Maximum Likelihood Estimation in Compound Geometric Failure Model with Changing Parameters From Type-I Two-Stage Progressively Censored and Group Censored Samples

Boardman and Kendell (1970 Boardman , T. J. , Kendell , P. J. ( 1970 ). Estimation in compound failure models . Technometrics 12 : 891 – 908 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) considered the problem of estimation with respect to Type-I censoring when an item is subjected to only one of the two causes of failure assuming exponential model. Patel and Gajjar (1992 Patel , M. N. , Gajjar , A. V. ( 1992 ). Maximum likelihood estimation in compound exponential failure model with changing failure rates from Type-I progressively censored and group censored samples . Commun. Statist. Theor. Meth. 21 ( 10 ): 2899 – 2908 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) considered extension of the Boardman and Kendell's results in case of two-stage progressive censoring. Here we have considered geometric competing risk failure model with two independent causes of failures. Maximum likelihood estimation of the parameters is carried out using Type-I two-stage progressively censored and group censored samples. Asymptotic standard errors of the estimators are obtained for both the cases. Two illustrative examples are cited for ungroup and group competing risk models.     

Improved Predictive Inference for Responses Based on Location Parameter Median from the Rayleigh Model Given a Doubly Censored Sample

The purpose of this article is to investigate the predictive inference for responses from the location parameter mean as well as from the median given a doubly censored sample from the two-parameter Rayleigh model. The predictive results by Khan et al. (2010 Khan , H. M. R. , Provost , S. B. , Ashima , S. ( 2010 ). Predictive Inference from a Two-Parameter Rayleigh Life Model Given a Doubly Censored Sample . Commun. Statist. Theor. Meth. 39 ( 7 ): 1237 – 1246 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are used to obtain the predictive inference for responses from the median, where Khan et al. (2010 Khan , H. M. R. , Provost , S. B. , Ashima , S. ( 2010 ). Predictive Inference from a Two-Parameter Rayleigh Life Model Given a Doubly Censored Sample . Commun. Statist. Theor. Meth. 39 ( 7 ): 1237 – 1246 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) obtained the future estimates from the mean. A numerical example representing 66 liver cancer patients is used for predictive analysis. It is concluded that the predictive inference from the median gives precise results as compared with the location parameter mean.     

A Multivariate Threshold Varying Conditional Correlations Model

In this article, a multivariate threshold varying conditional correlation (TVCC) model is proposed. The model extends the idea of Engle (2002 Engle , R. F. ( 2002 ). Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models . Journal of Business and Economic Statistics 20 ( 3 ): 339 – 350 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Tse and Tsui (2002 Tse , Y. K. , Tsui , A. K. C. ( 2002 ). A multivariate GARCH model with time-varying correlations . Journal of Business and Economic Statistics , July 2002 , 20 ( 3 ): 351 – 362 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to a threshold framework. This model retains the interpretation of the univariate threshold GARCH model and allows for dynamic conditional correlations. Techniques of model identification, estimation, and model checking are developed. Some simulation results are reported on the finite sample distribution of the maximum likelihood estimate of the TVCC model. Real examples demonstrate the asymmetric behavior of the mean and the variance in financial time series and the ability of the TVCC model to capture these phenomena.     

Spurious Rejections with Endogenous Break Unit Root Tests in the Presence of Outliers and Breaks

Unit root tests with structural break developed by Zivot and Andrews (1992 Zivot , E. , Andrews , D. W. K. ( 1992 ). Further evidence on the great crash, the oil price shock and the unit root hypothesis . Journal of Business and Economic Statistics 10 : 251 – 270 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Perron and Rodriguez (2003 Perron , P. , Rodriguez , G. ( 2003 ). GLS detrending, efficient unit root tests and structural change . Journal of Econometrics 115 : 1 – 27 .[Crossref], [Web of Science ®] , [Google Scholar]) in the presence of additive outliers and breaks are studied by simulation experiments. The results show that the Zivot–Andrews test appears to have size distortions due to the additive outliers whereas the Perron–Rodriguez test exhibits good properties of size and power. However, the two tests are biased when a second break is present but not taken into account. Furthermore, these endogenous break unit root tests tend to determine the break point incorrectly at one period behind the true break point, leading to spurious rejections of the unit root null hypothesis.     

