Approximate conditional inference and the linear functional model

Approximate conditional inference is developed for the slope parameter of the linear functional model with two variables. It is shown that the model can be transformed so that the slope parameter becomes an angle and nuisance parameters are radial distances. If the nuisance parameters are known an exact confidence interval based on a location-type conditional distribution is available for the angle. More gen¬erally, confidence distributions are used to average the conditional distribution over the nuisance parameters yielding an approximate conditional confidence interval that reflects the precision indicated by the data. An example is analyzed.     

On the accuracy of approximate studentization

With a parametric model, a measure of departure for an interest parameter is often easily constructed but frequently depends in distribution on nuisance parameters; the elimination of such nuisance parameter effects is a central problem of statistical inference. Fraser & Wong (1993) proposed a nuisance-averaging or approximate Studentization method for eliminating the nuisance parameter effects. They showed that, for many standard problems where an exact answer is available, the averaging method reproduces the exact answer. Also they showed that, if the exact answer is unavailable, as say in the gamma-mean problem, the averaging method provides a simple approximation which is very close to that obtained from third order asymptotic theory. The general asymptotic accuracy, however, of the method has not been examined. In this paper, we show in a general asymptotic context that the averaging method is asymptotically a second order procedure for eliminating the effects of nuisance parameters.     

Bounded Estimation in the Presence of Nuisance Parameters

The aim of this paper is to extend in a natural fashion the results on the treatment of nuisance parameters from the profile likelihood theory to the field of robust statistics. Similarly to what happens when there are no nuisance parameters, the attempt is to derive a bounded estimating function for a parameter of interest in the presence of nuisance parameters. The proposed method is based on a classical truncation argument of the theory of robustness applied to a generalized profile score function. By means of comparative studies, we show that this robust procedure for inference in the presence of a nuisance parameter can be used successfully in a parametric setting.     

Sample Tests for Detection of Size-Biased Sampling Mechanism

The aim of this article is to present a new test for the detection of size-bias in a sample with or without censored observations. The test is simple in the form and demands only the knowledge of consistent estimators of any nuisance parameters appeared in the model. With the use of simulated samples from the Weibull distribution, we show the advantages of the new test compared to the Likelihood Ratio and the Wald test.     

SUBSAMPLING THE JOHANSEN TEST WITH STABLE INNOVATIONS

By taking into account the thick-tail property of the errors, cointegration analysis in vector error-correction models with infinite-variance stable errors is a natural generalization of cointegration analysis in error-correction models with normally distributed errors. We study the Johansen test for cointegrated systems under symmetric stable innovations with discrete spectral measures. The results show that the distributions of the Johansen test statistics under these innovations involve nuisance parameters. To overcome the problem of nuisance parameters, we implement a nonparametric subsampling procedure. We document some subsampling simulation results and demonstrate in an empirical example how the test can be used in practice.     

Randomization in a complete block design with few blocks

It is known that the distributions of statistics commonly used in experimental design (notably the F statistic) involve certain nuisance parameters which depend on the model and the design. Raudomization was a technique introduced by R.A. Fisher to eliminate some of these parameter and produce a usable distribution function. We will show that there is a close relationship between the analytical properties of the non-randomized distribution and the more combinatorial properties of the nuisance parameters. This relationship allows us to determine theoretically, and practically in some cases, how good an approximation the central F distribution is to the randomized distribution.     

A Score Test for the Multivariate Burr and Other Weibull Mixture Distributions

The p -variate Burr distribution has been derived, developed, discussed and deployed by various authors. In this paper a score statistic for testing independence of the components, equivalent to testing for p independent Weibull against a p -variate Burr alternative, is obtained. Its null and non-null properties are investigated with and without nuisance parameters and including the possibility of censoring. Two applications to real data are described. The test is also discussed in the context of other Weibull mixture models.     

Robust Bayesian hypothesis testing in the presence of nuisance parameters

Robust Bayesian testing of point null hypotheses is considered for problems involving the presence of nuisance parameters. The robust Bayesian approach seeks answers that hold for a range of prior distributions. Three techniques for handling the nuisance parameter are studied and compared. They are (i) utilize a noninformative prior to integrate out the nuisance parameter; (ii) utilize a test statistic whose distribution does not depend on the nuisance parameter; and (iii) use a class of prior distributions for the nuisance parameter. These approaches are studied in two examples, the univariate normal model with unknown mean and variance, and a multivariate normal example.     

