THE MEDIAN ESTIMATE OF THE AUTOREGRESSIVE LOCATION PARAMETER

ABSTRACT

The concept of generalized order statistics was introduced by Kamps (1995 Kamps , U. ( 1995 ) A Concept of Generalized Order Statistics . Germany : B. G. Teubner Stuttgart [Crossref] , [Google Scholar]) to unify several concepts that have been used in statistics such as order statistics, record values, and sequential order statistics. Estimation of the parameters of the Burr type XII distribution are obtained based on generalized order statistics. The maximum likelihood and Bayes methods of estimation are used for this purposes. The Bayes estimates are derived by using the approximation form of Lindley (1980 Lindley , D. V. ( 1980 ). Approximate Bayesian methods . J. Trabajos de Estadistica 31 : 223 – 237 .[Crossref] , [Google Scholar]). Estimation based on upper records from the Burr model is obtained and compared by using Monte Carlo simulation study. Our results are specialized to the results of AL-Hussaini and Jaheen (1992 AL-Hussaini , E. K. , Jaheen , Z. F. ( 1992 ). Bayesian estimation of the parameters, reliability and failure rate functions of the Burr type XII failure model . J. Statist. Comput. Simul. 41 : 31 – 40 .[Taylor &; Francis Online] , [Google Scholar]) which are based on ordinary order statistics.     

Uniqueness of the Maximum Likelihood Estimate of the Weibull Distribution Tampered Failure Rate Model

Abstract

Bhattacharyya and Soejoeti (Bhattacharyya, G. K., Soejoeti, Z. A. (1989 Bhattacharyya, G. K. and Soejoeti, Z. A. 1989. Tampered failure rate model for step-stress accelerated life test. Commun. Statist.—Theory Meth., 18(5): 1627–1643. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Tampered failure rate model for step-stress accelerated life test. Commun. Statist.—Theory Meth. 18(5):1627–1643.) pro- posed the TFR model for step-stress accelerated life tests. Under the TFR model, this article proves that the maximum likelihood estimate of the shape parameters is unique for the Weibull distribution in a multiple step-stress accelerated life test, and investigates the accuracy of the maximum likelihood estimate using the Monte-Carlo simulation.     

SIMULTANEOUS CONFIDENCE REGIONS IN AN EXTENDED GROWTH CURVE MODEL WITH k HIERARCHICAL WITHIN-INDIVIDUALS DESIGN MATRICES

ABSTRACT

We consider an extended growth curve model with k hierarchical within-individuals design matrices. The model includes the one whose mean structure consists of polynomial growth curves with k different degrees. First we propose certain simple estimators of the mean and covariance parameters which are closely related to the MLE's. Using these estimators we construct simultaneous confidence regions for each or all of k growth curves which is an extension of Fujikoshi.[2] Fujikoshi, Y. 1999. Simultaneous Confidence Intervals in an Extended Growth Curves. Commun. Statist.–-Theor. Meth., 28: 671–682. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] A numerical example with k = 3 is also given.     

A Class of Improved Heteroskedasticity-Consistent Covariance Matrix Estimators

Abstract

The heteroskedasticity-consistent covariance matrix estimator proposed by White [White, H. A. (1980 White, H. A. 1980. Heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48: 817–838. [Crossref], [Web of Science ®] , [Google Scholar]). Heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48:817–838], also known as HC0, is commonly used in practical applications and is implemented into a number of statistical software. Cribari–Neto et al. [Cribari–Neto, F., Ferrari, S. L. P., Cordeiro, G. M. (2000 Cribari–Neto, F., Ferrari, S. L. P. and Cordeiro, G. M. 2000. Improved heteroscedasticity–consistent covariance matrix estimators. Biometrika, 87: 907–918. [Crossref], [Web of Science ®] , [Google Scholar]). Improved heteroscedasticity–consistent covariance matrix estimators. Biometrika 87:907–918] have developed a bias-adjustment scheme that delivers bias-corrected White estimators. There are several variants of the original White estimator that are also commonly used by practitioners. These include the HC1, HC2, and HC3 estimators, which have proven to have superior small-sample behavior relative to White's estimator. This paper defines a general bias-correction mechamism that can be applied not only to White's estimator, but to variants of this estimator as well, such as HC1, HC2, and HC3. Numerical evidence on the usefulness of the proposed corrections is also presented. Overall, the results favor the sequence of improved HC2 estimators.     

