Curve Fitting Under Jump and Peak Irregularities Using Local Linear Regression

In Desmet and Gijbels (2009 Desmet , L. , Gijbels , I. ( 2009 ). Local linear fitting and improved estimation near peaks . Canad. J. Statist. 37 ( 3 ): 453 – 475 .[Crossref] , [Google Scholar]), the problem of curve fitting on functions with peaks was addressed and a method was proposed that was building further on the one used in Gijbels et al. (2007 Gijbels , I. , Lambert , A. , Qiu , P. ( 2007 ). Jump-preserving regression and smoothing using local linear fitting: a compromise . Ann. Instit. Statist. Math. 59 : 235 – 272 .[Crossref], [Web of Science ®] , [Google Scholar]) when dealing with functions that have jump discontinuities. In this article, we propose a common framework for both estimation problems and an integrated procedure where an appropriate diagnostic quantity is a key ingredient. In particular we discuss practical procedure parameter selection considering target functions with an unknown number of irregularities of unknown type. The method is illustrated both on simulated and real life data sets.     

On Nonparametric Estimation of a Reliability Function

This article considers the properties of a nonparametric estimator developed for a reliability function which is used in many reliability problems. Properties such as asymptotic unbiasedness and consistency are proven for the estimator and using U-statistics, weak convergence of the estimator to a normal distribution is shown. Finally, numerical examples based on an extensive simulation study are presented to illustrate the theory and compare the estimator developed in this article with another based directly on the ratio of two empirical distributions studied in Zardasht and Asadi (2010 Zardasht , V. , Asadi , M. ( 2010 ). Evaluation of P(X _t  > Y _t ) when both X _t and Y _t are residual lifetimes of two systems . Statistica Neerlandica 64 : 460 – 481 .[Crossref], [Web of Science ®] , [Google Scholar]).     

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.     

Is Each NPMLE of a Continuous Bivariate Distribution Function with Singly Right-Censored Data Really Inconsistent?

We consider non-parametric estimation of a continuous cdf of a random vector (X ₁, X ₂). With bivariate RC data, it is stated in van der Laan (1996 Van der Laan , M. J. ( 1996 ) Efficient estimation in the bivariate censoring model and repairing NPMLE . Ann. Statist. 24 : 596 – 627 .[Crossref], [Web of Science ®] , [Google Scholar], p. 598¹⁰, Ann. Statist.), Quale et al. (2006 Quale , C. M. , van der Laan , M. J. , Robins , J. R. ( 2006 ). Locally efficient estimation with bivariate right-censored data . JASA. 101 : 1076 – 1084 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], JASA) etc. that “it is well known that the NPMLE for continuous data is inconsistent (Tsai et al. (1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar])).” The claim is based on a result in Tsai et al. (1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar], p.1352, Ann. Statist.) that if X ₁ is right censored but not X ₂, then common ways for defining one NPMLE lead to inconsistency. If X ₁ is right censored and X ₂ is type I right-censored (which includes the case in Tsai et al.), we present a consistent NPMLE. The result corrects a common misinterpretation of Tsai's example (Tsai et al., 1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar], Ann. Statist.).     

DSGE Models with Student-t Errors

This paper deals with Dynamic Stochastic General Equilibrium (DSGE) models under a multivariate student-t distribution for the structural shocks. Based on the solution algorithm of Klein (2000) and the gamma-normal representation of the t-distribution, the TaRB-MH algorithm of Chib and Ramamurthy (2010 Chib , S. , Ramamurthy , S. ( 2010 ). Tailored randomized block MCMC methods with application to DSGE models . Journal of Econometrics 108 : 19 – 38 .[Crossref], [Web of Science ®] , [Google Scholar]) is used to estimate the model. A technique for estimating the marginal likelihood of the DSGE student-t model is also provided. The methodologies are illustrated first with simulated data and then with the DSGE model of Ireland (2004 Ireland , P. N. ( 2004 ). Technology shocks in the new keynesian model . Review of Economics and Statistics 86 ( 4 ): 923 – 936 .[Crossref], [Web of Science ®] , [Google Scholar]) where the results support the t-error model in relation to the Gaussian model.     

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.     