Asymptotic properties of a conditional maximum-likelihood estimator

Inference for a scalar interest parameter in the presence of nuisance parameters is considered in terms of the conditional maximum-likelihood estimator developed by Cox and Reid (1987). Parameter orthogonality is assumed throughout. The estimator is analyzed by means of stochastic asymptotic expansions in three cases: a scalar nuisance parameter, m nuisance parameters from m independent samples, and a vector nuisance parameter. In each case, the expansion for the conditional maximum-likelihood estimator is compared with that for the usual maximum-likelihood estimator. The means and variances are also compared. In each of the cases, the bias of the conditional maximum-likelihood estimator is unaffected by the nuisance parameter to first order. This is not so for the maximum-likelihood estimator. The assumption of parameter orthogonality is crucial in attaining this result. Regardless of parametrization, the difference in the two estimators is first-order and is deterministic to this order.     

Simplified Likelihood Based Goodness-of-fit Tests for the Weibull Distribution

The aim of this paper is to present new likelihood based goodness-of-fit tests for the two-parameter Weibull distribution. These tests consist in nesting the Weibull distribution in three-parameter generalized Weibull families and testing the value of the third parameter by using the Wald, score, and likelihood ratio procedures. We simplify the usual likelihood based tests by getting rid of the nuisance parameters, using three estimation methods. The proposed tests are not asymptotic. A comprehensive comparison study is presented. Among a large range of possible GOF tests, the best ones are identified. The results depend strongly on the shape of the underlying hazard rate.     

Improved inference for the shape-scale family of distributions under type-II censoring

The presence of a nuisance parameter may often perturb the quality of the likelihood-based inference for a parameter of interest under small to moderate sample sizes. The article proposes a maximal scale invariant transformation for likelihood-based inference for the shape in a shape-scale family to circumvent the effect of the nuisance scale parameter. The transformation can be used under complete or type-II censored samples. Simulation-based performance evaluation of the proposed estimator for the popular Weibull, Gamma and Generalized exponential distribution exhibits markedly improved performance in all types of likelihood-based inference for the shape under complete and type-II censored samples. The simulation study leads to a linear relation between the bias of the classical maximum likelihood estimator (MLE) and the transformation-based MLE for the popular Weibull and Gamma distributions. The linearity is exploited to suggest an almost unbiased estimator of the shape parameter for these distributions. Allied estimation of scale is also discussed.     

Bayesian and frequentist confidence intervals via adjusted likelihoods under prior specification on the interest parameter

Suppose a prior is specified only on the interest parameter and a posterior distribution, free from nuisance parameters, is considered on the basis of the profile likelihood or an adjusted version thereof. In this setup, we derive higher order asymptotic results on the construction of confidence intervals that have approximately correct posterior as well as frequentist coverage. Apart from meeting both Bayesian and frequentist objectives under prior specification on the interest parameter alone, these results allow a comparison with their counterpart arising when the nuisance parameters are known, and hence provide additional justification for the Cox and Reid adjustment from a Bayesian-cum-frequentist perspective, with regard to neutralization of unknown nuisance parameters.     

Dropping observations without affecting posterior and predictive distributions

Suppose in a distribution problem, the sample information W is split into two pieces W ₁ and W ₂, and the parameters involved are split into two sets, π containing the parameters of interest, and θ containing nuisance parameters. It is shown that, under certain conditions, the posterior distribution of π does not depend on the data W ₂, which can thus be ignored. This also has consequences for the predictive distribution of future (or missing) observations. In fact, under similar conditions, the predictive distributions using W or just W ₁ are identical.     

Approximate jackknife empirical likelihood method for estimating equations

It is known that the profile empirical likelihood method based on estimating equations is computationally intensive when the number of nuisance parameters is large. Recently, Li, Peng, & Qi (2011) proposed a jackknife empirical likelihood method for constructing confidence regions for the parameters of interest by estimating the nuisance parameters separately. However, when the estimators for the nuisance parameters have no explicit formula, the computation of the jackknife empirical likelihood method is still intensive. In this paper, an approximate jackknife empirical likelihood method is proposed to reduce the computation in the jackknife empirical likelihood method when the nuisance parameters cannot be estimated explicitly. A simulation study confirms the advantage of the new method. The Canadian Journal of Statistics 40: 110–123; 2012 © 2012 Statistical Society of Canada     

Corrections for Bias in Maximum Likelihood Parameter Estimates Due to Nuisance Parameters