Two Statistics for Unconditional Exact Tests in Matched-Pairs Design

Abstract

In a recent article Hsueh et al. (Hsueh, H.-M., Liu, J.-P., Chen, J. J. (2001 Hsueh, H.-M., Liu, J.-P. and Chen, J. J. 2001. Unconditional exact tests for equivalence or noninferiority for paired binary endpoints. Biometrics, 57: 478–483. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Unconditional exact tests for equivalence or noninferiority for paired binary endpoints. Biometrics 57:478–483.) considered unconditional exact tests for paired binary endpoints. They suggested two statistics one of which is based on the restricted maximum-likelihood estimator. Properties of these statistics and the related tests are treated in this article.     

Evaluating the relative merits of competing models based on empirical likelihood ratio test

Competing models arise naturally in many research fields, such as survival analysis and economics, when the same phenomenon of interest is explained by different researcher using different theories or according to different experiences. The model selection problem is therefore remarkably important because of its great importance to the subsequent inference; Inference under a misspecified or inappropriate model will be risky. Existing model selection tests such as Vuong's tests [26 Q.H. Vuong, Likelihood ratio test for model selection and non-nested hypothesis, Econometrica 57 (1989), pp. 307–333. doi: 10.2307/1912557[Crossref], [Web of Science ®] , [Google Scholar]] and Shi's non-degenerate tests [21 X. Shi, A non-degenerate Vuong test, Quant. Econ. 6 (2015), pp. 85–121. doi: 10.3982/QE382[Crossref], [Web of Science ®] , [Google Scholar]] suffer from the variance estimation and the departure of the normality of the likelihood ratios. To circumvent these dilemmas, we propose in this paper an empirical likelihood ratio (ELR) tests for model selection. Following Shi [21 X. Shi, A non-degenerate Vuong test, Quant. Econ. 6 (2015), pp. 85–121. doi: 10.3982/QE382[Crossref], [Web of Science ®] , [Google Scholar]], a bias correction method is proposed for the ELR tests to enhance its performance. A simulation study and a real-data analysis are provided to illustrate the performance of the proposed ELR tests.     

On an Extended Version of Skew Generalized Normal Distribution and Some of its Properties