Multivariate Aging with Archimedean Dependence Structures in High Dimensions

Bivariate aging notions for a vector X of lifetimes based on stochastic comparisons between X and X _t, where X _t is the multivariate residual lifetime after time t > 0, have been studied in Pellerey (2008 Pellerey , F. ( 2008 ). On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas . Kybernetika 44 : 795 – 806 .[Web of Science ®] , [Google Scholar]) under the assumption that the dependence structure in X is described by an Archimedean survival copula. Similar stochastic comparisons between X _t and X _t+s, for all t; s > 0, were considered in Mulero and Pellerey (2010 Mulero , J. , Pellerey , F. ( 2010 ). Bivariate aging properties under Archimedean dependence structures . Commun. Statist. Theor. Meth. 39 : 3108 – 3121 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). In this article, these results are generalized and extended to the multivariate case. Two illustrative examples are also provided.     

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.     

Some Remarks on Testing Overdispersion in Zero-Inflated Poisson and Binomial Regression Models

This note extends the score test statistics for overdispersion in Poisson and binomial regression models (Dean, 1992 Dean , C. B. ( 1992 ). Testing for overdispersion in Poisson and binomial regression models . J. Amer. Statist. Assoc. 87 : 451 – 457 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to the zero-inflated models. Some general results are obtained, and examples illustrate the application of the extended results.     

Efficient Block Designs for Dependent Observations—A Computer-Aided Search

In this article, the exchange and interchange algorithm of Zergaw (1989 Zergaw , G. ( 1989 ). A sequential method of constructing optimal block designs . Austral. J. Statist. 31 : 333 – 342 .[Crossref] , [Google Scholar]) and Martin and Eccleston (1992 Martin , R. J. , Eccleston , J. A. ( 1992 ). Recursive formulae for constructing block designs with dependent errors . Biometrika 79 : 426 – 430 .[Crossref], [Web of Science ®] , [Google Scholar]) have been modified and used for searching efficient block designs for making all possible pairwise treatment comparisons when observations are dependent. The lower bounds to the A- and D-efficiencies of the designs in a given class of the designs have been obtained for correlated observation structure and the procedure of computing lower bounds to A- and D-efficiencies has been incorporated in the algorithm. The algorithm has been translated into a computer program using Microsoft Visual C++. Using this program, a search for efficient designs for making all possible pairwise treatment comparisons has been made for v ≤ 10, b ≤ 33, k ≤ 10 such that bk ≤ 100 and v > k. The block designs considered are usual block designs (rectangular block designs) and circular block designs. Nearest neighbor (NN), autoregressive of order 1 (AR(1)) correlation structures are studied. The ranges of correlation coefficients for different correlation structures investigated are |ρ|≤0.50 for NN correlation structure in rectangular blocks, |ρ|≤0.45 for NN correlation structure in circular blocks, and |ρ|≤0.95 for AR(1) correlation structure. For these ranges, the matrix of correlation coefficients among observations within a block is positive definite. Robustness aspects of designs that are efficient for a given value of correlation have been investigated against other values of correlation coefficients. Robustness aspects of designs that are efficient for independent observations have also been studied for experimental situations with dependent observations.     

A Comparison of Procedures for Controlling the False Discovery Rate in the Presence of Small Variance Genes: A Simulation Study

The Significance Analysis of Microarrays (SAM; Tusher et al., 2001 Tusher , V. G. , Tibshirani , R. , Chu , G. ( 2001 ). Significance analysis of microarrys applied to the ionizing radiation response . Proceedings of the National Academy of Sciences 98 : 5116 – 5121 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) method is widely used in analyzing gene expression data while controlling the FDR by using resampling-based procedure in the microarray setting. One of the main components of the SAM procedure is the adjustment of the test statistic. The introduction of the fudge factor to the test statistic aims at deflating the large value of test statistics due to the small standard error of gene-expression. Lin et al. (2008 Lin , D. , Shkedy , Z. , Burzykowski , T. , Göhlmann , H. W. H. , De Bondt , A. , Perera , T. , Geerts , T. , Bijnens , L. ( 2008 ). Significance analysis of microarray (SAM) for comparisons of several treatments with one control . Biometric Journal, MCP 50 ( 5 ): 801 – 823 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) pointed out that the fudge factor does not effectively improve the power and the control of the FDR as compared to the SAM procedure without the fudge factor in the presence of small variance genes. Motivated by the simulation results presented in Lin et al. (2008 Lin , D. , Shkedy , Z. , Burzykowski , T. , Göhlmann , H. W. H. , De Bondt , A. , Perera , T. , Geerts , T. , Bijnens , L. ( 2008 ). Significance analysis of microarray (SAM) for comparisons of several treatments with one control . Biometric Journal, MCP 50 ( 5 ): 801 – 823 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), in this article, we extend our study to compare several methods for choosing the fudge factor in the modified t-type test statistics and use simulation studies to investigate the power and the control of the FDR of the considered methods.     

