Statistical Analysis of the Log-Normal Distribution under Type-II Progressive Hybrid Censoring Schemes

In this article, we obtain the maximum likelihood estimators (MLEs) and approximate maximum likelihood estimators (AMLEs) of the parameters, from a two-parameter log-normal distribution based on the adaptive Type-II progressive hybrid censoring scheme, which was introduced by Ng et al. (2009 Ng , H. K. T. , Kundu , D. , Chan , P. S. ( 2009 ). Statistical analysis of exponential lifetimes under an adaptive Type-II progressively censoring scheme . Naval Research Logistics 56 : 687 – 698 .[Crossref], [Web of Science ®] , [Google Scholar]) for life testing or reliability experiment. In order to compare the results, we calculate corresponding estimators of the Type-II progressive hybrid censoring scheme. In particular, we provide computational formulas of the expected total test time and the expected number of failures for each scheme. We also compute the observed Fisher information matrix and use them to obtain the asymptotic confidence intervals. A simulation study carries out to evaluate the bias and mean square error of the MLEs and AMLEs from the two above-mentioned schemes. Finally, we present a numerical example to illustrate the methods of inference discussed here.     

Sample Partitioning Estimation for Ergodic Diffusions

In this article, we present a new technique to obtain estimators for parameters of ergodic processes. When a diffusion is ergodic its transition density converges to the invariant density Durett (1996 Durett, R. (1996). Stochastic Calculus. A Practical Introduction. New York: CRC Press. [Google Scholar]). This convergence enabled us to introduce a sample partitioning technique that gives, in each subsample, observations that can be treated as independent and identically distributed. Within this framework, is possible the construction of estimators like maximum likelihood estimators or others.     

Bias-Corrected Maximum Likelihood Estimators in Nonlinear Heteroscedastic Models

Nonlinear heteroscedastic models are widely used in econometrics and statistical applications. We derive matrix formulae for the second-order biases of the maximum likelihood estimators of the parameters in the mean and variance response which generalize previous results by Cook et al. (1986 Cook , D. R. , Tsai , C. L. , Wei , B. C. ( 1986 ). Bias in nonlinear regression . Biometrika 73 : 615 – 623 .[Crossref], [Web of Science ®] , [Google Scholar]) and Cordeiro (1993 Cordeiro , G. M. ( 1993 ). Bartlett corrections and bias correction for two heteroscedastic regression models . Commun. Statist. Theor. Meth. 22 : 169 – 188 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The biases of the estimators are easily obtained as vectors of regression coefficients from suitable weighted linear regressions. The practical use of such biases is illustrated in a simulation study and in an application to a real data set.     

Minimum Divergence,Generalized Empirical Likelihoods,and Higher Order Expansions

This article studies the minimum divergence (MD) class of estimators for econometric models specified through moment restrictions. We show that MD estimators can be obtained as solutions to a tractable lower dimensional optimization problem. This problem is similar to the one solved by the generalized empirical likelihood estimators of Newey and Smith (2004 Newey , W. K. , Smith , R. J. ( 2004 ). Higher order properties of GMM and Generalized Empirical Likelihood estimators . Econometrica 72 : 219 – 255 .[Crossref], [Web of Science ®] , [Google Scholar]), but it is equivalent to it only for a subclass of divergences. The MD framework provides a coherent testing theory: tests for overidentification and parametric restrictions in this framework can be interpreted as semiparametric versions of Pearson-type goodness of fit tests. The higher order properties of MD estimators are also studied and it is shown that MD estimators that have the same higher order bias as the empirical likelihood (EL) estimator also share the same higher order mean square error and are all higher order efficient. We identify members of the MD class that are not only higher order efficient, but also, unlike the EL estimator, well behaved when the moment restrictions are misspecified.     

