Presence Of Nuisance Parameters Research Articles

Objective Priors: An Introduction for Frequentists

Bayesian methods are increasingly applied in these days in the theory and practice of statistics. Any Bayesian inference depends on a likelihood and a prior. Ideally one would like to elicit a prior from related sources of information or past data. However, in its absence, Bayesian methods need to rely on some "objective" or "default" priors, and the resulting posterior inference can still be quite valuable. Not surprisingly, over the years, the catalog of objective priors also has become prohibitively large, and one has to set some specific criteria for the selection of such priors. Our aim is to review some of these criteria, compare their performance, and illustrate them with some simple examples. While for very large sample sizes, it does not possibly matter what objective prior one uses, the selection of such a prior does influence inference for small or moderate samples. For regular models where asymptotic normality holds, Jeffreys' general rule prior, the positive square root of the determinant of the Fisher information matrix, enjoys many optimality properties in the absence of nuisance parameters. In the presence of nuisance parameters, however, there are many other priors which emerge as optimal depending on the criterion selected. One new feature in this article is that a prior different from Jeffreys' is shown to be optimal under the chi-square divergence criterion even in the absence of nuisance parameters. The latter is also invariant under one-to-one reparameterization.

Generalized P-values and Bayesian evidence in the one-sided testing problems under exponential distributions

The problem of comparing the frequentist evidence and the Bayesian evidence in the one-sided testing problems has been widely treated and many researches revealed that these two methods can reach an agreement approximately. However, most of the previous work dealt mainly with situations without nuisance parameters. Since the presence of nuisance parameters is very common in practice, whether these two kinds of evidence still reach an agreement is a problem worthy of study. In this article, we establish in a systematic way under the exponential distributions the agreement of the Bayesian evidence and the generalized frequentist evidence (the generalized P-value) for a variety of one-sided testing problems where the nuisance parameters are involved.

Model complexity and information in the data: Could it be a house built on sand?

Heisey et al. (2010), in an interesting paper, try to address a very difficult problem of analyzing spatially referenced, age specific prevalence data. The general goal of the analysis is to understand how force of infection changes as a function of age, time, and space. To further complicate matters, all the data considered in the paper are censored observations. Binary data are notoriously difficult to analyze, especially when latent processes are involved and prevalence is very low. Frankly, I was surprised by the complexity of the models they consider and the limited amount of information available to fit these models. I would like to congratulate them for trying to address such a difficult problem and in the process bringing to the attention of the ecologists some important statistical models in survival analysis. How does one generally deal with the conflicting issues of lack of information and desire to conduct inference about complex underlying processes? The standard approach is to compensate for lack of information by adding assumptions. This is done routinely in most statistical analyses by assuming a parametric model. For example, one can conduct inference in ANOVA without assuming any specific relationship between the treatment means if replicate observations are available at each treatment level. If such replicate data are not available, instead of giving up, we assume that there is a linear (or, some parametric) relationship between the covariates and the response, the regression approach. This is a smoothing assumption. Similarly, in one of the fundamental papers on statistical inference in the presence of nuisance parameters, Kiefer and Wolfowitz (1956) showed that simply assuming that the nuisance parameters arise from a distribution is enough of a smoothing assumption to estimate not only the parameters of interest but also the distribution function from which nuisance parameters are assumed to have arisen. Heisey et al. (2010) try to get away with the limited information available in the prevalence data, where all observations are censored, by imposing constraints on the log-hazard, a smoothing assumption of another sort. This is the easy part. The real questions are: (1) Given the limited amount of information in the data, what assumptions do we need until some inference is feasible? and (2) Are these inferences primarily driven by the data or by the assumptions? Technically, the answer to the first question is straightforward: add assumptions until the parameters in the model, at least the ones that are of scientific interest, are estimable given the data. The second question is qualitative. It is partially addressed by studying the sensitivity of the inferences on the parameters of interest (assuming they are identifiable) to the violations of the assumptions. I will discuss these issues in the remainder of the commentary. I assume readers are familiar with the basic descriptions in Heisey et al. (2010). Perhaps the easiest way out of the limited information in the prevalence data is to assume a specific parametric model for the log-hazard function. This does not guarantee that the parameters will be identifiable but it has the best chance. Heisey et al. (2010) do not take this easy way out. They aspire to assume less about the form of the log-hazard function. As they point out, the ‘‘nonparametric’’ MLE of the log-hazard is very choppy and unstable. It is generally not consistent, at least not at the usual ffiffiffi