A Variance Estimator for Systematic Sampling from a Deliberately Ordered Population

ABSTRACT

The systematic sampling (SYS) design (Madow and Madow, 1944 Madow , L. H. , Madow , W. G. ( 1944 ). On the theory of systematic sampling . Ann. Math. Statist. 15 : 1 – 24 .[Crossref] , [Google Scholar]) is widely used by statistical offices due to its simplicity and efficiency (e.g., Iachan, 1982 Iachan , R. ( 1982 ). Systematic sampling a critical review . Int. Statist. Rev. 50 : 293 – 303 .[Crossref], [Web of Science ®] , [Google Scholar]). But it suffers from a serious defect, namely, that it is impossible to unbiasedly estimate the sampling variance (Iachan, 1982 Iachan , R. ( 1982 ). Systematic sampling a critical review . Int. Statist. Rev. 50 : 293 – 303 .[Crossref], [Web of Science ®] , [Google Scholar]) and usual variance estimators (Yates and Grundy, 1953 Yates , F. , Grundy , P. M. ( 1953 ). Selection without replacement from within strata with probability proportional to size . J. Roy. Statist. Soc. Series B 1 : 253 – 261 . [Google Scholar]) are inadequate and can overestimate the variance significantly (Särndal et al., 1992 Särndal , C. E. , Swenson , B. , Wretman , J. H. ( 1992 ). Model Assisted Survey Sampling . New York : Springer-Verlag , Ch. 3 .[Crossref] , [Google Scholar]). We propose a novel variance estimator which is less biased and that can be implemented with any given population order. We will justify this estimator theoretically and with a Monte Carlo simulation study.     

A Generalized Time-Dependent Conditional Linear Model with Left-Truncated and Right-Censored Data

Consider the model φ(S(y | X)) = β(y) ^T X, where φ is a known link function, S(· | X) is the survival function of a response Y given a covariate X = (1, X, X ²,…, X ^p ), and β(y) is an unknown vector of time-dependent regression coefficients. The response Y is subject to left truncation and right censoring. We assume that given X, Y is independent of (C, T) where C and T are censoring and truncation variables with P(C ≥ T) = 1. In this article, with some modification of the assumptions in Lemmas 5 and 6 of Iglesias-Pérez and González-Manteiga (1999 Iglesias-Pérez , C. J. , González-Manteiga , W. G. ( 1999 ). Strong representation of a generalized product-limit estimator for truncated and censored data with some application . J. Nonparametric Statist. 10 : 213 – 244 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), we present an almost sure representation for the generalized product-limit estimator (GPL) of S(y | X). Based on the GPL and the approach of Teodorescu et al. (2010 Teodorescu , B. , Keilegom , I. V. , Cao , R. ( 2010 ). Generalized time-dependent conditional linear models under left truncation and right censoring . Ann. Instit. Statist. Math. 62 : 465 – 485 .[Crossref], [Web of Science ®] , [Google Scholar]), a least squares estimator of β(y) is obtained and a bootstrap procedure is proposed to choose the optimum bandwidth.     

D-Optimal Chemical Balance Weighing Designs with N ≡ 0 (mod 4) and 3 Objects

In this article, the problem of estimation of the individual weights of three objects using a chemical balance weighing design is considered. We use the criterion of D-optimality. We assume that the covariance matrix of errors is the matrix of first-order autoregressive process. Such problems were discussed in Li and Yang (2005 Li , C. H. , Yang , S. Y. ( 2005 ). On a conjecture in D-optimal designs with n ≡ 0 (mod 4) . Lin. Alg. Applic. 400 : 279 – 290 .[Crossref], [Web of Science ®] , [Google Scholar]) and also in Yeh and Lo Huang (2005 Yeh , H. G. , Lo Huang , M. N. ( 2005 ). On exact D-optimal designs with 2 two-level factors and n autocorrelated observations . Metrika 61 : 261 – 275 .[Crossref], [Web of Science ®] , [Google Scholar]). We present some results of D-optimal designs in certain class of designs with the design matrix X  ∈ M _n×3(±1) such that each column of matrix X has at least one 1 and one ?1.     

