ASYMPTOTIC DISTRIBUTIONS OF SEMIPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATORS WITH ESTIMATING EQUATIONS FOR GROUP-CENSORED DATA

Semiparametric maximum likelihood estimation with estimating equations (SMLE) is more flexible than traditional methods; it has fewer restrictions on distributions and regression models. The required information about distribution and regression structures is incorporated in estimating equations of the SMLE to improve the estimation quality of non‐parametric methods. The likelihood of SMLE for censored data involves complicated implicit functions without closed‐form expressions, and the first derivatives of the log‐profile‐likelihood cannot be expressed as summations of independent and identically distributed random variables; it is challenging to derive asymptotic properties of the SMLE for censored data. For group‐censored data, the paper shows that all the implicit functions are well defined and obtains the asymptotic distributions of the SMLE for model parameters and lifetime distributions. With several examples the paper compares the SMLE, the regular non‐parametric likelihood estimation method and the parametric MLEs in terms of their asymptotic efficiencies, and illustrates application of SMLE. Various asymptotic distributions of the likelihood ratio statistics are derived for testing the adequacy of estimating equations and a partial set of parameters equal to some known values.     

Modified likelihood ratio statistics for inflated beta regressions

The class of inflated beta regression models generalizes that of beta regressions [S.L.P. Ferrari and F. Cribari-Neto, Beta regression for modelling rates and proportions, J. Appl. Stat. 31 (2004), pp. 799–815] by incorporating a discrete component that allows practitioners to model data on rates and proportions with observations that equal an interval limit. For instance, one can model responses that assume values in (0, 1]. The likelihood ratio test tends to be quite oversized (liberal, anticonservative) in inflated beta regressions estimated with a small number of observations. Indeed, our numerical results show that its null rejection rate can be almost twice the nominal level. It is thus important to develop alternative testing strategies. This paper develops small-sample adjustments to the likelihood ratio and signed likelihood ratio test statistics in inflated beta regression models. The adjustments do not require orthogonality between the parameters of interest and the nuisance parameters and are fairly simple since they only require first- and second-order log-likelihood cumulants. Simulation results show that the modified likelihood ratio tests deliver much accurate inference in small samples. An empirical application is presented and discussed.     

THE REDUCED-RANK GROWTH CURVE MODEL FOR DISCRIMINANT ANALYSIS OF LONGITUDINAL DATA

This paper presents a method of discriminant analysis especially suited to longitudinal data. The approach is in the spirit of canonical variate analysis (CVA) and is similarly intended to reduce the dimensionality of multivariate data while retaining information about group differences. A drawback of CVA is that it does not take advantage of special structures that may be anticipated in certain types of data. For longitudinal data, it is often appropriate to specify a growth curve structure (as given, for example, in the model of Potthoff & Roy, 1964). The present paper focuses on this growth curve structure, utilizing it in a model-based approach to discriminant analysis. For this purpose the paper presents an extension of the reduced-rank regression model, referred to as the reduced-rank growth curve (RRGC) model. It estimates discriminant functions via maximum likelihood and gives a procedure for determining dimensionality. This methodology is exploratory only, and is illustrated by a well-known dataset from Grizzle & Allen (1969).     

On testing inference in beta regressions

This article deals with testing inference in the class of beta regression models with varying dispersion. We focus on inference in small samples. We perform a numerical analysis in order to evaluate the sizes and powers of different tests. We consider the likelihood ratio test, two adjusted likelihood ratio tests proposed by Ferrari and Pinheiro [Improved likelihood inference in beta regression, J. Stat. Comput. Simul. 81 (2011), pp. 431–443], the score test, the Wald test and bootstrap versions of the likelihood ratio, score and Wald tests. We perform tests on the parameters that index the mean submodel and also on the parameters in the linear predictor of the precision submodel. Overall, the numerical evidence favours the bootstrap tests. It is also shown that the score test is considerably less size-distorted than the likelihood ratio and Wald tests. An application that uses real (not simulated) data is presented and discussed.     

NEIGHBOUR ANALYSIS OF UNIFORMITY DATA

The neighbour model of Williams (1986) is studied using barley uniformity data. Particular attention is paid to the validity of the model. This is done by comparing the variety and residual mean squares in the intra-N analysis.     

The likelihood ratio test for high-dimensional linear regression model

The paper considers a significance test of regression variables in the high-dimensional linear regression model when the dimension of the regression variables p, together with the sample size n, tends to infinity. Under two sightly different cases, we proved that the likelihood ratio test statistic will converge in distribution to a Gaussian random variable, and the explicit expressions of the asymptotical mean and covariance are also obtained. The simulations demonstrate that our high-dimensional likelihood ratio test method outperforms those using the traditional methods in analyzing high-dimensional data.     