A Unified Approach to Ordering Comparison of GPS Distributions with their Mixtures

Recently, several authors have been concerned with ordering comparison of known distributions of the family of generalized power series (GPS) distributions with their mixtures in various senses. In this article, we shall employ a unified approach and obtain similar results, more generally, for all members of the class of the GPS distributions. Some of the previous findings of Misra et al. (2003 Misra , N. , Singh , H. , Harner , E. J. ( 2003 ). Stochastic comparisons of Poisson and binomial random variables with their mixtures . Statist. Probab. Lett. 65 : 279 – 290 .[Crossref], [Web of Science ®] , [Google Scholar]), Alamatsaz and Abbasi (2008 Alamatsaz , M. H. , Abbasi , S. ( 2008 ). Ordering comparison of negative binomial random variables with their mixtures . Statist. Probab. Lett. 78 : 2234 – 2239 . [Google Scholar]), and Aghababaei Jazi and Alamatsaz (2010 Aghababaei Jazi , M. , Alamatsaz , M. H. ( 2010 ). Ordering comparison of logarithmic series random variables with their mixtures . Commun. Statist. Theor. Meth. 39 : 3252 – 3263 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in this connection, then, follow as corollaries. Further, we have derived some more ordering comparison results.     

Symmetric quantiles and their applications

To develop estimators with stronger efficiencies than the trimmed means which use the empirical quantile, Kim (1992) Kim, S. J. 1992. The metrically trimmed means as a robust estimator of location. Annals of Statistics, 20: 1534–1547. [Crossref], [Web of Science ®] [Google Scholar] and Chen & Chiang (1996) Chen, L. A. and Chiang, Y. C. 1996. Symmetric type quantile and trimmed means for location and linear regression model. Journal of Nonparametric Statistics, 7: 171–185. [Taylor & Francis Online] [Google Scholar], implicitly or explicitly used the symmetric quantile, and thus introduced new trimmed means for location and linear regression models, respectively. This study further investigates the properties of the symmetric quantile and extends its application in several aspects. (a) The symmetric quantile is more efficient than the empirical quantiles in asymptotic variances when quantile percentage α is either small or large. This reveals that for any proposal involving the α th quantile of small or large α s, the symmetric quantile is the right choice; (b) a trimmed mean based on it has asymptotic variance achieving a Cramer-Rao lower bound in one heavy tail distribution; (c) an improvement of the quantiles-based control chart by Grimshaw & Alt (1997) Grimshaw, S. D. and Alt, F. B. 1997. Control charts for quantile function values. Journal of Quality Technology, 29: 1–7. [Taylor & Francis Online] [Google Scholar] is discussed; (d) Monte Carlo simulations of two new scale estimators based on symmetric quantiles also support this new quantile.     

Revisiting Beal's Confidence Intervals for the Difference of Two Binomial Proportions

Confidence interval construction for the difference of two independent binomial proportions is a well-known problem with a full panoply of proposed solutions. In this paper, we focus largely on the family of intervals proposed by Beal (1987 Beal , S. ( 1987 ). Asymptotic confidence intervals for the difference between two binomial parameters for use with small samples . Biometrics 43 : 941 – 950 . [CSA] [CROSSREF] [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). This family, which includes the Haldane and Jeffreys–Perks intervals as special cases, assumes a symmetric prior distribution for the population proportions p ₁ and p ₂. We propose new methods that allow the currently observed data to set the prior distribution by taking a parametric empirical-Bayes approach; in addition, we also provide an investigation of the new interval' behaviors in small-sample situations. Unlike other solutions, our intervals can be used adaptively for experiments conducted in multiple stages over time. We illustrate this notion using data from an Argentinean study involving the Mal Rio Cuarto virus and its transmission to susceptible maize crops.     

Estimation of Ordered Means of Two Poisson Distributions

In this article, we discuss the estimation of linear functions of two Poisson means, on which an order restriction is given. We give a necessary and sufficient condition on the coefficients of the linear function for the maximum likelihood estimator (MLE) which satisfies the order restriction to dominate the unbiased estimator under squared error loss. Furthermore, simultaneous estimation of two ordered Poisson means is considered and we suggest the Clevenson–Zidek type modification of MLE which dominates the MLE under normalized squared error loss. We also improve the estimator proposed by Clevenson and Zidek (1975 Clevenson , M. , Zidek , J. ( 1975 ). Simultaneous estimation of the means of independent Poisson laws . J. Amer. Statist. Assoc. 70 : 698 – 705 . [CSA] [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) which ignores the order restriction.     

