Showing papers in "Statistics in 2015"
TL;DR: In this article, the authors provide some exponential inequalities for extended negatively dependent (END) random variables and investigate the complete consistency for the estimator of nonparametric regression model based on end errors.
Abstract: In this paper, we provide some exponential inequalities for extended negatively dependent (END) random variables. By using these exponential inequalities and the truncated method, we investigate the complete consistency for the estimator of nonparametric regression model based on END errors. As an application, the complete consistency for the nearest neighbour estimator is obtained.
70 citations
TL;DR: In this paper, a stochastic differential equation involving standard and fractional Brownian motion with unknown drift parameter is considered, and the authors investigate the standard maximum likelihood estimate of the drift parameter, two non-standard estimates and three estimates for the sequential estimation.
Abstract: We consider a stochastic differential equation involving standard and fractional Brownian motion with unknown drift parameter to be estimated. We investigate the standard maximum likelihood estimate of the drift parameter, two non-standard estimates and three estimates for the sequential estimation. Model strong consistency and some other properties are proved. The linear model and Ornstein–Uhlenbeck model are studied in detail. As an auxiliary result, an asymptotic behaviour of the fractional derivative of the fractional Brownian motion is established.
47 citations
TL;DR: In this paper, modified cumulative sum (CUSUM) procedures have been proposed for the detection of abrupt changes in the regression parameter of multiple time series regression models, that show a higher stability with respect to the time of change than ordinary CUSUM procedures.
Abstract: In a variety of different settings cumulative sum (CUSUM) procedures have been applied for the sequential detection of structural breaks in the parameters of stochastic models. Yet their performance depends strongly on the time of change and is best under early change scenarios. For later changes their finite sample behavior is rather questionable. We therefore propose modified CUSUM procedures for the detection of abrupt changes in the regression parameter of multiple time series regression models, that show a higher stability with respect to the time of change than ordinary CUSUM procedures. The asymptotic distributions of the test statistics and the consistency of the procedures are provided. In a simulation study it is shown that the proposed procedures behave well in finite samples. Finally the procedures are applied to a set of capital asset pricing data related to the Fama–French extension of the CAPM.
33 citations
TL;DR: In this article, asymptotic properties of maximum likelihood estimators for Heston models based on continuous time observations of the log-price process are studied. But the authors distinguish three cases: subcritical (also called ergodic), critical and supercritical.
Abstract: We study asymptotic properties of maximum-likelihood estimators for Heston models based on continuous time observations of the log-price process. We distinguish three cases: subcritical (also called ergodic), critical and supercritical. In the subcritical case, asymptotic normality is proved for all the parameters, while in the critical and supercritical cases, non-standard asymptotic behaviour is described.
30 citations
TL;DR: In this article, a simple EM-type algorithm for iteratively computing maximum-likelihood estimates of the parameters of the censored linear regression model with normal errors to Student-t errors is presented.
Abstract: In this paper, we extend the censored linear regression model with normal errors to Student-t errors. A simple EM-type algorithm for iteratively computing maximum-likelihood estimates of the parameters is presented. To examine the performance of the proposed model, case-deletion and local influence techniques are developed to show its robust aspect against outlying and influential observations. This is done by the analysis of the sensitivity of the EM estimates under some usual perturbation schemes in the model or data and by inspecting some proposed diagnostic graphics. The efficacy of the method is verified through the analysis of simulated data sets and modelling a real data set first analysed under normal errors. The proposed algorithm and methods are implemented in the R package CensRegMod.
28 citations
TL;DR: In this paper, the problem of variable selection in high-dimensional partial linear regression under some model allowing for possibly functional variable is considered, and the procedure studied is that of nonconcave-penalized least squares.
Abstract: The problem of variable selection is considered in high-dimensional partial linear regression under some model allowing for possibly functional variable. The procedure studied is that of nonconcave-penalized least squares. It is shown the existence of a √n/sn-consistent estimator for the vector of pn linear parameters in the model, even when pn tends to ∞ as the sample size n increases (sn denotes the number of influential variables). An oracle property is also obtained for the variable selection method, and the nonparametric rate of convergence is stated for the estimator of the nonlinear functional component of the model. Finally, a simulation study illustrates the finite sample size performance of our procedure.
28 citations
TL;DR: In this article, the authors proposed a one sample test for repeated measures designs and derived its limit distribution for the situation where both the sample size n as well as the dimension d of the observations go to infinity.
