Improved P-Values for Testing Marginal Homogeneity in 2 × 2 Contingency Tables

This article considers the problem of testing marginal homogeneity in a 2 × 2 contingency table. We first review some well-known conditional and unconditional p-values appeared in the statistical literature. Then we treat the p-value as the test statistic and use the unconditional approach to obtain the modified p-value, which is shown to be valid. For a given nominal level, the rejection region of the modified p-value test contains that of the original p-value test. Some nice properties of the modified p-value are given. Especially, under mild conditions the rejection region of the modified p-value test is shown to be the Barnard convex set as described by Barnard (1947 Barnard , G. A. ( 1947 ). Significance tests for 2 × 2 tables . Biometrika 34 : 123 – 138 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). If the one-sided null hypothesis has two nuisance parameters, we show that this result can reduce the dimension of the nuisance parameter space from two to one for computing modified p-values and sizes of tests. Numerical studies including an illustrative example are given. Numerical comparisons show that the sizes of the modified p-value tests are closer to a nominal level than those of the original p-value tests for many cases, especially in the case of small to moderate sample sizes.     

Identification-robust moment-based tests for Markov switching in autoregressive models

ABSTRACT

This paper develops tests of the null hypothesis of linearity in the context of autoregressive models with Markov-switching means and variances. These tests are robust to the identification failures that plague conventional likelihood-based inference methods. The approach exploits the moments of normal mixtures implied by the regime-switching process and uses Monte Carlo test techniques to deal with the presence of an autoregressive component in the model specification. The proposed tests have very respectable power in comparison with the optimal tests for Markov-switching parameters of Carrasco et al. (2014 Carrasco, M., Hu, L., Ploberger, W. (2014). Optimal test for Markov switching parameters. Econometrica 82(2):765–784.[Crossref], [Web of Science ®] , [Google Scholar]), and they are also quite attractive owing to their computational simplicity. The new tests are illustrated with an empirical application to an autoregressive model of USA output growth.     

A Confidence Interval and Test for the Mean of an Asymmetric Distribution

ABSTRACT

A confidence interval and test are obtained for the mean of an asymmetric distribution using a random sample of size n. The method is based on N. J. Johnson's (1978 Johnson , N. J. ( 1978 ). Modified, t tests and confidence intervals for asymmetrical populations. J. Amer. Statist. Assoc. 73 ( 363 ): 536 – 544 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) modified t-test, where terms of Cornish–Fisher expansions involving the third moment are used to adjust the conventional statistic to have more closely a Student's t-distribution with n ? 1 degrees of freedom. Johnson's (1978 Johnson , N. J. ( 1978 ). Modified, t tests and confidence intervals for asymmetrical populations. J. Amer. Statist. Assoc. 73 ( 363 ): 536 – 544 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) test cannot be inverted uniquely, so a corresponding confidence interval for the mean may be disjointed. However, an artificial term of small order can be added to make inversion of the test a uniquely defined operation, which prevents such disjointedness. The resulting one-sided and two-sided intervals perform better than others in the literature with skewed distributions, and have good performance with a normal distribution. The two-sided interval may be recommended for general use if the sample size is 10 or more and the nominal confidence coefficient is 95% or less, or if the sample size is 30 or more and the confidence coefficient is 99% or less.     

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.     

Testing for Unit Roots and the Impact of Quadratic Trends,with an Application to Relative Primary Commodity Prices

In practice a degree of uncertainty will always exist concerning what specification to adopt for the deterministic trend function when running unit root tests. While most macroeconomic time series appear to display an underlying trend, it is often far from clear whether this component is best modeled as a simple linear trend (so that long-run growth rates are constant) or by a more complicated nonlinear trend function which may, for instance, allow the deterministic trend component to evolve gradually over time. In this article, we consider the effects on unit root testing of allowing for a local quadratic trend, a simple yet very flexible example of the latter. Where a local quadratic trend is present but not modeled, we show that the quasi-differenced detrended Dickey–Fuller-type test of Elliott et al. (1996 Elliott , G. , Rothenberg , T. J. , Stock , J. H. ( 1996 ). Efficient tests for an autoregressive unit root . Econometrica 64 : 813 – 836 .[Crossref], [Web of Science ®] , [Google Scholar]) has both size and power which tend to zero asymptotically. An extension of the Elliott et al. (1996 Elliott , G. , Rothenberg , T. J. , Stock , J. H. ( 1996 ). Efficient tests for an autoregressive unit root . Econometrica 64 : 813 – 836 .[Crossref], [Web of Science ®] , [Google Scholar]) approach to allow for a quadratic trend resolves this problem but is shown to result in large power losses relative to the standard detrended test when no quadratic trend is present. We consequently propose a simple and practical approach to dealing with this form of uncertainty based on a union of rejections-based decision rule whereby the unit root is rejected whenever either of the detrended or quadratic detrended unit root tests rejects. A modification of this basic strategy is also suggested which further improves on the properties of the procedure. An application to relative primary commodity price data highlights the empirical relevance of the methods outlined in this article. A by-product of our analysis is the development of a test for the presence of a quadratic trend which is robust to whether the data admit a unit root.     