Regression M-estimators with One Modified Form of Doubly Censored Data

In this article, we consider the M-estimators for the linear regression model when both response and covariate variables are subject to double censoring. The proposed estimators are constructed as some functional of three types of estimators for a bivariate survival distribution. The first two estimators are the generalizations of the Campbell and Földes (1982 Campbell, G. and Földes, A. 1982. “Large sample properties of nonparametric statistical inference”. In Nonparametric Statistical Inference., Edited by: Gnredenko, B. V., Puri, M. L. and Vineze, I. 103–122. Amsterdam: North-Holland.  [Google Scholar]) and Dabrowska (1988 Dabrowska, D. M. 1988. Kaplan-Meier estimate on the plane. Annals of Statistics, 18: 1475–1489. [Crossref], [Web of Science ®] , [Google Scholar]) estimators proposed by Shen (2009 Shen, P. S. 2009. Nonparametric estimation of the bivariate survival function one modified form of doubly censored data. Computational Statistics, 25: 203–313. [Crossref], [Web of Science ®] , [Google Scholar]). The third estimator is the generalization of the Prentice and Cai (1992 Prentice, R. L. and Cai, J. 1992. Covariance and survivor function estimation using censored multivariate failure time data. Biometrika, 79: 495–512. [Crossref], [Web of Science ®] , [Google Scholar]) estimator. The consistency of the proposed M-estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators. Furthermore, the simple bootstrap methods are used to estimate standard deviations and construct interval estimators.     

Aspects of Dependence in Generalized Farlie-Gumbel-Morgenstern Distributions

In this article, we obtained a dependence measure for generalized Farlie-Gumbel-Morgenstern (FGM) family in view of Kochar and Gupta (1987 Kochar , S. G. , Gupta , R. P. ( 1987 ). Competitors of Kendall-tau test for testing independence against PQD . Biometrika 74 ( 3 ): 664 – 669 .[Crossref], [Web of Science ®] , [Google Scholar]) and then compared this measure with Spearman's rho and Kendall's tau in FGM family. Moreover, we evaluated the empirical power of the class of distribution-free tests proposed by Kochar and Gupta (1987 Kochar , S. G. , Gupta , R. P. ( 1987 ). Competitors of Kendall-tau test for testing independence against PQD . Biometrika 74 ( 3 ): 664 – 669 .[Crossref], [Web of Science ®] , [Google Scholar], 1990 Kochar , S. G. , Gupta , R. P. ( 1990 ). Distribution-free tests based on sub-sample extrema for testing against positive dependence . Australian Journal of Statistics 32 : 45 – 51 .[Crossref] , [Google Scholar]) based on exact distribution of a U-statistics. This is derived via a simulation study for sample of sizes n = 6, 8, 10, 12, 16, and 20. Also, we compared our simulation results with those achieved by Amini et al. (2010 Amini , M. , Jabbari , H. , Mohtashami Borzadaran , G. R. , Azadbakhsh , M. ( 2010 ). Power comparison of independence test for the Farlie-Gumbel-Moregenstern family . Communications of the Korean Statistical Society 17 ( 4 ): 493 – 505 .[Crossref] , [Google Scholar]) and Güven and Kotz (2008 Güven , B. , Kotz , S. ( 2008 ). Test of independence for generalized Farlie-Gumbel-Morgenstern distributions . Journal of Computational and Applied Mathematics 212 : 102 – 111 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Performance Evaluation of Different Tests for Location Parameters