Abstract

In his Fisher Lecture, Efron (Efron, B. R. A. (1998 Efron, B. R. A. 1998. Fisher in the 21st century (with discussion). Statistical Science, 13: 95–122. [Crossref], [Web of Science ®] , [Google Scholar]). Fisher in the 21st Century (with discussion). Statistical Science 13:95–122) pointed out that maximum likelihood estimates (MLE) can be badly biased in certain situations involving many nuisance parameters. He predicted that with modern computing equipment a computer-modified version of the MLE that was less biased could become the default estimator of choice in applied problems in the 21st century. This article discusses three modifications—Lindsay's conditional likelihood, integrated likelihood, and Bartlett's bias-corrected estimating function. Each is evaluated through a study of the bias and MSE of the estimates in a stratified Weibull model with a moderate number of nuisance parameters. In Lindsay's estimating equation, three different methods for estimation of the nuisance parameters are evaluated—the restricted maximum likelihood estimate (RMLE), a Bayes estimator, and a linear Bayes estimator. In our model, the conditional likelihood with RMLE of the nuisance parameters is equivalent to Bartlett's bias-corrected estimating function. In the simulation we show that Lindsay's conditional likelihood is in general preferred, irrespective of the estimator of the nuisance parameters. Although the integrated likelihood has smaller MSE when the precise nature of the prior distribution of the nuisance parameters is known, this approach may perform poorly in cases where the prior distribution of the nuisance parameters is not known, especially using a non-informative prior. In practice, Lindsay's method using the RMLE of the nuisance parameters is recommended.     

On the integrated maximum likelihood estimators for a closed population capture–recapture model with unequal capture probabilities

Nuisance parameter elimination is a central problem in capture–recapture modelling. In this paper, we consider a closed population capture–recapture model which assumes the capture probabilities varies only with the sampling occasions. In this model, the capture probabilities are regarded as nuisance parameters and the unknown number of individuals is the parameter of interest. In order to eliminate the nuisance parameters, the likelihood function is integrated with respect to a weight function (uniform and Jeffrey's) of the nuisance parameters resulting in an integrated likelihood function depending only on the population size. For these integrated likelihood functions, analytical expressions for the maximum likelihood estimates are obtained and it is proved that they are always finite and unique. Variance estimates of the proposed estimators are obtained via a parametric bootstrap resampling procedure. The proposed methods are illustrated on a real data set and their frequentist properties are assessed by means of a simulation study.     

Tests of hypotheses based on ranks in the general linear model

A unified approach is developed for testing hypotheses in the general linear model based on the ranks of the residuals. It complements the nonparametric estimation procedures recently reported in the literature. The testing and estimation procedures together provide a robust alternative to least squares. The methods are similar in spirit to least squares so that results are simple to interpret. Hypotheses concerning a subset of specified parameters can be tested, while the remaining parameters are treated as nuisance parameters. Asymptotically, the test statistic is shown to have a chi-square distribution under the null hypothesis. This result is then extended to cover a sequence of contiguous alternatives from which the Pitman efficacy is derived. The general application of the test requires the consistent estimation of a functional of the underlying distribution and one such estimate is furnished.     

Generalised Smooth Tests of Goodness of Fit Utilising L‐moments

In this paper we present a semiparametric test of goodness of fit which is based on the method of L‐moments for the estimation of the nuisance parameters. This test is particularly useful for any distribution that has a convenient expression for its quantile function. The test proceeds by investigating equality of the first few L‐moments of the true and the hypothesised distributions. We provide details and undertake simulation studies for the logistic and the generalised Pareto distributions. Although for some distributions the method of L‐moments estimator is less efficient than the maximum likelihood estimator, the former method has the advantage that it may be used in semiparametric settings and that it requires weaker existence conditions. The new test is often more powerful than competitor tests for goodness of fit of the logistic and generalised Pareto distributions.     

Bayesian Analysis of Masked Series System Lifetime Data

The problem of analyzing series system lifetime data with masked or partial information on cause of failure is recent, compared to that of the standard competing risks model. A generic Gibbs sampling scheme is developed in this article towards a Bayesian analysis for a general parametric competing risks model with masked cause of failure data. The masking probabilities are not subjected to the symmetry assumption and independent Dirichlet priors are used to marginalize these nuisance parameters. The developed methodology is illustrated for the case where the components of a series system have independent log-Normal life distributions by employing independent Normal-Gamma priors for these component lifetime parameters. The Gibbs sampling scheme developed for the required analysis can also be used to provide a Bayesian analysis of data arising from the conventional competing risks model of independent log-Normals, which interestingly has so far remained by and large neglected in the literature. The developed methodology is deployed to analyze a masked lifetime data of PS/2 computer systems.