Abstract

A generalization of Chauvenet's test (see Bol'shev, L. N. 1969 Bol'shev, L. N. 1969. On tests for rejecting outlying observations. Trudy In-ta prikladnoi Mat. Tblissi Gosudart. univ., 2: 159–177. (In Russian) [Google Scholar]. On tests for rejecting outlying observations. Trudy In-ta prikladnoi Mat. Tblissi Gosudart. univ. 2:159–177. (In Russian); Voinov, V. G., Nikulin, M. N. 1996 Voinov, V. G. and Nikulin, M. N. 1996. Unbaised Estimators and Their Applications Vol. 2, Kluwer Academic Publishers.  [Google Scholar]. Unbaised Estimators and Their Applications. Vol. 2. Kluwer Academic Publishers.) suitable to applied the problem of detecting r outliers in an univariate data set is proposed. In the exponential case, the Chauvenet's test can be used. Various modifications of this test were considered by Bol'shev, Ibrakimov and Khalfina (Ibrakimov, I. A., Khalfina 1978 Ibrakimov, I. A. and Khalfina. 1978. Some asymptotic results concerning the Chauvenet test. Ter. Veroyatnost. i Primenen., 23(3): 593–597.  [Google Scholar]. Some asymptotic results concerning the Chauvenet test. Ter. Veroyatnost. i Primenen. 23(3):593–597.), Greenwood and Nikulin (Greenwood, Nikulin, P. E. 1996 Greenwood and Nikulin, P. E. 1996. A Guide to Chi-Squared Testing New York: John Wiley and Sons, Inc..  [Google Scholar]. A Guide to Chi-Squared Testing. New York: John Wiley and Sons, Inc.) depending on the choice of the estimation method used: MLE or MVUE. As procedures for testing one outlier in exponential model have been investigated by a number of authors including Chikkagoudar and Kunchur (Chikkagoudar, M. S., Kunchur, S. H. 1983 Chikkagoudar, M. S. and Kunchur, S. H. 1983. Distribution of test statistics for multiple outliers in exponential samples. Comm. Stat. Theory. and Meth., 12: 2127–2142. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]. Distribution of test statistics for multiple outliers in exponential samples. Comm. Stat. Theory. and Meth. 12:2127–2142.), Lewis and Fieller (Lewis, T., Fiellerm N. R. J. 1979 Lewis, T. and Fiellerm, N. R. J. 1979. A recursive algorithm for null distribution for outliers: I. Gamma samples. Technometrics, 21: 371–376. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]. A recursive algorithm for null distribution for outliers : I. Gamma samples. Technometrics 21:371–376.), Likes (Likes, J. 1966 Likes, J. 1966. Distribution of Dixon's statistics in the case of an exponential population. Metrika, 11: 46–54. (91, 96, 136, 198–200, 204, 209, 210)[Crossref] , [Google Scholar]. Distribution of Dixon's statistics in the case of an exponential population. Metrika 11:46–54. (91, 96, 136, 198–200, 204, 209, 210).) and Kabe (Kabe, D. G. 1970 Kabe, D. G. 1970. Testing outliers from an exponential population. Metrika, 15: 15–18. [Crossref], [Web of Science ®] , [Google Scholar]. Testing outliers from an exponential population. Metrika 15:15–18.); only two types of statistics for testing multiple outliers exist. First is Dixon's while the second is based on the ratio of the sum of the observations suspected to be outliers to the sum of all observations of the sample. In fact, most of these authors have considered a general case of gamma model and the results for exponential model are given a special case. The object of the present communication is to focus on alternative models, namely slippage alternatives (see Barnett, Vic., Toby Lewis 1978 Barnett, Vic. and Toby, Lewis. 1978. Outlier in Statistical Data New York: John Wiley and Sons, Inc..  [Google Scholar]. Outlier in Statistical Data. New York: John Wiley and Sons, Inc.) in exponential samples. We propose a statistic different from the well known Dixon's statistic D_r to test for multiple outliers. Distribution of the test based on this new statistic under slippage alternatives is obtained and hence the tables of critical values are given, for various n (size of the sample) and r (the number of outliers). The power of the new test is also calculated, it is compared to the power of the Dixon's statistic (Chikkagoudar, M. S., Kunchur, S. H. 1983 Chikkagoudar, M. S. and Kunchur, S. H. 1983. Distribution of test statistics for multiple outliers in exponential samples. Comm. Stat. Theory. and Meth., 12: 2127–2142. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]. Distribution of test statistics for multiple outliers in exponential samples. Comm. Stat. Theory. and Meth. 12:2127–2142.). Notice that the new statistic based test power is greater the Dixon's statistic based test one.     

Random Effects,Fixed Effects and Hausman's Test for the Generalized Mixed Regressive Spatial Autoregressive Panel Data Model

This article suggests random and fixed effects spatial two-stage least squares estimators for the generalized mixed regressive spatial autoregressive panel data model. This extends the generalized spatial panel model of Baltagi et al. (2013 Baltagi, B. H., Egger, P., Pfaffermayr, M. (2013). A generalized spatial panel data model with random effects. Econometric Reviews 32:650–685.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) by the inclusion of a spatial lag term. The estimation method utilizes the Generalized Moments method suggested by Kapoor et al. (2007 Kapoor, M., Kelejian, H. H., Prucha, I. R. (2007). Panel data models with spatially correlated error components. Journal of Econometrics 127(1):97–130.[Crossref], [Web of Science ®] , [Google Scholar]) for a spatial autoregressive panel data model. We derive the asymptotic distributions of these estimators and suggest a Hausman test a la Mutl and Pfaffermayr (2011 Mutl, J., Pfaffermayr, M. (2011). The Hausman test in a Cliff and Ord panel model. Econometrics Journal 14:48–76.[Crossref], [Web of Science ®] , [Google Scholar]) based on the difference between these estimators. Monte Carlo experiments are performed to investigate the performance of these estimators as well as the corresponding Hausman test.     