Improved likelihood inference for the shape parameter in Weibull regression

We obtain adjustments to the profile likelihood function in Weibull regression models with and without censoring. Specifically, we consider two different modified profile likelihoods: (i) the one proposed by Cox and Reid [Cox, D.R. and Reid, N., 1987, Parameter orthogonality and approximate conditional inference. Journal of the Royal Statistical Society B, 49, 1–39.], and (ii) an approximation to the one proposed by Barndorff–Nielsen [Barndorff–Nielsen, O.E., 1983, On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70, 343–365.], the approximation having been obtained using the results by Fraser and Reid [Fraser, D.A.S. and Reid, N., 1995, Ancillaries and third-order significance. Utilitas Mathematica, 47, 33–53.] and by Fraser et al. [Fraser, D.A.S., Reid, N. and Wu, J., 1999, A simple formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 655–661.]. We focus on point estimation and likelihood ratio tests on the shape parameter in the class of Weibull regression models. We derive some distributional properties of the different maximum likelihood estimators and likelihood ratio tests. The numerical evidence presented in the paper favors the approximation to Barndorff–Nielsen's adjustment.     

AN ALGORITHM FOR THE DESIGN OF 2? FACTORIAL EXPERIMENTS ON CONTINUOUS PROCESSES

Saunders & Eccleston (1992) presented an approach to the design of 2-level factorial experiments for continuous processes. It determined sets of contrasts between the observations that could be well estimated, and then selected a design so that those contrasts estimated the parameters of interest. This paper shows that a well-estimated contrast must have a large number of changes of sign or level, and also be ‘paired’ in a particular sense. It develops an algorithm which constructs designs that must have a large number of changes of sign, evenly spread among the contrasts and optimal or near optimal. When such designs exist they are often preferable to those produced by the reverse foldover algorithm of Cheng & Steinberg (1991).     

Multivariate Life Testing in Variably Scaled Environments

This paper examines modeling and inference questions for experiments in which different subsets of a set of k possibly dependent components are tested in r different environments. In each environment, the failure times of the set of components on test is assumed to be governed by a particular type of multivariate exponential (MVE) distribution. For any given component tested in several environments, it is assumed that its marginal failure rate varies from one environment to another via a change of scale between the environments, resulting in a joint MVE model which links in a natural way the applicable MVE distributions describing component behavior in each fixed environment. This study thus extends the work of Proschan and Sullo (1976) to multiple environments and the work of Kvam and Samaniego (1993) to dependent data. The problem of estimating model parameters via the method of maximum likelihood is examined in detail. First, necessary and sufficient conditions for the identifiability of model parameters are established. We then treat the derivation of the MLE via a numerically-augmented application of the EM algorithm. The feasibility of the estimation method is demonstrated in an example in which the likelihood ratio test of the hypothesis of equal component failure rates within any given environment is carried out.     

ANALYSIS OF MASKED DATA IN A DEPENDENT COMPETING RISKS MODEL UNDER UNKNOWN ENVIRONMENT

ABSTRACT

System failure data is often analyzed to estimate component reliabilities. Due to cost and time constraints, the exact component causing the failure of the system cannot be identified in some cases. This phenomenon is called masking. Further, it is sometimes necessary for us to take account of the influence of the operating environment. Here we consider a series system, operating under unknown environment, of two components whose failure times follow the Marshall-Olkin bivariate exponential distribution. We present a maximum likelihood approach for obtaining estimators from the masked data for this system. From a simulation study, we found that the relative errors of the estimates are almost well behaved even for small or moderate expected number of systems whose cause of failure is identified.     

TEST FOR TREATMENT EFFECT BASED ON BINARY DATA WITH RANDOM SAMPLE SIZES

The problem of testing for treatment effect based on binary response data is considered, assuming that the sample size for each experimental unit and treatment combination is random. It is assumed that the sample size follows a distribution that belongs to a parametric family. The uniformly most powerful unbiased tests, which are equivalent to the likelihood ratio tests, are obtained when the probability of the sample size being zero is positive. For the situation where the sample sizes are always positive, the likelihood ratio tests are derived. These test procedures, which are unconditional on the random sample sizes, are useful even when the random sample sizes are not observed. Some examples are presented as illustration.     

A SEMIPARAMETRIC MIXTURE MODEL FOR THE ANALYSIS OF COMPETING RISKS DATA

This paper deals with the regression analysis of failure time data when there are censoring and multiple types of failures. We propose a semiparametric generalization of a parametric mixture model of Larson & Dinse (1985), for which the marginal probabilities of the various failure types are logistic functions of the covariates. Given the type of failure, the conditional distribution of the time to failure follows a proportional hazards model. A marginal like lihood approach to estimating regression parameters is suggested, whereby the baseline hazard functions are eliminated as nuisance parameters. The Monte Carlo method is used to approximate the marginal likelihood; the resulting function is maximized easily using existing software. Some guidelines for choosing the number of Monte Carlo replications are given. Fixing the regression parameters at their estimated values, the full likelihood is maximized via an EM algorithm to estimate the baseline survivor functions. The methods suggested are illustrated using the Stanford heart transplant data.     