Step-Up Procedure for Multiple Comparison with a Control for Multivariate Normal Means

In this study, we consider the multiple comparison with a control for multivariate normal means. Specifically, we construct a step-up procedure by referring to Dunnett and Tamhane (1992 Dunnett , C. W. , Tamhane , A. C. ( 1992 ). A step-up multiple test procedure . Journal of the American Statistical Association 87 : 162 – 170 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We derive recursive formulae for determining the critical values of the step-up procedure for a specified significance level. Then we formulate the power of the test. Finally, we compare the step-up procedure with the single-step procedure proposed by Nakamura and Imada (2005 Nakamura , T. , Imada , T. ( 2005 ). Multiple comparison procedure of Dunnett's type for multivariate normal means . Journal of the Japanese Society of Computational Statistics 18 : 21 – 32 .[Crossref] , [Google Scholar]) and the step-down procedure proposed by Imada and Douke (2007 Imada , T. , Douke , H. ( 2007 ). Step down procedure for comparing several treatments with a control based on multivariate normal response . Biometrical Journal 49 ( 1 ): 18 – 29 .[Crossref], [Web of Science ®] , [Google Scholar]) in terms of numerical examples regarding the power of the test.     

Comment on Ridge Estimation to the Restricted Linear Model

In this note, we show that the estimator and the following results given by Zhong and Yang (2007 Zhong , Z. , Yang , H. ( 2007 ). Ridge estimation to the restricted linear model . Commun. Statist. Theor. Meth. 36 : 2099 – 2115 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are the same with that of Groß (2003 Groß , J. ( 2003 ). Restricted ridge estimation . Statist. Probab. Lett. 65 : 57 – 64 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Stirling's Formula and Its Extensions: Heuristic Approaches

Walsh (1995 Walsh , D. P. ( 1995 ). Equating Poisson and normal probability functions to derive Stirling's formula . Amer. Statist. 49 : 270 – 271 .[Taylor & Francis Online] , [Google Scholar]) introduced a heuristic approach to motivate Stirling's formula by equating a Poisson probability to an analogous value from a normal density function. We explore similar heuristics to derive approximations for various binomial, negative binomial, and multinomial coefficients. Also, using heuristics markedly different from those of Walsh, we develop an approximation of (nk)! for positive integers n (large) and k. These heuristics are then used to validate Stirling's formula for Γ(nα) where α is a positive real number. To derive each of our approximations we use a different probability distribution, and hence each section may serve as pedagogical module.     

Confidence Interval of the Difference of Two Independent Binomial Proportions Using Weighted Profile Likelihood

Interval estimation of the difference of two independent binomial proportions is an important problem in many applied settings. Newcombe (1998 Newcombe , R. G. ( 1998 ). Interval estimation for the difference between independent proportions: comparison of seven methods . Statistics in Medicine 17 : 873 – 890 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) compared the performance of several existing asymptotic methods, and based on the results obtained, recommended a method known as Wilson's method, a modified version of a method originally proposed for single binomial proportion. In this article, we propose a method based on profile likelihood, where the likelihood is weighted by noninformative Jeffrey' prior. By doing extensive simulations, we find that the proposed method performs well compared to Wilson's method. A SAS/IML program implementing this method is also given with this article.     

Additive Mixed Effect Model for Clustered Doubly Censored Data

Cai and Zeng (2011 Cai, J. and Zeng, D. 2011. Additive mixed effect model for clustered failure time data. Biometrics, 67(4): 1340–1351. [Crossref], [PubMed] , [Google Scholar]) proposed an additive mixed effect model to analyze clustered right-censored data. In this article, we demonstrate that the approach of Cai and Zeng (2011 Cai, J. and Zeng, D. 2011. Additive mixed effect model for clustered failure time data. Biometrics, 67(4): 1340–1351. [Crossref], [PubMed] , [Google Scholar]) can be extended to clustered doubly censored data. Furthermore, when both left- and right-censoring variables are always observed, we propose alternative estimators using the approach of Cai and Cheng (2004 Cai, T. and Cheng, S. C. 2004. Semiparametric regression analysis for doubly censored data. Biometrika, 91: 277–290. [Crossref], [Web of Science ®] , [Google Scholar]). A simulation study is conducted to investigate the performance of the proposed estimators.     

D s -Optimal Designs for Quadratic Log Contrast Model for Experiments with Mixtures

The approximate D _s -optimal design for discriminating between linear and quadratic log contrast models for experiments with mixtures suggested by Aitchison and Bacon-Shone (1984 Aitchison , J. , Bacon-Shone , J. ( 1984 ). Log contrast models for experiments with mixtures . Biometrika 71 : 323 – 330 .[Crossref], [Web of Science ®] , [Google Scholar]) is investigated, where the experimental domain is restricted further as in Chan (1992 Chan , L. Y. ( 1992 ). D-Optimal design for a quadratic log contrast model for experiments with mixtures . Commun. Statist. Theor. Meth. 21 ( 10 ): 2909 – 2930 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). It is found that for a symmetric subspace of the finite dimensional simplex, there is a D _s -optimal design with the nice structure that puts a weight 1/2 ^k?1 on the centroid of this subspace and the remaining weight is uniformly distributed on the vertices of the experimental domain. Finally, the D _s -efficiency of the D-optimal design for quadratic model and the design given by Aitchison and Bacon-Shone (1984 Aitchison , J. , Bacon-Shone , J. ( 1984 ). Log contrast models for experiments with mixtures . Biometrika 71 : 323 – 330 .[Crossref], [Web of Science ®] , [Google Scholar]) are also discussed.     