An Approach to the Conditional Error Rate Principle with Nuisance Parameters

In the presence of nuisance parameters, the conditional error rate principle is difficult to apply because of the dependency of the conditional error function of the preplanned test on nuisance parameters. To use the conditional error rate principle with nuisance parameters, we propose to search among tests that guarantee overall error control for the test that maximizes a weighted combination of the conditional error rates over possible values of the nuisance parameters. We show that the optimization problem that defines such a test can be solved efficiently by existing algorithms.

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.

Data-dependent probability matching priors of the second order

We consider second-order probability matching priors that ensure frequentist validity of posterior quantiles with margin of error o(n −1), where n is the sample size. It is well known that there are many models of interest where data-free second-order probability matching priors do not exist. We explore how this problem can be resolved via consideration of data-dependent priors. This is done both in the absence and presence of nuisance parameters.

On construction of the smallest one-sided confidence interval for the difference of two proportions

For any class of one-sided 1−α confidence intervals with a certain monotonicity ordering on the random confidence limit, the smallest interval, in the sense of the set inclusion for the difference of two proportions of two independent binomial random variables, is constructed based on a direct analysis of coverage probability function. A special ordering on the confidence limit is developed and the corresponding smallest confidence interval is derived. This interval is then applied to identify the minimum effective dose (MED) for binary data in dose-response studies, and a multiple test procedure that controls the familywise error rate at level α is obtained. A generalization of constructing the smallest one-sided confidence interval to other discrete sample spaces is discussed in the presence of nuisance parameters.

Reference priors for exponential families with increasing dimension

In this article, we establish the asymptotic normality of the posterior distribution for the natural parameter in an exponential family based on independent and identically distributed data. The mode of convergence is expected Kullback-Leibler distance and the number of parameters p is increasing with the sample size n. Using this, we give an asymptotic expansion of the Shannon mutual information valid when p=pn increases at a sufficiently slow rate. The second term in the asymptotic expansion is the largest term that depends on the prior and can be optimized to give Jeffreys’ prior as the reference prior in the absence of nuisance parameters. In the presence of nuisance parameters, we find an analogous result for each fixed value of the nuisance parameter. In three examples, we determine the rates at which pn can be allowed to increase while still retaining asymptotic normality and the reference prior property.

A note on multiple testing for composite null hypotheses

Multiple hypothesis testing literature has recently experienced a growing development with particular attention to the control of the false discovery rate (FDR) based on p-values. While these are not the only methods to deal with multiplicity, inference with small samples and large sets of hypotheses depends on the specific choice of the p-value used to control the FDR in the presence of nuisance parameters. In this paper we propose to use the partial posterior predictive p-value [Bayarri, M.J., Berger, J.O., 2000. p-values for composite null models. J. Amer. Statist. Assoc. 95, 1127–1142] that overcomes this difficulty. This choice is motivated by theoretical considerations and examples. Finally, an application to a controlled microarray experiment is presented.