Change-point detection in a shape-restricted regression model

A regression model with a possible structural change and with a small number of measurements is considered. A priori information about the shape of the regression function is used to formulate the model as a linear regression model with inequality constraints and a likelihood ratio test for the presence of a change-point is constructed. The exact null distribution of the test statistic is given. Consistency of the test is proved when the noise level goes to zero. Numerical approximations to the powers against various alternatives are given and compared with the powers of the k-linear-r-ahead recursive residuals tests and CUSUM tests. Performance of four different estimators of the change-point is studied in a Monte Carlo experiment. An application of the procedures to some real data is also presented.     

Analysis of ordered covariate effects among groups with repeated measurements

Testing procedures for ordered covariate effects are developed in the repeated measures experiment. The maximum likelihood estimators of covariate effects under the ordered hypothesis are approximated by the isotonic regression of their unconstrained estimators. The asymptotic null distributions of the test statistics are chi-bar-square distributions which are mixtures of chi-square distributions. A Monte-Carlo simulation reveals that the proposed test for ordered covariate effects is seriously more powerful than the usual chi-square test that neglects the information on the order restriction. These testing methods are applied for analyzing the effect of vitamin E diet supplement on growth rate of animals.     

Sampling experiments and the quandt statistic

The distribution of the Quandt likelihood ratio λ_t. for a two-phase regression, has yet to be determined. In particular it is known that-2 log λ_t. is not distributed as chi-square (Quandt; 1960), for unknown switch point.

In this paper we describe sampling experiments which suggest that-log λ_t, has a Pearson Type III distribution, The parameters of the distribution appear to depend not only on the values of the x-vector (Feder; 1968) but also its dimension k.     

Testing the linearity of negative binomial regression models

The negative binomial (NB) is frequently used to model overdispersed Poisson count data. To study the effect of a continuous covariate of interest in an NB model, a flexible procedure is used to model the covariate effect by fixed-knot cubic basis-splines or B-splines with a second-order difference penalty on the adjacent B-spline coefficients to avoid undersmoothing. A penalized likelihood is used to estimate parameters of the model. A penalized likelihood ratio test statistic is constructed for the null hypothesis of the linearity of the continuous covariate effect. When the number of knots is fixed, its limiting null distribution is the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom. The smoothing parameter value is determined by setting a specified value equal to the asymptotic expectation of the test statistic under the null hypothesis. The power performance of the proposed test is studied with simulation experiments.     

Modified likelihood ratio tests for unit gamma regressions

Regression analyses are commonly performed with doubly limited continuous dependent variables; for instance, when modeling the behavior of rates, proportions and income concentration indices. Several models are available in the literature for use with such variables, one of them being the unit gamma regression model. In all such models, parameter estimation is typically performed using the maximum likelihood method and testing inferences on the model''s parameters are usually based on the likelihood ratio test. Such a test can, however, deliver quite imprecise inferences when the sample size is small. In this paper, we propose two modified likelihood ratio test statistics for use with the unit gamma regressions that deliver much more accurate inferences when the number of data points in small. Numerical (i.e. simulation) evidence is presented for both fixed dispersion and varying dispersion models, and also for tests that involve nonnested models. We also present and discuss two empirical applications.     

Empirical Likelihood Based Rank Regression Inference

Rank regression procedures have been proposed and studied for numerous research applications that do not satisfy the underlying assumptions of the more common linear regression models. This article develops confidence regions for the slope parameter of rank regression using an empirical likelihood (EL) ratio method. It has the advantage of not requiring variance estimation which is required for the normal approximation method. The EL method is also range respecting and results in asymmetric confidence intervals. Simulation studies are used to compare and evaluate normal approximation versus EL inference methods for various conditions such as different sample size or error distribution. The simulation study demonstrates our proposed EL method almost outperforms the traditional method in terms of coverage probability, lower-tail side error, and upper-tail side error. An application of stability analysis also shows the EL method results in shorter confidence intervals for real life data.     

EXTENDED ANALYSIS FOR RANKED DATA

Standard analyses for ranked data or data which are analyzed using ranks are extended to give better comparisons. The new analysis allows detection of significant quadratic or dispersion effects.     

Adaptive tests of regression functions via multiscale generalized likelihood ratios

Many applications of nonparametric tests based on curve estimation involve selecting a smoothing parameter. The author proposes an adaptive test that combines several generalized likelihood ratio tests in order to get power performance nearly equal to whichever of the component tests is best. She derives the asymptotic joint distribution of the component tests and that of the proposed test under the null hypothesis. She also develops a simple method of selecting the smoothing parameters for the proposed test and presents two approximate methods for obtaining its P‐value. Finally, she evaluates the proposed test through simulations and illustrates its application to a set of real data.