An Improved Class of Estimators in Two-Phase Sampling Using Two Auxiliary Variables

This article suggests an improved class of estimators under the general framework of two-phase sampling scheme in presence of two auxiliary variables. This class includes a large number of estimators (Chand, 1975 Chand , L. ( 1975 ). Some Ratio-Type Estimator Based on Two or More Auxiliary Variables. Unpublished Ph.D. dissertation, Iowa State University, Iowa . [Google Scholar]; Kiregyera, 1980 Kiregyera , B. ( 1980 ). A chain ratio-type estimator in finite population double sampling using two auxiliary variables . Metrika 27 : 217 – 223 .[Crossref] , [Google Scholar], 3; Mukharjee et al., 1987 Mukharjee , R. , Rao , T. J. , Vijayan , K. ( 1987 ). Regression-type estimators using multiple auxiliary information . Aust. J. Statist. 29 : 244 – 254 . [Google Scholar]) and also the class of estimators suggested by Sahoo et al. (1993 Sahoo , J. , Sahoo , L. N. , Mohanty , S. ( 1993 ). A regression approach to estimation in two phase sampling using two auxiliary variables . Curr. Sci. 65 ( 1 ): 73 – 75 . [Google Scholar]).     

Shrinkage Estimators for Prediction Out-of-Sample: Conditional Performance

We find that, in a linear model, the James–Stein estimator, which dominates the maximum-likelihood estimator in terms of its in-sample prediction error, can perform poorly compared to the maximum-likelihood estimator in out-of-sample prediction. We give a detailed analysis of this phenomenon and discuss its implications. When evaluating the predictive performance of estimators, we treat the regressor matrix in the training data as fixed, i.e., we condition on the design variables. Our findings contrast those obtained by Baranchik (1973 Baranchik , A. J. ( 1973 ). Inadmissibility of maximum likelihood estimators in some multiple regression problems with three or more independent variables . Ann. Statist. 1 ( 2 ): 312 – 321 .[Crossref], [Web of Science ®] , [Google Scholar]) and, more recently, by Dicker (2012 Dicker , L. ( 2012 ). Dense signals, linear estimators, and out-of-sample prediction for high-dimensional linear models. arXiv:1102.2952 [math.ST].  [Google Scholar]) in an unconditional performance evaluation.     

Hartley-Ross Type Estimators for Population Mean Using Known Parameters of Auxiliary Variate

This article proposes Hartley-Ross type unbiased estimators of finite population mean using information on known parameters of auxiliary variate when the study variate and auxiliary variate are positively correlated. The variances of the proposed unbiased estimators are obtained. It has been shown that the proposed estimators are more efficient than the simple mean estimator, usual ratio estimator and estimators proposed by Sisodia and Dwivedi (1981 Sisodia , B. V. S. , Dwivedi , V. K. ( 1981 ). A modified ratio estimator using coefficient of variation of auxiliary variable . J. Indian Soc. Agricultural Statist. 33 ( 1 ): 13 – 18 . [Google Scholar]), Kadilar and Cingi (2006 Kadilar , C. , Cingi , H. ( 2006 ). A new ratio estimator using correlation coefficient . Int. Statist. 1 – 11 . [Google Scholar]), and Kadilar et al. (2007 Kadilar , C. , Candan , M. , Cingi , H. ( 2007 ). Ratio estimators using robust regression . Hacet. J. Math. Statist. 36 ( 2 ): 181 – 188 .[Web of Science ®] , [Google Scholar]) under certain realistic conditions. Empirical studies are also carried out to demonstrate the merits of the proposed unbiased estimators over other estimators considered in this article.     

Maximum Likelihood Estimators in a Two Step Model for PLS

Univariate partial least squares regression (PLS1) is a method of modeling relationships between a response variable and explanatory variables, especially when the explanatory variables are almost collinear. The purpose is to predict a future response observation, although in many applications there is an interest to understand the contributions of each explanatory variable. It is an algorithmic approach. In this article, we are going to use the algorithm presented by Helland (1988 Helland , I. S. ( 1988 ). On the structure of partial least squares regression . Commun. Statist. Simul. Computat. 17 : 581 – 607 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The population PLS predictor is linked to a linear model including a Krylov design matrix and a two-step estimation procedure. For the first step, the maximum likelihood approach is applied to a specific multivariate linear model, generating tools for evaluating the information in the explanatory variables. It is shown that explicit maximum likelihood estimators of the dispersion matrix can be obtained where the dispersion matrix, besides representing the variation in the error, also includes the Krylov structured design matrix describing the mean.     