Abstract: We propose a novel one sample test for repeated measures designs and derive its limit distribution for the situation where both the sample size n as well as the dimension d of the observations go to infinity. This covers the high-dimensional case with d>n. The tests are based on a quadratic form which involve new unbiased and dimension-stable estimators of different traces of the underlying unrestricted covariance structure. It is shown that the asymptotic distribution of the statistic may be standard normal, standardized χ2-distributed, or even of weighted χ2-form in some situations. To this end, we suggest an approximation technique which is asymptotically valid in the first two cases and provides an accurate approximation for the latter. We motivate and illustrate the application with a sleep lab data set and also discuss the practical meaning of d→∞ in case of repeated measures designs. It turns out that the limit behaviour depends on how the number of repeated measures is increased which is crucial f...
27 citations
TL;DR: The so-called partition function is a sample moment statistic based on blocks of data and it is shown that its behaviour is strongly influenced by the tail of the distribution underlying the data both in independent identically distributed and weakly dependent cases.
Abstract: The so-called partition function is a sample moment statistic based on blocks of data and it is often used in the context of multifractal processes. It will be shown that its behaviour is strongly influenced by the tail of the distribution underlying the data both in independent identically distributed and weakly dependent cases. These results will be used to develop graphical and estimation methods for the tail index of a distribution. The performance of the tools proposed is analysed and compared with other methods by means of simulations and examples.
26 citations
TL;DR: The fractional Ornstein-Uhlenbeck process of the second kind (fOU(2)) is the solution of the Langevin equation with driving noise where B is a fractional Brownian motion with Hurst parameter H(0,... ) as mentioned in this paper.
Abstract: The fractional Ornstein-Uhlenbeck process of the second kind (fOU(2)) is the solution of the Langevin equation with driving noise where B is a fractional Brownian motion with Hurst parameter H(0, ...
26 citations
TL;DR: In this paper, a three-parameter extension of the exponential distribution is introduced and studied, which can be used effectively in modelling survival data, reliability problems, fatigue life studies and hydrological data.
Abstract: A three-parameter extension of the exponential distribution is introduced and studied in this paper. The new distribution is quite flexible and can be used effectively in modelling survival data, reliability problems, fatigue life studies and hydrological data. It can have constant, decreasing, increasing, upside-down bathtub (unimodal), bathtub-shaped and decreasing–increasing–decreasing hazard rate functions. We provide a comprehensive account of the mathematical properties of the new distribution and various structural quantities are derived. We discuss maximum likelihood estimation of the model parameters for complete sample and for censored sample. An empirical application of the new model to real data is presented for illustrative purposes. We hope that the new distribution will serve as an alternative model to other models available in the literature for modelling real data in many areas.
25 citations
TL;DR: In this paper, Shmueli et al. proved their conjecture to be true and discussed for what range of parameters the approximation can be accurately applied, although they conjectured it was also valid for non-integers.
Abstract: By adding a second parameter, Conway and Maxwell created a new distribution for situations where data deviate from the standard Poisson distribution. This new distribution contains a normalization constant expressed as an infinite sum whose summation has no known closed-form expression. Shmueli et al. produced an approximation for this sum but proved that it was valid only for integer values of the second parameter, although they conjectured it was also valid for non-integers. Here we prove their conjecture to be true and discuss for what range of parameters the approximation can be accurately applied.
TL;DR: In this paper, a new class of skew distributions by extending the alpha skew normal distribution proposed by Elal-Olivero was introduced, and the problem of estimating parameters on the basis of a random sample coming from this class of distribution is considered.
Abstract: This paper introduces a new class of skew distributions by extending the alpha skew normal distribution proposed by Elal-Olivero [Elal-Olivero, D. Alpha-skew-normal distribution. Proyecciones. 2010;29:224–240]. Statistical properties of the new family are studied in details. In particular, explicit expressions for the moments and the shape parameters including the skewness and the kurtosis coefficients and the moment generating function are derived. The problem of estimating parameters on the basis of a random sample coming from the new class of distribution is considered. To examine the performance of the obtained estimators, a Monte Carlo simulation study is conducted. Flexibility and usefulness of the proposed family of distributions are illustrated by analysing three real data sets.
TL;DR: HuHuang et al. as discussed by the authors proposed a multimodal skewed extension of normal distribution by applying the general method as in [Huang WJ, Chen YH], for the construction of skew-symmetric distributions by using a trigonometric periodic skew function.