Local influence diagnostics for generalized linear mixed models with overdispersion

Since the seminal paper by Cook and Weisberg [9 R.D. Cook and S. Weisberg, Residuals and Influence in Regression, Chapman &; Hall, London, 1982. [Google Scholar]], local influence, next to case deletion, has gained popularity as a tool to detect influential subjects and measurements for a variety of statistical models. For the linear mixed model the approach leads to easily interpretable and computationally convenient expressions, not only highlighting influential subjects, but also which aspect of their profile leads to undue influence on the model's fit [17 E. Lesaffre and G. Verbeke, Local influence in linear mixed models, Biometrics 54 (1998), pp. 570–582. doi: 10.2307/3109764[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]]. Ouwens et al. [24 M.J.N.M. Ouwens, F.E.S. Tan, and M.P.F. Berger, Local influence to detect influential data structures for generalized linear mixed models, Biometrics 57 (2001), pp. 1166–1172. doi: 10.1111/j.0006-341X.2001.01166.x[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]] applied the method to the Poisson-normal generalized linear mixed model (GLMM). Given the model's nonlinear structure, these authors did not derive interpretable components but rather focused on a graphical depiction of influence. In this paper, we consider GLMMs for binary, count, and time-to-event data, with the additional feature of accommodating overdispersion whenever necessary. For each situation, three approaches are considered, based on: (1) purely numerical derivations; (2) using a closed-form expression of the marginal likelihood function; and (3) using an integral representation of this likelihood. Unlike when case deletion is used, this leads to interpretable components, allowing not only to identify influential subjects, but also to study the cause thereof. The methodology is illustrated in case studies that range over the three data types mentioned.     

SECOND-ORDER ASYMPTOTICS FOR SCORE TESTS IN HETEROSKEDASTIC t REGRESSION MODELS

ABSTRACT

This paper develops corrected score tests for heteroskedastic t regression models, thus generalizing results by Cordeiro, Ferrari and Paula[1] Cordeiro, G.M., Ferrari, S.L.P. and Paula, G.A. 1993. Improved Score Tests for Generalized Linear Models. Journal of the Royal Statistical Society B, 55: 661–674.  [Google Scholar] and Cribari-Neto and Ferrari[2] Cribari-Neto, F. and Ferrari, S.L.P. 1995. Second-order Asymptotics for Score Tests in Generalised Linear Models. Biometrika, 82: 426–432. [Crossref], [Web of Science ®] , [Google Scholar] for normal regression models and by Ferrari and Arellano-Valle[3] Ferrari, S.L.P. and Arellano-Valle, R. 1996. Modified Likelihood Ratio and Score Tests in Linear Regression Models Using the t Distribution. Brazilian Journal of Probability and Statistics, 10: 15–33.  [Google Scholar] for homoskedastic t regression models. We present, in matrix notation, Bartlett-type correction formulae to improve score tests in this class of models. The corrected score statistics have a chi-squared distribution to order n ^?1, where n is the sample size. We apply our main result to a few special models and present simulation results comparing the performance of the usual score tests and their corrected versions.     

A new cure rate model based on the Yule–Simon distribution with application to a melanoma data set

In this paper, a new survival cure rate model is introduced considering the Yule–Simon distribution [12 H.A. Simon, On a class of skew distribution functions, Biometrika 42 (1955), pp. 425–440.[Crossref], [Web of Science ®] , [Google Scholar]] to model the number of concurrent causes. We study some properties of this distribution and the model arising when the distribution of the competing causes is the Weibull model. We call this distribution the Weibull–Yule–Simon distribution. Maximum likelihood estimation is conducted for model parameters. A small scale simulation study is conducted indicating satisfactory parameter recovery by the estimation approach. Results are applied to a real data set (melanoma) illustrating the fact that the model proposed can outperform traditional alternative models in terms of model fitting.     

A Generalized Log-Normal Model for Grouped Survival Data

It is common to have experiments in which it is not possible to observe the exact lifetimes but only the interval where they occur. This sort of data presents a high number of ties and it is called grouped or interval-censored survival data. Regression methods for grouped data are available in the statistical literature. The regression structure considers modeling the probability of a subject's survival past a visit time conditional on his survival at the previous visit. Two approaches are presented: assuming that lifetimes come from (1) a continuous proportional hazards model and (2) a logistic model. However, there may be situations in which none of the models are adequate for a particular data set. This article proposes the generalized log-normal model as an alternative model for discrete survival data. This model was introduced by Chen (1995 Chen , G. ( 1995 ). Generalized Log-normal distributions with reliability application . Comput. Stat. Data Anal. 19 : 300 – 319 . [Google Scholar]) and it is extended in this article for grouped survival data. A real example related to a Chagas disease illustrates the proposed model.     