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.     

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]).     

Nonparametric Estimators of the Distribution Function for One Modified Model of Current Status Data

In this article, we consider the estimation of distribution function for one modified form of current status data. An inverse-probability-weighted (IPW) estimator and a self-consistent estimator (SCE) are proposed. The asymptotic properties of the IPW estimator are derived. A simulation study is conducted to compare the performances among the IPW estimator, SCE, and the product-limit estimator proposed by Patilea and Rolin (2006 Patilea , V. , Rolin , J.-M. (2006). Product-limit estimators of the survival function for two modified forms of current-status data. Bernoulli 12(5):801–819.[Crossref], [Web of Science ®] , [Google Scholar]). Simulation results indicate that when right censoring is light and left censoring is heavy, both IPW estimator and SCE can outperform the product-limit estimator. The performances of the IPW estimator and SCE are close to each other.     

Estimating Prediction Error: Cross-Validation vs. Accumulated Prediction Error

We study the validation of prediction rules such as regression models and classification algorithms through two out-of-sample strategies, cross-validation and accumulated prediction error. We use the framework of Efron (1983 Efron , B. ( 1983 ). Estimating the error rate of a prediction rule: improvement on cross-validation . Journal of the American Statistical Association 78 : 316 – 331 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) where measures of prediction errors are defined as sample averages of expected errors and show through exact finite sample calculations that cross-validation and accumulated prediction error yield different smoothing parameter choices in nonparametric regression. The difference in choice does not vanish as sample size increases.     

A Generalization to Bivariate Mittag–Leffler and Bivariate Discrete Mittag–Leffler Autoregressive Processes

A generalization of the Gaver and Lewis (1980 Gaver , D. P. , Lewis , P. A. W. ( 1980 ). First order autoregressive gamma sequences and point processes . Adv. Appl. Probab. 12 : 727 – 745 .[Crossref], [Web of Science ®] , [Google Scholar]) model of first-order autoregressive process with marginals as bivariate Mittag–Leffler distribution is obtained. A necessary and sufficient condition for stationarity of the process is established. Autoregressive process with marginals follow bivariate discrete Mittag–Leffler distribution is also developed. The unknown parameters of the processes are estimated and some numerical results of the estimations are given.     

Block–Savits Characterization and Star Ordering of Exponential Mixtures

Block and Savits (1980 Block , H. W. , Savits , T. H. ( 1980 ). Laplace transforms for classes of life distributions . Ann. Probab. 8 : 465 – 474 .[Crossref], [Web of Science ®] , [Google Scholar]) established a characterization of life distributions using the Laplace transform. In this article, we remark that one of the necessary conditions to be IFRA distribution is equivalent to the star ordering of exponential mixtures. It leads to the definition of two new classes of life distributions, called LIFR and LIFRA, and their dual classes: LDFR and LDFRA. It occurs that these classes have many useful aging properties and preserve known reliability operations. Properties of the classes are studied and relations with known classes are established.     