In this article, our objective is to evaluate the performance of different tests which are used to compare the equality of more than two location parameters. We have considered six tests (including some commonly used) in this study, one of which is parametric and the others are nonparametric. These tests include the usual F test (Fisher and Mackenzie, 1923 Fisher , R. A. , Mackenzie , M. A. ( 1923 ). Studies in crop variation. II. The manurial response of different potato . Journal of Agricultural Science 13 : 311 – 320 .[Crossref], [Web of Science ®] , [Google Scholar]), Kruskal–Wallis test (Kruskall and Wallis, 1952 Kruskall , W. H. , Wallis , W. A. ( 1952 ). Use of ranks in one-criterion variance analysis . Journal of American Statistical Association 47 : 583 – 621 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Kolmogorov–Smirnov test (David, 1958 David , H. T. ( 1958 ). Three-sample Kolnogorov–Smirnov test . The Annals of Mathematical Statistics 29 : 842 – 851 .[Crossref] , [Google Scholar]), the g test (Stekler, 1987 Stekler , H. O. ( 1987 ). Who forecasters better? Journal of Business and Economic Statistics 5 : 155 – 158 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), f test (Batchelor, 1990 Batchelor , R. A. ( 1990 ). All forecasters are equal . Journal of Business and Economic Statistics 8 : 143 – 144 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), and Extension of Median test (as given in Daniel, 1990 Daniel , W. W. ( 1990 ). Applied Nonparametric Statistics. , 2nd ed. Duxbury Classic Series , Boston . [Google Scholar]). Performance of these tests are compared under different symmetric, skewed and contaminated probability distributions that include Normal, Cauchy, Uniform, Laplace, Lognormal, Exponential, Weibull, Gamma, t, Chi-square, Half Normal, Mixed Weibull, and Mixed Normal. Performances of these tests are measured in terms of power. We have suggested appropriate tests which may perform better under different situations. It is expected that researchers will find these results useful in decision making.     

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.     

Shortest Expected Length Confidence Interval for the Power of the t-Test

Tarasińska (2005 Tarasińska , J. ( 2005 ). Confidence intervals for the power of student's t-test . Statistics & Probability Letters 73 ( 2 ): 125 – 130 .[Crossref], [Web of Science ®] , [Google Scholar]) considered a method to construct the shortest length confidence interval on the power of the t-test using a confidence interval for the population standard deviation in the non centrality parameter. Gilliland and Li (2008 Gilliland , D. , Li , M. ( 2008 ). A note on confidence intervals for the power of t-test . Statistics & Probability Letters 78 ( 5 ): 488 – 489 .[Crossref], [Web of Science ®] , [Google Scholar]) used simulations to show that this confidence interval has less than the nominal coverage, particularly in small samples. We propose to find the shortest expected length confidence interval for the power of the t-test by accounting for the variation in the sample standard deviation, and provide the necessary constants for its implementation for some selected sample and shift sizes. It is seen that the proposed interval is reasonably robust to the specification of the population standard deviation and maintains the nominal coverage.     

Regression Analysis with Covariates Missing at Random: A Piece-wise Nonparametric Model for Missing Covariates

Statistical analysis for the regression model f _β(y | x, z) with missing values in the covariate vector X requires modeling of the covariate distribution g(x | z). Likelihood methods, including Ibrahim (1990 Ibrahim , J. G. ( 1990 ). Incomplete data in generalized linear models . J. Amer. Statist. Assoc. 85 : 765 – 769 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Chen (2004 Chen , H. Y. (2004). Nonparametric and semiparametric models for missing covariates in parametric regression. J. Amer. Statist. Assoc. 99:1176–1189.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), and Zhao (2005 Zhao , Y. ( 2005 ). Design and Efficient Estimation in Regression Analysis with Missing Data in Two-Phase Studies. Ph.D. thesis , University of Waterloo . [Google Scholar]), need either X or Z to be discrete. This article considers extending the likelihood methods to deal with cases where both X and Z may be continuous. We propose modeling the covariate distribution g(x | z) using a piece-wise nonparametric model, then a maximum likelihood estimate (MLE) of β can be computed following the maximum likelihood estimating procedure of Chen (2004 Chen , H. Y. (2004). Nonparametric and semiparametric models for missing covariates in parametric regression. J. Amer. Statist. Assoc. 99:1176–1189.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) or Zhao (2005 Zhao , Y. ( 2005 ). Design and Efficient Estimation in Regression Analysis with Missing Data in Two-Phase Studies. Ph.D. thesis , University of Waterloo . [Google Scholar]). The resulting estimation method is easy to implement and the asymptotic properties of the MLE follow under certain conditions. Extensive simulation studies for different models indicate that the proposed method is acceptable for practical implementation. A real data example is used to illustrate the method.