Flexible parametric modelling of the hazard function in breast cancer studies

In cancer research, study of the hazard function provides useful insights into disease dynamics, as it describes the way in which the (conditional) probability of death changes with time. The widely utilized Cox proportional hazard model uses a stepwise nonparametric estimator for the baseline hazard function, and therefore has a limited utility. The use of parametric models and/or other approaches that enables direct estimation of the hazard function is often invoked. A recent work by Cox et al. [6 Cox, C., Chu, H., Schneider, M. F. and Munoz, A. 2007. Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Stat. Med., 26: 4352–4374. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]] has stimulated the use of the flexible parametric model based on the Generalized Gamma (GG) distribution, supported by the development of optimization software. The GG distribution allows estimation of different hazard shapes in a single framework. We use the GG model to investigate the shape of the hazard function in early breast cancer patients. The flexible approach based on a piecewise exponential model and the nonparametric additive hazards model are also considered.     

Longitudinal profiles of bounded outcome scores as predictors for disease activity in rheumatoid arthritis patients: a joint modeling approach

A variety of statistical approaches have been suggested in the literature for the analysis of bounded outcome scores (BOS). In this paper, we suggest a statistical approach when BOSs are repeatedly measured over time and used as predictors in a regression model. Instead of directly using the BOS as a predictor, we propose to extend the approaches suggested in [16 E. Lesaffre, D. Rizopoulos, and R. Tsonaka, The logistics-transform for bounded outcome scores, Biostatistics 8 (2007), pp. 72–85. doi: 10.1093/biostatistics/kxj034[Crossref], [PubMed], [Web of Science ®] , [Google Scholar],21 M. Molas and E. Lesaffre, A comparison of the three random effects approaches to analyse repeated bounded outcome scores with an application in a stroke revalidation study, Stat. Med. 27 (2008), pp. 6612–6633. doi: 10.1002/sim.3432[Crossref], [PubMed], [Web of Science ®] , [Google Scholar],28 R. Tsonaka, D. Rizopoulos, and E. Lesaffre, Power and sample size calculations for discrete bounded outcome scores, Stat. Med. 25 (2006), pp. 4241–4252. doi: 10.1002/sim.2679[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]] to a joint modeling setting. Our approach is illustrated on longitudinal profiles of multiple patients’ reported outcomes to predict the current clinical status of rheumatoid arthritis patients by a disease activities score of 28 joints (DAS28). Both a maximum likelihood as well as a Bayesian approach is developed.     

Weighting variables in Kohonen competitive learning algorithms

This paper presents a new variable weight method, called the singular value decomposition (SVD) approach, for Kohonen competitive learning (KCL) algorithms based on the concept of Varshavsky et al. [18 R. Varshavsky, A. Gottlieb, M. Linial, and D. Horn, Novel unsupervised feature filtering of bilogical data, Bioinformatics 22 (2006), pp. 507–513.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]]. Integrating the weighted fuzzy c-means (FCM) algorithm with KCL, in this paper, we propose a weighted fuzzy KCL (WFKCL) algorithm. The goal of the proposed WFKCL algorithm is to reduce the clustering error rate when data contain some noise variables. Compared with the k-means, FCM and KCL with existing variable-weight methods, the proposed WFKCL algorithm with the proposed SVD's weight method provides a better clustering performance based on the error rate criterion. Furthermore, the complexity of the proposed SVD's approach is less than Pal et al. [17 S.K. Pal, R.K. De, and J. Basak, Unsupervised feature evaluation: a neuro-fuzzy approach, IEEE. Trans. Neural Netw. 11 (2000), pp. 366–376.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]], Wang et al. [19 X.Z. Wang, Y.D. Wang, and L.J. Wang, Improving fuzzy c-means clustering based on feature-weight learning, Pattern Recognit. Lett. 25 (2004), pp. 1123–1132.[Crossref], [Web of Science ®] , [Google Scholar]] and Hung et al. [9 W. -L. Hung, M. -S. Yang, and D. -H. Chen, Bootstrapping approach to feature-weight selection in fuzzy c-means algorithms with an application in color image segmentation, Pattern Recognit. Lett. 29 (2008), pp. 1317–1325.[Crossref], [Web of Science ®] , [Google Scholar]].     