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.     

On Power Approximations and Comparison of Several Asymptotic Tests to Detect a Specified Difference between Two Proportions

A large sample test is proposed for a problem of testing for a specified difference between two binomial proportions. The test is compared to the tests by Falk and Koch (1998 Falk , R. W. , Koch , G. G. ( 1998 ). Testing a specified difference between proportions . Biometrics 54 ( 4 ): 1602 – 1614 .[Crossref] , [Google Scholar]), and Parmet and Schechtman (2007 Parmet , Y. , Schechtman , E. ( 2007 ). On a test of the difference between two binomial proportions . Communications in Statistics – Theory and Methods 36 : 887 – 895 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), and is shown to dominate in terms of the Type I error rate control. Asymptotic power is derived for each test and is shown to result in values quite proximate to the simulated power values. In addition, formulas to perform sample size estimation are provided. These methods are expected to be especially valuable in the design stage when obtaining the correct power/sample size estimation is essential.     

Tests for Multiple Outliers in an Exponential Sample

By applying the recursion of Huffer (1988 Huffer, F. 1988. Divided differences and the joint distribution of linear combinations of spacings. Journal of Applied Probability, 25: 346–354. [Crossref], [Web of Science ®] , [Google Scholar]) repeatedly, we propose an algorithm for evaluating the null joint distribution of Dixon-type test statistics for testing discordancy of k upper outliers in exponential samples. By using the critical values of Dixon-type test statistics determined from the proposed algorithm and those of Cochran-type test statistics presented earlier by Lin and Balakrishnan (2009 Lin, C. T. and Balakrishnan, N. 2009. Exact computation of the null distribution of a test for multiple outliers in an exponential sample. Computational Statistics & Data Analysis, 53: 3281–3290. [Crossref], [Web of Science ®] , [Google Scholar]), we carry out an extensive Monte Carlo study to investigate the powers and the error probabilities for the effects of masking and swamping when the number of outliers k = 2 and 3. Based on our empirical findings, we recommend Rosner’s (1975 Rosner, B. 1975. On the detection of many outliers. Technometrics, 17: 221–227. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) sequential test procedure based on Dixon-type test statistics for testing multiple outliers from an exponential distribution.     

Second-Order Covariance Matrix Formula for Heteroskedastic Generalized Linear Models

For the first time, we provide a matrix formula for second-order covariances of maximum likelihood estimates in heteroskedastic generalized linear models, thus generalizing the results 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]) and Cordeiro et al. (2006 Cordeiro , G. M. , Barroso , L. P. , Botter , D. A. (2006). Covariance matrix formula for generalized linear models with unknown dispersion. Commun. Statist. Theor. Meth. 35:113–120.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) related to the generalized linear models with known and unknown dispersion parameter, respectively. The covariance matrix formula does not involve cumulants of log-likelihood derivatives and can be easily obtained using simple matrix operations. We apply our main result to a simple model. Some simulations show that the second-order covariances can be quite pronounced in small to moderate samples. The usual covariances of the maximum likelihood estimates can be corrected by these second-order covariances.     

Comments on “Improvement of generalized difference-based mixed Liu estimator in partially linear model”

Abstract

We make some comments about the article of Wu (2018 Wu, J. B. 2018. Improvement of generalized difference-based mixed Liu estimator in partially linear model. Communications in Statistics: Theory and Methods 47 (18):4433–42.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and correct the theorems in that article.     

Correlation Structure and Model Selection for Negative Binomial Distribution in GEE

A new model selection criterion, termed as the “quasi-likelihood under the independence model criterion” (QIC), was proposed by Pan (2001 Pan , W. ( 2001 ). Akaike's information criterion in generalized estimating equations . Biometrics 57 : 120 – 125 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) for GEE models. Cui (2007 Cui , J. ( 2007 ). QIC program and model selection in GEE analyses . Stata Journal 7 : 209 – 220 .[Web of Science ®] , [Google Scholar]) developed a general computing program to implement the QIC method for a range of statistical distributions. However, only a special case of the negative binomial distribution was considered in Cui (2007 Cui , J. ( 2007 ). QIC program and model selection in GEE analyses . Stata Journal 7 : 209 – 220 .[Web of Science ®] , [Google Scholar]), where the dispersion parameter equals to unity. This article introduces a new computing program that can be applied for the general negative binomial model, where the dispersion parameter can be any fixed value. An example is also given in this article.