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.     

Linear calibra and conditional inference

Approximate conditional inference is developed for the linear calibration problem. It is shown that this problem can be transformed so that the primary parameter is an angle, the nuisance parameter is a radial distance, and the density is rotationally symmetric. Were the nuisance parameter known, exact location confidence intervals would be available by location of structural arguments. A confidence distribution is used to average out the nuisance parameter yielding an approximate confidence interval that involves a precision indicator derived from the radial distance. Some difficulties with the ordinary solution are avoided by the conditional procedure.     

Frailty Models for Arbitrarily Censored and Truncated Data

In this paper, we propose a frailty model for statistical inference in the case where we are faced with arbitrarily censored and truncated data. Our results extend those of Alioum and Commenges (1996), who developed a method of fitting a proportional hazards model to data of this kind. We discuss the identifiability of the regression coefficients involved in the model which are the parameters of interest, as well as the identifiability of the baseline cumulative hazard function of the model which plays the role of the infinite dimensional nuisance parameter. We illustrate our method with the use of simulated data as well as with a set of real data on transfusion-related AIDS.     

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.     

Bayesian Analysis in Regression Models Using Pseudo-Likelihoods

This article deals with the issue of using a suitable pseudo-likelihood, instead of an integrated likelihood, when performing Bayesian inference about a scalar parameter of interest in the presence of nuisance parameters. The proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals. Moreover, it is particularly useful when it is difficult, or even impractical, to write the full likelihood function.

We focus on Bayesian inference about a scalar regression coefficient in various regression models. First, in the context of non-normal regression-scale models, we give a theroetical result showing that there is no loss of information about the parameter of interest when using a posterior distribution derived from a pseudo-likelihood instead of the correct posterior distribution. Second, we present non trivial applications with high-dimensional, or even infinite-dimensional, nuisance parameters in the context of nonlinear normal heteroscedastic regression models, and of models for binary outcomes and count data, accounting also for possibile overdispersion. In all these situtations, we show that non Bayesian methods for eliminating nuisance parameters can be usefully incorporated into a one-parameter Bayesian analysis.     

Integrated likelihoods in parametric survival models for highly clustered censored data

In studies that involve censored time-to-event data, stratification is frequently encountered due to different reasons, such as stratified sampling or model adjustment due to violation of model assumptions. Often, the main interest is not in the clustering variables, and the cluster-related parameters are treated as nuisance. When inference is about a parameter of interest in presence of many nuisance parameters, standard likelihood methods often perform very poorly and may lead to severe bias. This problem is particularly evident in models for clustered data with cluster-specific nuisance parameters, when the number of clusters is relatively high with respect to the within-cluster size. However, it is still unclear how the presence of censoring would affect this issue. We consider clustered failure time data with independent censoring, and propose frequentist inference based on an integrated likelihood. We then apply the proposed approach to a stratified Weibull model. Simulation studies show that appropriately defined integrated likelihoods provide very accurate inferential results in all circumstances, such as for highly clustered data or heavy censoring, even in extreme settings where standard likelihood procedures lead to strongly misleading results. We show that the proposed method performs generally as well as the frailty model, but it is superior when the frailty distribution is seriously misspecified. An application, which concerns treatments for a frequent disease in late-stage HIV-infected people, illustrates the proposed inferential method in Weibull regression models, and compares different inferential conclusions from alternative methods.     

Optimality of marginal likelihood estimating equations

In the presence of nuisance parameters, Godambe (1976) showed an optimality property of the conditional efficient score. We derive analogous results for marginal likelihood when the remainder of the likelihood is conditionally complete. The method of deriving this result is shown to be useful for deriving other optimality results. including optimality of the partial score.     

Approximate Studentization with marginal and conditional inference

Many inference problems lead naturally to a marginal or conditional measure of departure that depends on a nuisance parameter. As a device for first-order elimination of the nuisance parameter, we suggest averaging with respect to an exact or approximate confidence distribution function. It is shown that for many standard problems where an exact answer is available by other methods, the averaging method reproduces the exact answer. Moreover, for the gamma-mean problem, where the exact answer is not explicitly available, the averaging method gives results that agree closely with those obtained from higher-order asymptotic methods. Examples are discussed; detailed asymptotic calculations will be examined elsewhere.     

Bootstrapping one way analysis of rao's quadratic entropy

We establish a bootstrap approximation for the one way analysis of quadratic entropy in this paper. The quadratic entropy was introduced by Rao in 1982. It generalizes the concepts of variance and diversity measures and is useful in statistical inference. Our results in this paper provide a computational method to get rid of the nuisance parameters in the asymptotic distribution of the analysis of quadratic entropy (ANOQE) statistic.     

Composite transformation models: A fiducial perspective

This paper discusses inferences for the parameters of a transformation model in the presence of a scalar nuisance parameter that describes the shape of the error distribution. The development is from the point of view of conditional inference and thus is an attempt to extend the classical fiducial (or structural inference) argument. For known shape parameter it is straightforward to derive a fiducial distribution of the transformation parameters from which confidence points can be obtained. For unknown shape parameter, the paper discusses a certain average of these fiducial distributions. The weights used in this averaging process are naturally induced by the action of the underlying group of transformations and correspond to a noninformative prior for the nuisance parameter. This results in a confidence distribution for the transformation parameters which in some cases has good frequentist properties. The method is illustrated by some examples.     