Some Classes of Estimators for the Population Median Using Auxiliary Information

This article considers some classes of estimators of the population median of the study variable using information on an auxiliary variable with their properties under large sample approximation. Asymptotic optimum estimator (AOE) in each class of estimators has been investigated along with the approximate mean square error formulae. It has been shown that the proposed classes of estimators are better than these considered by Gross (1980 Gross , T. S. ( 1980 ). Median estimation in sample surveys. Proc. Surv. Res. Meth. Sect. Amer. Statist. Assoc. 181–184 . [Google Scholar]), Kuk and Mak (1989 Kuk , A. Y. C. , Mak , T. K. ( 1989 ). Median estimation in the presence of auxiliary information . J. Roy. Statist. Soc. Ser. B51 : 261 – 269 . [Google Scholar]), Singh et al. (2003a Singh , H. P. , Singh , S. , Joarder , A. H. ( 2003a ). Estimation of population median when mode of an auxiliary variable is known . J. Statist. Res. 37 ( 1 ): 57 – 63 . [Google Scholar]), and Al and Cingi (2009 Al , S. , Cingi , H. ( 2009 ). New estimators for the population median in simple random sampling. Tenth Islamic Countries Conference on Statistical Sciences, held in New Cairo, Egypt . [Google Scholar]). An empirical study is carried out to judge the merits of the suggested class of estimators over other existing estimators.     

Estimation of the Bias of the Maximum Likelihood Estimators in an Extreme Value Context

Interest is centered on the maximum likelihood (ML) estimators of the parameters of the Generalized Pareto Distribution in an extreme value context. Our aim consists of reducing the bias of these estimates for which no explicit expression is available. To circumvent this difficulty, we prove that these estimators are asymptotically equivalent to one-step estimators introduced by Beirlant et al. (2010 Beirlant , J. , Guillou , A. , Toulemonde , G. ( 2010 ). Peaks-over-threshold modeling under random censoring . Commun. Statist. Theor. Meth.  [Google Scholar]) in a right-censoring context. Then, using this equivalence property, we estimate the bias of these one-step estimators to approximate the asymptotic bias of the ML-estimators. Finally, a small simulation study and an application to a real data set are provided to illustrate that these new estimators actually exhibit reduced bias.     

On Maximum Likelihood Estimation for the Two Parameter Burr XII Distribution

This article considers three related aspects of maximum likelihood estimation of parameters in the two-parameter Burr XII distribution. Specifically, we first provide further clarification to some limiting results in Wingo (1993 Wingo , D. R. ( 1993 ). Maximum likelihood estimation of Burr XII distribution parameters under Type II censoring . Microelectron. Reliab. 33 : 1251 – 1257 .[Crossref], [Web of Science ®] , [Google Scholar]). We then focus on details in a proof of the uniqueness of the maximum likelihood estimators. Finally, we consider using the likelihood approach for data which does not satisfy Wingo's criterion, and show that this results in fitting either a Pareto distribution or an intuitively sensible degenerate distribution to the data. The discussion here is completely general, and not restricted to data obtained under Type II censoring.     

Covariance Matrix Formula for Exponential Family Nonlinear Models

This article gives a matrix formula for second-order covariances of maximum likelihood estimators in exponential family nonlinear models, thus generalizing the result of Cordeiro (2004 Cordeiro , G. M. ( 2004 ). Second-order covariance matrix of maximum likelihood estimates in generalized linear models . Statist. Probab. Lett. 66 : 153 – 160 .[Crossref], [Web of Science ®] , [Google Scholar]) valid for generalized linear models with known dispersion parameter. Some simulations show that the second-order covariances for exponential family nonlinear models can be quite pronounced in small to moderate sample sizes.     