Abstract: A multimodal skewed extension of normal distribution is proposed by applying the general method as in [Huang WJ, Chen YH. Generalized skew-Cauchy distribution. Stat Probab Lett. 2007;77:1137–1147] for the construction of skew-symmetric distributions by using a trigonometric periodic skew function. Some of its distributional properties are investigated. Properties of maximum likelihood estimation of the parameters are studied numerically by simulation. The suitability of the proposed distribution in empirical data modelling is investigated by carrying out comparative fitting of two real-life data sets.
TL;DR: In this article, the estimation of hidden periodicities in a non-linear regression model with stationary noise displaying cyclical dependence was studied and consistency and asymptotic normality were established for the least squares estimates.
Abstract: This paper deals with the estimation of hidden periodicities in a non-linear regression model with stationary noise displaying cyclical dependence. Consistency and asymptotic normality are established for the least-squares estimates.
TL;DR: In this article, the use of generalized inverses in Wald's-type quadratic forms of test statistics having singular normal limiting distributions does not guarantee to obtain chi-square limiting distributions.
Abstract: The use of generalized inverses in Wald's-type quadratic forms of test statistics having singular normal limiting distributions does not guarantee to obtain chi-square limiting distributions. In this article, the use of {2} -inverses for that problem is investigated. Alternatively, Imhof-based test statistics can also be defined, which converge in distribution to weighted sum of chi-square variables. The asymptotic distributions of these test statistics under the null and alternative hypotheses are discussed. Under fixed and local alternatives, the asymptotic powers are compared theoretically. Simulation studies are also performed to compare the exact powers of the test statistics in finite samples. A data analysis on the temperature and precipitation variability in the European Alps illustrates the proposed methods.
TL;DR: In this paper, an augmented inverse probability weighted-type empirical likelihood ratio for the parameters of interest is defined such that this ratio is asymptotically standard chi-squared.
Abstract: In this paper, we investigate the empirical-likelihood-based inference for the construction of confidence intervals and regions of the parameters of interest in single index models with missing covariates at random. An augmented inverse probability weighted-type empirical likelihood ratio for the parameters of interest is defined such that this ratio is asymptotically standard chi-squared. Our approach is to directly calibrate the empirical log-likelihood ratio, and does not need multiplication by an adjustment factor for the original ratio. Our bias-corrected empirical likelihood is self-scale invariant and no plug-in estimator for the limiting variance is needed. Some simulation studies are carried out to assess the performance of our proposed method.
TL;DR: In this paper, a new characterization of the Pareto distribution is presented and goodness-of-fit tests based on it are considered, for small sample sizes, and the power of those tests with some common goodness of fit tests is compared.
Abstract: In this paper we present a new characterization of the Pareto distribution and consider goodness-of-fit tests based on it. We provide an integral and Kolmogorov–Smirnov-type statistics based on U-statistics and we calculate Bahadur efficiency for various alternatives. We find locally optimal alternatives for those tests. For small sample sizes, we compare the power of those tests with some common goodness-of-fit tests.
TL;DR: In this article, the influence function, the asymptotic variance and the MSE for penalized M-estimators and the sparse least trimmed squares (LTS) estimator have been proposed.
Abstract: To perform regression analysis in high dimensions, lasso or ridge estimation are a common choice. However, it has been shown that these methods are not robust to outliers. Therefore, alternatives as penalized M-estimation or the sparse least trimmed squares (LTS) estimator have been proposed. The robustness of these regression methods can be measured with the influence function. It quantifies the effect of infinitesimal perturbations in the data. Furthermore, it can be used to compute the asymptotic variance and the mean-squared error (MSE). In this paper we compute the influence function, the asymptotic variance and the MSE for penalized M-estimators and the sparse LTS estimator. The asymptotic biasedness of the estimators make the calculations non-standard. We show that only M-estimators with a loss function with a bounded derivative are robust against regression outliers. In particular, the lasso has an unbounded influence function.
TL;DR: In this article, a parametric class of distributions designed to match the behavior of the partially observed screened data is proposed, obtained from the nontruncated marginal of the rectangle-truncated multivariate normal distributions.
Abstract: A screening problem is tackled by proposing a parametric class of distributions designed to match the behavior of the partially observed screened data. This class is obtained from the nontruncated marginal of the rectangle-truncated multivariate normal distributions. Motivations for the screened distribution as well as some of the basic properties, such as its characteristic function, are presented. These allow us a detailed exploration of other important properties that include closure property in linear transformation, in marginal and conditional operations, and in a mixture operation as well as the first two moments and some sampling distributions. Various applications of these results to the statistical modelling and data analysis are also provided.