A computer algebra package for approximate conditional inference

This paper presents a set of REDUCE procedures that make a number of existing higher-order asymptotic results available for both theoretical and practical research. Attention has been restricted to the context of exact and approximate inference for a parameter of interest conditionally either on an ancillary statistic or on a statistic partially sufficient for the nuisance parameter. In particular, the procedures apply to regression-scale models and multiparameter exponential families. Most of them support algebraic computation as well as numerical calculation for a given data set. Examples illustrate the code.     

On comparing two methods for approximate conditional inference

Approximate conditional inference for a real parameter in the presence of nuisance parameters was examined from a sample-space differential viewpoint in Fraser and Reid (1988) and a conditional inference procedure was proposed. Conditional likelihood-based inference in the same setting was discussed in Cox and Reid (1987), where emphasis was placed on orthogonalizing the nuisance parameter to the parameter of interest. In this paper the sample-space partitions of the two methods are examined for the case that the minimal sufficient statistic has the same dimension as the parameter space. The methods are identical if observed and expected information gives the same orthogonality; an example indicates how they can differ more generally. A specially chosen reparameterization provides some geometrical insight to the methods and allows a comparison in terms of score functions and locally defined orthogonal parameters.     

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.     

The analysis of ratios and cross-product ratios of poisson variates with application to incidence rates

The incidence of most diseases is low enough that in. large populations the number of new cases may be considered a Poisson variate. This paper explores models and methods for analyzing such data Specific cases are the estimation and testing of ratios and the cross-product ratios, both simple and stratified* We assume the Poisson means are exponential functions of the relevant parameters. The resulting sets of sufficient statistics are partitioned into a test statistic and a vector of statistics related to the nuisance parameters . The methods derived are based on the conditional distribution of the test statistic given the other sufficient statistics. The analyses of stratified cross-product ratios are seen to be analogues of the noncentral distribution associated with theanalysis of the common odds ratio in several 2×2 tables. The various methods are illustrated in numerical examples involving incidence rates of cancer in two metropolitan areas adjusting for both age and sex.     

A consistent method of estimation for the parameters of the three-parameter inverse Gaussian distribution

In this paper, we propose a consistent method of estimation for the parameters of the three-parameter inverse Gaussian distribution. We then discuss some properties of these estimators and show by means of a Monte Carlo simulation study that the proposed estimators perform better than some other prominent estimators in terms of bias and root mean squared error. Finally, we present two real-life examples to illustrate the method of inference developed here.     

Estimating generalized semiparametric additive models using parameter cascading

Elimination of nuisance parameters is a central but difficult problem in statistical inference. We propose the parameter cascading method to estimate statistical models that involve nuisance parameters, structural parameters, and complexity parameters. The parameter cascading method has several unique aspects. First, we consider functional relationships between parameters, quantitatively described using analytical derivatives. These functional relationships can be explicit or implicit, and in the latter case the Implicit Function Theorem is applied to obtain the required derivatives. Second, we can express the gradients and Hessian matrices analytically, which is essential for fast and stable computation. Third, we develop the unconditional variance estimates for parameters, which include the uncertainty coming from other parameter estimates. The parameter cascading method is demonstrated by estimating generalized semiparametric additive models (GSAMs), where the response variable is allowed to be from any distribution. The practical necessity and empirical performance of the parameter cascading method are illustrated through a simulation study, and two applied example, one on finding the effect of air pollution on public health, and the other on the management of a retirement fund. The results demonstrate that the parameter cascading method is a good alternative to traditional methods.     

Estimating systems of trending variables

A survey is given of some results on inference in cointegrated systems. We discuss some regression methods, and contrast them with the analysis of the vector autoregressive model. We discuss determination of cointegrating rank and estimation of parameters, as well as asymptotic inference. The problems are treated for 1(1) and for 1(2) variables.     

Estimating systems of trending variables

A survey is given of some results on inference in cointegrated systems. We discuss some regression methods, and contrast them with the analysis of the vector autoregressive model. We discuss determination of cointegrating rank and estimation of parameters, as well as asymptotic inference. The problems are treated for 1(1) and for 1(2) variables.     

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.     

Upper probabilities based only on the likelihood function

In the problem of parametric statistical inference with a finite parameter space, we propose some simple rules for defining posterior upper and lower probabilities directly from the observed likelihood function, without using any prior information. The rules satisfy the likelihood principle and a basic consistency principle ('avoiding sure loss'), they produce vacuous inferences when the likelihood function is constant, and they have other symmetry, monotonicity and continuity properties. One of the rules also satisfies fundamental frequentist principles. The rules can be used to eliminate nuisance parameters, and to interpret the likelihood function and to use it in making decisions. To compare the rules, they are applied to the problem of sampling from a finite population. Our results indicate that there are objective statistical methods which can reconcile three general approaches to statistical inference: likelihood inference, coherent inference and frequentist inference.