Using Implied Probabilities to Improve the Estimation of Unconditional Moment Restrictions for Weakly Dependent Data

In this article, we investigate the use of implied probabilities (Back and Brown, 1993) to improve estimation in unconditional moment conditions models. Using the seminal contributions of Bonnal and Renault (2001 Bonnal, H., Renault, E. (2001). Minimal Chi-Square Estimation with Conditional Moment Restrictions, Document de Travail, CESG, September 2001. [Google Scholar]) and Antoine et al. (2007 Antoine, B., Bonnal, H., Renault, E. (2007). On the efficient use of the informational content of estimating equations: Implied probabilities and euclidean empirical likelihood. Journal of Econometrics 138(2):461–487.[Crossref], [Web of Science ®] , [Google Scholar]), we propose two three-step Euclidian empirical likelihood (3S-EEL) estimators for weakly dependent data. Both estimators make use of a control variates principle that can be interpreted in terms of implied probabilities in order to achieve higher-order improvements relative to the traditional two-step GMM estimator. A Monte Carlo study reveals that the finite and large sample properties of the three-step estimators compare favorably to the existing approaches: the two-step GMM and the continuous updating estimator.     

Best Spatial Two‐Stage Least Squares Estimators for a Spatial Autoregressive Model with Autoregressive Disturbances

Abstract

Estimation of a cross‐sectional spatial model containing both a spatial lag of the dependent variable and spatially autoregressive disturbances are considered. [Kelejian and Prucha (1998)] Kelejian, H. H. and Prucha, I. R. 1998. A generalized spatial two‐stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances. J. Real Estate Financ. and Economics, 17: 99–121. [Crossref], [Web of Science ®] , [Google Scholar]described a generalized two‐stage least squares procedure for estimating such a spatial model. Their estimator is, however, not asymptotically optimal. We propose best spatial 2SLS estimators that are asymptotically optimal instrumental variable (IV) estimators. An associated goodness‐of‐fit (or over identification) test is available. We suggest computationally simple and tractable numerical procedures for constructing the optimal instruments.     

Darling-Erdös-type test for change detection in parameters and variance for stationary VAR models

The aim of this article is the construction of the test statistic for the detection of changes in vector autoregressive (AR) models where both AR parameters and the variance matrix of the error term are the subjects of a change. The approximating distribution of the proposed statistic is the Gumbel distribution. The proof stands on the approximation of weakly dependent random vectors by independent ones and by application of Horváth’s extension of Darling-Erdös extremal result for random vectors, see Darling and Erdös (1956) Darling, D.A., Erdös, P. (1956). A limit theorem for the maximum of normalized sums of independent random variables. Duke Math. J. 23:143–155.[Crossref], [Web of Science ®] , [Google Scholar] and Horváth (1993) Horváth, L. (1993). The maximum likelihood method for testing changes in the parameters of normal observations. Ann. Stat. 21(2):671–680.[Crossref], [Web of Science ®] , [Google Scholar]. The test statistic is a modification of the likelihood ratio.     

On generalized exponential chain ratio estimators under stratified two-phase random sampling

Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]) have suggested generalized exponential chain ratio estimators under stratified two-phase sampling scheme for estimating the finite population mean. However, the bias and mean square error (MSE) expressions presented in that work need some corrections, and consequently the study based on efficiency comparison also requires corrections. In this article, we revisit Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]) estimator and provide the correct bias and MSE expressions of their estimator. We also propose an estimator which is more efficient than several competing estimators including the classes of estimators in Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]). Three real datasets are used for efficiency comparisons.     

