary is O(n^?8/9) is also discussed. Finally, numerical studies based on both simulated and real data sets are presented.     

On a new goodness‐of‐fit process for families of copulas

A goodness‐of‐fit procedure is proposed for parametric families of copulas. The new test statistics are functionals of an empirical process based on the theoretical and sample versions of Spearman's dependence function. Conditions under which this empirical process converges weakly are seen to hold for many families including the Gaussian, Frank, and generalized Farlie–Gumbel–Morgenstern systems of distributions, as well as the models with singular components described by Durante [Durante ( 2007 ) Comptes Rendus Mathématique. Académie des Sciences. Paris, 344, 195–198]. Thanks to a parametric bootstrap method that allows to compute valid P‐values, it is shown empirically that tests based on Cramér–von Mises distances keep their size under the null hypothesis. Simulations attesting the power of the newly proposed tests, comparisons with competing procedures and complete analyses of real hydrological and financial data sets are presented. The Canadian Journal of Statistics 37: 80‐101; 2009 © 2009 Statistical Society of Canada     

General results for the Kumaraswamy-G distribution

For any continuous baseline G distribution [G.M. Cordeiro and M. de Castro, A new family of generalized distributions, J. Statist. Comput. Simul. 81 (2011), pp. 883–898], proposed a new generalized distribution (denoted here with the prefix ‘Kw-G’ (Kumaraswamy-G)) with two extra positive parameters. They studied some of its mathematical properties and presented special sub-models. We derive a simple representation for the Kw-G density function as a linear combination of exponentiated-G distributions. Some new distributions are proposed as sub-models of this family, for example, the Kw-Chen [Z.A. Chen, A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statist. Probab. Lett. 49 (2000), pp. 155–161], Kw-XTG [M. Xie, Y. Tang, and T.N. Goh, A modified Weibull extension with bathtub failure rate function, Reliab. Eng. System Safety 76 (2002), pp. 279–285] and Kw-Flexible Weibull [M. Bebbington, C.D. Lai, and R. Zitikis, A flexible Weibull extension, Reliab. Eng. System Safety 92 (2007), pp. 719–726]. New properties of the Kw-G distribution are derived which include asymptotes, shapes, moments, moment generating function, mean deviations, Bonferroni and Lorenz curves, reliability, Rényi entropy and Shannon entropy. New properties of the order statistics are investigated. We discuss the estimation of the parameters by maximum likelihood. We provide two applications to real data sets and discuss a bivariate extension of the Kw-G distribution.     

Asymptotically optimal shrinkage estimates for non-normal data

Motivated by several practical issues, we consider the problem of estimating the mean of a p-variate population (not necessarily normal) with unknown finite covariance. A quadratic loss function is used. We give a number of estimators (for the mean) with their loss functions admitting expansions to the order of p ^?1/2 as p→∞. These estimators contain Stein's [Inadmissibility of the usual estimator for the mean of a multivariate normal population, in Proceedings of the Third Berkeley Symposium in Mathematical Statistics and Probability, Vol. 1, J. Neyman, ed., University of California Press, Berkeley, 1956, pp. 197–206] estimate as a particular case and also contain ‘multiple shrinkage’ estimates improving on Stein's estimate. Finally, we perform a simulation study to compare the different estimates.     

FACTORIAL ANALYSIS OF RECIDIVIST DATA

This paper presents a method of fitting factorial models to recidivism data consisting of the (possibly censored) time to ‘fail’ of individuals, in order to test for differences between groups. Here ‘failure’ means rearrest, reconviction or reincarceration, etc. A proportion P of the sample is assumed to be ‘susceptible’ to failure, i.e. to fail eventually, while the remaining 1-P are ‘immune’, and never fail. Thus failure may be described in two ways: by the probability P that an individual ever fails again (‘probability of recidivism’), and by the rate of failure Λ for the susceptibles. Related analyses have been proposed previously: this paper argues that a factorial approach, as opposed to regression approaches advocated previously, offers simplified analysis and interpretation of these kinds of data. The methods proposed, which are also applicable in medical statistics and reliability analyses, are demonstrated on data sets in which the factors are Parole Type (released to freedom or on parole), Age group (≤ 20 years, 20–40 years, > 40 years), and Marital Status. The outcome (failure) is a return to prison following first or second release.     

A variable selection proposal for multiple linear regression analysis

Variable selection in multiple linear regression models is considered. It is shown that for the special case of orthogonal predictor variables, an adaptive pre-test-type procedure proposed by Venter and Steel [Simultaneous selection and estimation for the some zeros family of normal models, J. Statist. Comput. Simul. 45 (1993), pp. 129–146] is almost equivalent to least angle regression, proposed by Efron et al. [Least angle regression, Ann. Stat. 32 (2004), pp. 407–499]. A new adaptive pre-test-type procedure is proposed, which extends the procedure of Venter and Steel to the general non-orthogonal case in a multiple linear regression analysis. This new procedure is based on a likelihood ratio test where the critical value is determined data-dependently. A practical illustration and results from a simulation study are presented.     

Adaptive bootstrap tests and its competitors in the c-sample scale problem

This paper deals with a study of different types of tests for the two-sided c-sample scale problem. We consider the classical parametric test of Bartlett [M.S. Bartlett, Properties of sufficiency and statistical tests, Proc. R. Stat. Soc. Ser. A. 160 (1937), pp. 268–282] several nonparametric tests, especially the test of Fligner and Killeen [M.A. Fligner and T.J. Killeen, Distribution-free two-sample tests for scale, J. Amer. Statist. Assoc. 71 (1976), pp. 210–213], the test of Levene [H. Levene, Robust tests for equality of variances, in Contribution to Probability and Statistics, I. Olkin, ed., Stanford University Press, Palo Alto, 1960, pp. 278–292] and a robust version of it introduced by Brown and Forsythe [M.B. Brown and A.B. Forsythe, Robust tests for the equality of variances, J. Amer. Statist. Assoc. 69 (1974), pp. 364–367] as well as two adaptive tests proposed by Büning [H. Büning, Adaptive tests for the c-sample location problem – the case of two-sided alternatives, Comm. Statist.Theory Methods. 25 (1996), pp. 1569–1582] and Büning [H. Büning, An adaptive test for the two sample scale problem, Nr. 2003/10, Diskussionsbeiträge des Fachbereich Wirtschaftswissenschaft der Freien Universität Berlin, Volkswirtschaftliche Reihe, 2003]. which are based on the principle of Hogg [R.V. Hogg, Adaptive robust procedures. A partial review and some suggestions for future applications and theory, J. Amer. Statist. Assoc. 69 (1974), pp. 909–927]. For all the tests we use Bootstrap sampling strategies, too. We compare via Monte Carlo Methods all the tests by investigating level α and power β of the tests for distributions with different strength of tailweight and skewness and for various sample sizes. It turns out that the test of Fligner and Killeen in combination with the bootstrap is the best one among all tests considered.