Parametric Nonlinear Regression with Endogenous Switching

Based on the insightful work of Olsen (1980 Olsen , R. J. ( 1980 ). A least squares correction for selectivity bias . Econometrica 48 : 1815 – 1820 .[Crossref], [Web of Science ®] , [Google Scholar]) for the linear context, a generic and unifying framework is developed that affords a simple extension of the classical method of Heckman (1974 Heckman , J. ( 1974 ). Shadow prices, market wages, and labor supply . Econometrica 42 : 679 – 694 .[Crossref], [Web of Science ®] , [Google Scholar], 1976 Heckman , J. ( 1976 ). The common structure of statistical models of truncation sample selection and limited dependent variables and a simple estimator for such models . Annals of Economic and Social Measurement 5 : 475 – 492 . [Google Scholar], 1978 Heckman , J. ( 1978 ). Dummy endogenous variables in a simultaneous equation system . Econometrica 46 : 931 – 959 .[Crossref], [Web of Science ®] , [Google Scholar], 1979 Heckman , J. ( 1979 ). Sample selection bias as a specification error . Econometrica 47 : 153 – 161 .[Crossref], [Web of Science ®] , [Google Scholar]) to a broad class of nonlinear regression models involving endogenous switching and its two most common incarnations, endogenous sample selection and endogenous treatment effects. The approach should be appealing to applied researchers for three reasons. First, econometric applications involving endogenous switching abound. Secondly, the approach requires neither linearity of the regression function nor full parametric specification of the model. It can, in fact, be applied under the minimal parametric assumptions—i.e., specification of only the conditional means of the outcome and switching variables. Finally, it is amenable to relatively straightforward estimation methods. Examples of applications of the method are discussed.     

A Note on Testing Homogeneity of Several Exponential Location Parameters

In this article, the problems of testing homogeneity of several exponential location parameters against simple and tree ordered alternatives are considered separately. Test procedures for both the alternatives are proposed using restricted maximum likelihood estimators (RMLE) of exponential location parameters under the respective orderings. Critical constants for the implementation of the proposed procedures are tabulated. Power comparison of the proposed test procedure under the simple ordered alternative with the procedure of Chen (1982 Chen , H. J. ( 1982 ). A new range statistic for comparisons of several exponential location parameters . Biometrika 69 ( 1 ): 257 – 260 .[Crossref], [Web of Science ®] , [Google Scholar]) and of Dhawan and Gill (1999 Dhawan , A. K. , Gill , A. N. ( 1999 ). A one-sided test for testing homogeneity of scale parameters against ordered alternative . Communication in Statistics—Theory and Methods 28 ( 10 ): 2417 – 2439 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is carried out using Monte-Carlo simulation.     

Interval Estimation for the Scaled Half-Logistic Distribution Under Progressive Type-II Censoring

In this article, we propose an exact confidence interval and an exact test for the scale parameter of the scaled half-logistic distribution based on progressively Type-II censored sample. Using the Monte Carlo method, we compare the expected length of the proposed confidence interval with one of confidence interval proposed by Balakrishnan and Asgharzadeh (2005 Balakrishnan , N. , Asgharzadeh , A. ( 2005 ). Inference for the scaled half-logistic distribution based on progressively Type-II censored sample . Commun. Statist. Theor. Meth. 34 : 73 – 87 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The simulation results show that the proposed confidence interval performs better. Finally, we present a numerical example to illustrate the proposed procedures.     

Model selection in regression based on pre-smoothing

In this paper, we investigate the effect of pre-smoothing on model selection. Christóbal et al 6 Christóbal Christóbal, J. A., Faraldo Roca, P. and González Manteiga, W. 1987. A class of linear regression parameter estimators constructed by nonparametric estimation. Ann. Statist.,, 15: 603–609. [Crossref], [Web of Science ®] [Google Scholar] showed the beneficial effect of pre-smoothing on estimating the parameters in a linear regression model. Here, in a regression setting, we show that smoothing the response data prior to model selection by Akaike's information criterion can lead to an improved selection procedure. The bootstrap is used to control the magnitude of the random error structure in the smoothed data. The effect of pre-smoothing on model selection is shown in simulations. The method is illustrated in a variety of settings, including the selection of the best fractional polynomial in a generalized linear model.     

Tuning Parameter Selection and Various Good Fitting Characteristics for the Liu-Type Estimator in Linear Regression

Liu (2003 Liu , K. ( 2003 ). Using Liu-Type estimator to combat collinearity . Commun. Statist. Theor. Meth. 32 ( 5 ): 1009 – 1020 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) proposed the Liu-Type estimator (LTE) to combat the well-known multicollinearity problem in linear regression. In this article, various better fitting characteristics of the LTE than those of the ordinary ridge regression estimator (Hoerl and Kennard, 1970 Hoerl , A. E. , Kennard , R. W. ( 1970 ). Ridge regression: Biased estimation for non-orthogonal problems . Technometrics 12 : 55 – 67 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are considered. In particular, we derived two methods to determine the parameter d for the LTE and find that the ridge parameter k could serve for regularization of an ill-conditioned design matrix, while the other parameter d could be used for tuning the fit quality. In addition, the coefficients of regression, coefficient of multiple determination, residual error variance, and generalized cross validation (GCV) of the prediction quality are very stable, and as the ridge parameter increases they eventually reach asymptotic levels, which produces robust regression models. Furthermore, a Monte Carlo evaluation of these features is also given to illustrate some of the theoretical results.     