TL;DR: In this paper, the authors proposed a bivariate log-linear model for the Birnbaum-Saunders distribution, which has five parameters and can be obtained using Gaussian copula.
Abstract: Univariate Birnbaum–Saunders distribution has received a considerable amount of attention in recent years. Rieck and Nedelman (A log-linear model for the Birnbaum–Saunders distribution. Technometrics, 1991;33:51–60) introduced a log Birnbaum–Saunders distribution. The main aim of this paper is to introduce bivariate log Birnbaum–Saunders distribution. The proposed model is symmetric and it has five parameters. It can be obtained using Gaussian copula. Different properties can be obtained using copula structure. It is observed that the maximum likelihood estimators (MLEs) cannot be obtained explicitly. Two-dimensional profile likelihood approach may be adopted to compute the MLEs. We propose some alternative estimators also, which can be obtained quite conveniently. The analysis of one data set is performed for illustrative purposes. Finally, it is observed that this model can be used as a bivariate log-linear model, and its multivariate generalization is also quite straight forward.
TL;DR: In observational studies, the overall aim when fitting a model for the propensity score is to reduce bias for an estimator of the causal effect as discussed by the authors, and the assumption of an unconfounded treatment...
Abstract: In observational studies, the overall aim when fitting a model for the propensity score is to reduce bias for an estimator of the causal effect. To make the assumption of an unconfounded treatment ...
TL;DR: In this paper, a link between the bivariate L-moments (BLM) and the underlying bivariate copula functions is established, which provides a new estimate of dependence parameters of bivariate statistical data.
Abstract: Serfling and Xiao [A contribution to multivariate L-moments, L-comoment matrices. J Multivariate Anal. 2007;98:1765–1781] extended the L-moment theory to the multivariate setting. In the present paper, we focus on the two-dimensional random vectors to establish a link between the bivariate L-moments (BLM) and the underlying bivariate copula functions. This connection provides a new estimate of dependence parameters of bivariate statistical data. Extensive simulation study is carried out to compare estimators based on the BLM, the maximum likelihood, the minimum distance and a rank approximate Z-estimation. The obtained results show that, when the sample size increases, BLM-based estimation performs better as far as the bias and computation time are concerned. Moreover, the root-mean-squared error is quite reasonable and less sensitive in general to outliers than those of the above cited methods. Further, the proposed BLM method is an easy-to-use tool for the estimation of multiparameter copula models. A g...
TL;DR: In this article, the authors propose test procedures for the hypothesis that a given set of discrete observations may be formulated as a particular time series of counts with a specific conditional law, and the test statistics incorporate the empirical probability-generating function computed from the observations.
Abstract: We propose testing procedures for the hypothesis that a given set of discrete observations may be formulated as a particular time series of counts with a specific conditional law. The new test statistics incorporate the empirical probability-generating function computed from the observations. Special emphasis is given to the popular models of integer autoregression and Poisson autoregression. The asymptotic properties of the proposed test statistics are studied under the null hypothesis as well as under alternatives. A Monte Carlo power study on bootstrap versions of the new methods is included as well as real-data examples.
TL;DR: In this article, a mixture model under multiplicative censoring is considered, and a new adaptive estimator based on wavelets and a hard thresholding rule is constructed for this problem.
Abstract: In this paper, a mixture model under multiplicative censoring is considered. We investigate the estimation of a component of the mixture (a density) from the observations. A new adaptive estimator based on wavelets and a hard thresholding rule is constructed for this problem. Under mild assumptions on the model, we study its asymptotic properties by determining an upper bound of the mean integrated squared error over a wide range of Besov balls. We prove that the obtained upper bound is sharp.
TL;DR: In this article, three different types of multivariate chi-square distributions are considered and their members play important roles as limiting distributions of vectors of test statistics in several applications of multiple hypotheses testing.
Abstract: We are concerned with three different types of multivariate chi-square distributions. Their members play important roles as limiting distributions of vectors of test statistics in several applications of multiple hypotheses testing. We explain these applications and consider the computation of multiplicity-adjusted p-values under the respective global hypothesis. By means of numerical examples, we demonstrate how much gain in level exhaustion or, equivalently, power can be achieved with corresponding multivariate multiple tests compared with approaches which are only based on univariate marginal distributions and do not take the dependence structure among the test statistics into account. As a further contribution of independent value, we provide an overview of essentially all analytic formulas for computing multivariate chi-square probabilities of the considered types which are available up to present. These formulas were scattered in the previous literature and are presented here in a unified manner.