A Note on Two-Way ECM Estimation of SUR Systems on Unbalanced Panel Data

This article considers the two-way error components model (ECM) estimation of seemingly unrelated regressions (SUR) on unbalanced panel by generalized least squares (GLS). As suggested by Biørn (2004 Biørn , E. ( 2004 ). Regression systems for unbalanced panel data: a stepwise maximum likelihood procedure . Journal of Econometrics 122 : 281 – 291 . [Google Scholar]) for the one-way case, in order to use the standard results for the balanced case the individuals are arranged in groups according to the number of times they are observed. Thus, the GLS estimator can be interpreted as a matrix weighted average of the group specific GLS estimators with weights equal to the inverse of their respective covariance matrices.     

Control charts for generalized exponential distribution percentiles

The generalized exponential (GE) distribution, which was introduced by Mudholkar and Srivastava in 1993 Mudholkar, G. S., Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure data. IEEE Transactions on Reliability 42:299–302. [Google Scholar], has been studied for various applications of lifetime modelings. In this article, five control charts, that comprise the Shewhart-type chart and four parametric bootstrap charts based on maximum likelihood estimation method, the moment estimation method, probability plot method, and least-square error method for the GE percentiles, are investigated. An extensive Monte Carlo simulation study is conducted to compare the performance among all five control charts in terms of average run length. Finally, an example is given for illustration.     

NONNORMAL REGRESSION. II. SYMMETRIC DISTRIBUTIONS

Salient features of a family of short-tailed symmetric distributions, introduced recently by Tiku and Vaughan [1] Tiku, M. L. and Vaughan, D. C. 1999. “A family of short-tailed symmetric distributions”. In Technical Report Canada: McMaster University.  [Google Scholar], are enunciated. Assuming the error distribution to be one of this family, the methodology of modified likelihood is used to derive MML estimators of parameters in a linear regression model. The estimators are shown to be efficient, and robust to inliers. This paper is essentially the first to achieve robustness to inliers. The methodology is extended to long-tailed symmetric distributions and the resulting estimators are shown to be efficient, and robust to outliers. This paper should be read in conjunction with Islam et al. [2] Islam, Q., Tiku, M. L. and Yildirim, F. 2001. Nonnormal regression, Part I: Skew distributions. Commun. Stat.—Theory Meth., to appear [Google Scholar]who develop modified likelihood methodology for skew distributions in the context of linear regression.     

The Restricted and Unrestricted Two-Parameter Estimators

In this article, we introduce a new two-parameter estimator by grafting the contraction estimator into the modified ridge estimator proposed by Swindel (1976 Swindel , B. F. ( 1976 ). Good ridge estimators based on prior information . Commun. Statist. Theor. Meth. A5 : 1065 – 1075 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). This new two-parameter estimator is a general estimator which includes the ordinary least squares, the ridge, the Liu, and the contraction estimators as special cases. Furthermore, by setting restrictions Rβ = r on the parameter values we introduce a new restricted two-parameter estimator which includes the well-known restricted least squares, the restricted ridge proposed by Groß (2003 Groß , J. ( 2003 ). Restricted ridge estimation . Statist. Probab. Lett. 65 : 57 – 64 .[Crossref], [Web of Science ®] , [Google Scholar]), the restricted contraction estimators, and a new restricted Liu estimator which we call the modified restricted Liu estimator different from the restricted Liu estimator proposed by Kaç?ranlar et al. (1999 Kaç?ranlar , S. , Sakall?o?lu , S. , Akdeniz , F. , Styan , G. P. H. , Werner , H. J. ( 1999 ). A new biased estimator in linear regression and a detailed analysis of the widely-analysed dataset on Portland cement . Sankhya Ser. B., Ind. J. Statist. 61 : 443 – 459 . [Google Scholar]). We also obtain necessary and sufficient condition for the superiority of the new two-parameter estimator over the ordinary least squares estimator and the comparison of the new restricted two-parameter estimator to the new two-parameter estimator is done by the criterion of matrix mean square error. The estimators of the biasing parameters are given and a simulation study is done for the comparison as well as the determination of the biasing parameters.