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

ABSTRACT

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

Matrix mean squared error comparisons of some biased estimators with two biasing parameters

To deal with multicollinearity problem, the biased estimators with two biasing parameters have recently attracted much research interest. The aim of this article is to compare one of the last proposals given by Yang and Chang (2010 Yang, H., and X. Chang. 2010. A new two-parameter estimator in linear regression. Communications in Statistics: Theory and Methods 39 (6):923–34.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) with Liu-type estimator (Liu 2003 Liu, K. 2003. Using Liu-type estimator to combat collinearity. Communications in Statistics: Theory and Methods 32 (5):1009–20.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and k ? d class estimator (Sakallioglu and Kaciranlar 2008 Sakallioglu, S., and S. Kaciranlar. 2008. A new biased estimator based on ridge estimation. Statistical Papers 49:669–89.[Crossref], [Web of Science ®] , [Google Scholar]) under the matrix mean squared error criterion. As well as giving these comparisons theoretically, we support the results with the extended simulation studies and real data example, which show the advantages of the proposal given by Yang and Chang (2010 Yang, H., and X. Chang. 2010. A new two-parameter estimator in linear regression. Communications in Statistics: Theory and Methods 39 (6):923–34.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) over the other proposals with increasing multicollinearity level.     

Analysis of discrete lifetime data under middle-censoring and in the presence of covariates

‘Middle censoring’ is a very general censoring scheme where the actual value of an observation in the data becomes unobservable if it falls inside a random interval (L, R) and includes both left and right censoring. In this paper, we consider discrete lifetime data that follow a geometric distribution that is subject to middle censoring. Two major innovations in this paper, compared to the earlier work of Davarzani and Parsian [3 N. Davarzani and A. Parsian, Statistical inference for discrete middle-censored data, J. Statist. Plan. Inference 141 (2011), pp. 1455–1462. doi: 10.1016/j.jspi.2010.10.012[Crossref], [Web of Science ®] , [Google Scholar]], include (i) an extension and generalization to the case where covariates are present along with the data and (ii) an alternate approach and proofs which exploit the simple relationship between the geometric and the exponential distributions, so that the theory is more in line with the work of Iyer et al. [6 S.K. Iyer, S.R. Jammalamadaka, and D. Kundu, Analysis of middle censored data with exponential lifetime distributions, J. Statist. Plan. Inference 138 (2008), pp. 3550–3560. doi: 10.1016/j.jspi.2007.03.062[Crossref], [Web of Science ®] , [Google Scholar]]. It is also demonstrated that this kind of discretization of life times gives results that are close to the original data involving exponential life times. Maximum likelihood estimation of the parameters is studied for this middle-censoring scheme with covariates and their large sample distributions discussed. Simulation results indicate how well the proposed estimation methods work and an illustrative example using time-to-pregnancy data from Baird and Wilcox [1 D.D. Baird and A.J. Wilcox, Cigarette smoking associated with delayed conception, J, Am. Med. Assoc. 253 (1985), pp. 2979–2983. doi: 10.1001/jama.1985.03350440057031[Crossref], [Web of Science ®] , [Google Scholar]] is included.     

The estimation for Lévy processes in high frequency data

In this article, a generalized Lévy model is proposed and its parameters are estimated in high-frequency data settings. An infinitesimal generator of Lévy processes is used to study the asymptotic properties of the drift and volatility estimators. They are consistent asymptotically and are independent of other parameters making them better than those in Chen et al. (2010 Chen, S. X., Delaigle, A., Hall, P. (2010). Nonparametric estimation for a class of Lévy processes. Journal of Econometrics 157:257–271.[Crossref], [Web of Science ®] , [Google Scholar]). The estimators proposed here also have fast convergence rates and are simple to implement.     

Records Properties of Non Stationary Time Series

This article proposes an asymptotic expansion for the Studentized linear discriminant function using two-step monotone missing samples under multivariate normality. The asymptotic expansions related to discriminant function have been obtained for complete data under multivariate normality. The result derived by Anderson (1973 Anderson , T. W. ( 1973 ). An asymptotic expansion of the distribution of the Studentized classification statistic W . The Annals of Statistics 1 : 964 – 972 .[Crossref], [Web of Science ®] , [Google Scholar]) plays an important role in deciding the cut-off point that controls the probabilities of misclassification. This article provides an extension of the result derived by Anderson (1973 Anderson , T. W. ( 1973 ). An asymptotic expansion of the distribution of the Studentized classification statistic W . The Annals of Statistics 1 : 964 – 972 .[Crossref], [Web of Science ®] , [Google Scholar]) in the case of two-step monotone missing samples under multivariate normality. Finally, numerical evaluations by Monte Carlo simulations were also presented.