TL;DR: In this paper, Aly and Benkherouf presented a new family of distributions based on probability generating functions and derived a very useful representation for the Harris extended density function as an absolutely convergent power series of the survival function of the baseline distribution.
Abstract: A new method for generating new classes of distributions based on the probability-generating function is presented in Aly and Benkherouf [A new family of distributions based on probability generating functions. Sankhya B. 2011;73:70–80]. In particular, they focused their interest to the so-called Harris extended family of distributions. In this paper, we provide several general results regarding the Harris extended models such as the general behaviour of the failure rate function. We also derive a very useful representation for the Harris extended density function as an absolutely convergent power series of the survival function of the baseline distribution. Additionally, some stochastic order relations are established and limiting distributions of sample extremes are also considered for this model. These general results are illustrated in several special Harris extended models. Finally, we discuss estimation of the model parameters by the method of maximum likelihood and provide an application to real da...
TL;DR: In this paper, a sparse coefficient estimation and automated model selection procedure for autoregressive processes with heavy-tailed innovations based on penalized conditional maximum likelihood was proposed, which satisfies a strong consistency, OP(N−1/2) consistency, and the oracle properties, where N is the sample size.
Abstract: We propose a sparse coefficient estimation and automated model selection procedure for autoregressive processes with heavy-tailed innovations based on penalized conditional maximum likelihood. Under mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator satisfies a strong consistency, OP(N−1/2) consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator and smoothly clipped average deviation, are compared. The proposed method provides a distribution-based penalized inference to AR models, which is especially useful when the other estimation methods fail or under perform for AR processes with heavy-tailed innovations [Feigin, Resnick. Pitfalls of fitting autoregressive models for heavy-tailed time series. Extremes. 1999;1:391–422]. A simulation study confirms our theoretical results. At the ...
TL;DR: In this paper, the search for locally and maximin optimal designs for multi-factor nonlinear models from optimal design for sub-models of a lower dimension is considered, and sufficient conditions are given so that maximin D-optimal designs for additive multilinear non-linear models can be built from their submodels with a single factor.
Abstract: This paper considers the search for locally and maximin optimal designs for multi-factor nonlinear models from optimal designs for sub-models of a lower dimension. In particular, sufficient conditions are given so that maximin D-optimal designs for additive multi-factor nonlinear models can be built from maximin D-optimal designs for their sub-models with a single factor. Some examples of application are models involving exponential decay in several variables.
TL;DR: In this article, the likelihood function of small generalized Laplace laws and variance gamma Levy processes in the short time framework was investigated and the local asymptotic normality property in statistical inference for the variance gamma gamma Levy process under high-frequency sampling with its associated optimal convergence rate and Fisher information matrix.
Abstract: We investigate the likelihood function of small generalized Laplace laws and variance gamma Levy processes in the short time framework. We prove the local asymptotic normality property in statistical inference for the variance gamma Levy process under high-frequency sampling with its associated optimal convergence rate and Fisher information matrix. The location parameter is required to be given in advance for this purpose, while the remaining three parameters are jointly well behaved with an invertible Fisher information matrix. The results are discussed with relation to equivalent formulations of the variance gamma Levy process, that is, as a time-changed Brownian motion and as a difference of two independent gamma processes.
TL;DR: In this paper, a modified Wilcoxon rank-sum test was proposed to test the location parameter. But the exact critical value of the statistic is difficult when the sample sizes are increased, and the accuracy of various approximations to the probability of the modified Wil-coxon statistic was investigated.
Abstract: When testing hypotheses in two-sample problems, the Wilcoxon rank-sum test is often used to test the location parameter, and this test has been discussed by many authors over the years. One modification of the Wilcoxon rank-sum test was proposed by Tamura [On a modification of certain rank tests. Ann Math Stat. 1963;34:1101–1103]. Deriving the exact critical value of the statistic is difficult when the sample sizes are increased. The normal approximation, the Edgeworth expansion, the saddlepoint approximation, and the permutation test were used to evaluate the upper tail probability for the modified Wilcoxon rank-sum test given finite sample sizes. The accuracy of various approximations to the probability of the modified Wilcoxon statistic was investigated. Simulations were used to investigate the power of the modified Wilcoxon rank-sum test for the one-sided alternative with various population distributions for small sample sizes. The method was illustrated by the analysis of real data.