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.     

A Meta Analytic Approach to Testing for Panel Cointegration

We propose new tests for panel cointegration by extending the panel unit root tests of Choi (2001 Choi , I. ( 2001 ). Unit root tests for panel data . Journal of International Money and Finance 20 ( 2 ): 249 – 272 .[Crossref], [Web of Science ®] , [Google Scholar]) and Maddala and Wu (1999 Maddala , G. , Wu , S. ( 1999 ). A comparative study of unit root tests with panel data and a new simple test . Oxford Bulletin of Economics and Statistics 61 ( S1 ): 631 – 652 .[Crossref] , [Google Scholar]) to the panel cointegration case. The tests are flexible, intuitively appealing, and relatively easy to compute. We investigate the finite sample behavior in a simulation study. Several variants of the tests compare favorably in terms of both size and power with other widely used panel cointegration tests.     

The Asymptotic Estimate of Ruin Probability Under a Class of Risk Model in the Presence of Heavy Tails

In contrast with the classical Cramér–Lundberg model where the premium process is a linear function of time, we consider the ruin probability under the risk model where the aggregate premium consists of both a compound Poisson process and a linear process of time. Moreover, a constant interest force is also taken into account in our model. We restrict ourselves to the case where the claim size is heavy-tailed, i.e., the equilibrium distribution function of the claim size belongs to a wide subclass of the subexponential distribution. An asymptotic formula for the ruin probability is obtained by using the similar method of Kalashnikov and Konstantinides (2000 Kalashnikov , V. , Konstantinides , D. ( 2000 ). Ruin under interest force and subexponential claims: a simple treament . Insur. Math. Econ. 27 : 145 – 149 .[Crossref], [Web of Science ®] , [Google Scholar]). The asymptotic formula we get here is the same as the one in Asmussen (1998 Asmussen , S. ( 1998 ). Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distribution and first passage probabilities . Ann. Appl. Probab. 8 : 354 – 374 .[Crossref], [Web of Science ®] , [Google Scholar]), Klüppelberg and Stadtmüller (1998 Klüppelberg , C. , Stadtmüller , U. ( 1998 ). Ruin probabilities in the presence of heavy-tails and interest rates . Scand. Actuarial J. 1 : 49 – 58 .[Taylor & Francis Online] , [Google Scholar]), and Kalashnikov and Konstantinides (2000 Kalashnikov , V. , Konstantinides , D. ( 2000 ). Ruin under interest force and subexponential claims: a simple treament . Insur. Math. Econ. 27 : 145 – 149 .[Crossref], [Web of Science ®] , [Google Scholar]) which did not consider the stochastic premium.     

The Identification of Potential Longitudinal Biomarkers and Measurements of Effectiveness for Biomarkers as Surrogates in Multivariate Survival Data

A proposed method based on frailty models is used to identify longitudinal biomarkers or surrogates for a multivariate survival. This method is an extention of earlier models by Wulfsohn and Tsiatis (1997 Wulfsohn , M. S. , Tsiatis , A. A. ( 1997 ). A joint model for survival and longitudinal data measured with error . Biometrics 53 ( 1 ): 330 – 339 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Song et al. (2002 Song , X. , Davidian , M. , Tsiatis , A. A. ( 2002 ). A Semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data . Biometrics 58 ( 4 ): 742 – 753 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). In this article, similar to Henderson et al. (2002 Henderson , R. , Diggle , P. J. , Dobson , A. ( 2002 ). Identification and efficacy of longitudinal markers for survival . Biostatistics 3 ( 1 ): 33 – 50 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), a joint likelihood function combines the likelihood functions of the longitudinal biomarkers and the multivariate survival times. We use simulations to explore how the number of individuals, the number of time points per individual and the functional form of the random effects from the longitudianl biomarkers influence the power to detect the association of a longitudinal biomarker and the multivariate survival time. The proposed method is illustrate by using the gastric cancer data.