Prior Distributions From Pseudo-Likelihoods in the Presence of Nuisance Parameters

Consider a model parameterized by θ == (ψ, λ), where ψ is the parameter of interest. The problem of eliminating the nuisance parameter λ can be tackled by resorting to a pseudo-likelihood function L**(ψ) for ψ—namely, a function of ψ only and the data y with properties similar to those of a likelihood function. If one treats L**(ψ) as a true likelihood, the posterior distribution π**(ψ | y) ∝ π(ψ)L**(ψ) for ψ can be considered, where π(ψ) is a prior distribution on ψ. The goal of this article is to construct probability matching priors for a scalar parameter of interest only (i.e., priors for which Bayesian and frequentist inference agree to some order of approximation) to be used in π**(ψ || y). When L**(ψ) is a marginal, a conditional, or a modification of the profile likelihood, we show that π(ψ) is simply proportional to the square root of the inverse of the asymptotic variance of the pseudo-maximum likelihood estimator. The proposed priors are compared with the reference or Jeffreys’ priors in four examples.

Structural instability of China inflation dynamics

This article employs recently developed unknown structural break tests to investigate intrinsic structural instability in China inflation dynamics over 1981–2007. In order to capture accurately the statistical nature of potential structural beak, we use asymptotic p-value function under the non-standard distribution condition to compute the p-values for structural change tests in the presence of nuisance parameter. Empirical results suggest that China inflation dynamics witnessed a significant structural change at the end of 1994 and the instability appears to be originated from the dynamic parameters in the underlying model. The paper discusses important policy implications of the empirical findings through impulse response analyses.

Conditional Information in Projections of Gaussian Vectors

Conditional information measures the information in a sample for an interest parameter in the presence of nuisance parameter. In the context of Gaussian likelihoods this paper first derives conditions under which a projection of the data may reduce conditional information to zero. These are then applied in the context of time series regressions, and inference on a covariance parameter, such as with either autoregressive or moving average errors. It is shown that regressing out very common regressors, such as a linear trend or dummy variable, can imply that conditional information is zero in the case of non-stationary autoregressions or non-invertible moving averages, respectively.

Conditional properties of unconditional parametric bootstrap procedures for inference in exponential families

Higher-order inference about a scalar parameter in the presence of nuisance parameters can be achieved by bootstrapping, in circumstances where the parameter of interest is a component of the canonical parameter in a full exponential family. The optimal test, which is approximated, is a conditional one based on conditioning on the sufficient statistic for the nuisance parameter. A bootstrap procedure that ignores the conditioning is shown to have desirable conditional properties in providing third-order relative accuracy in approximation of p-values associated with the optimal test, in both continuous and discrete models. The bootstrap approach is equivalent to third-order analytical approaches, and is demonstrated in a number of examples to give very accurate approximations even for very small sample sizes.

Empirical likelihood for linear models in the presence of nuisance parameters

We propose a simple alternative empirical likelihood (EL) method in linear regression which requires the same conditions of the ordinary profile EL but overcomes the challenge of maximizing the likelihood in the presence of high dimensional nuisance parameters. We adapt the idea of added variable plots. We regress the response and the independent variables of main interest on the ancillary variables and construct the likelihood based on the residuals. The hence constructed EL ratio has constraints only pertaining to the parameters of interest and has a standard χ 2 limiting distribution. Numerical results are included.

On the Use of Bayes Factor in Frequentist Testing of a Precise Hypothesis

This article deals with Bayes factors as useful Bayesian tools in frequentist testing of a precise hypothesis. A result and several examples are included to justify the definition of Bayes factor for point null hypotheses, without merging the initial distribution with a degenerate distribution on the null hypothesis. Of special interest is the problem of testing a proportion (joint with a natural criterion to compare different tests), the possible presence of nuisance parameters, or the influence of Bayesian sufficiency on this problem. The problem of testing a precise hypothesis under a Bayesian perspective is also considered and two alternative methods to deal with are given.

On Optimal Estimating Functions in the Presence of Nuisance Parameters

When there is only one interesting parameter θ1 and one nuisance parameter θ2, Godambe and Thompson (1974) showed that the optimal estimating function for θ1 essentially is a linear function of the θ1-score, the square of the θ2-score, and the derivative of θ2-score with respect to θ2. Mukhopadhyay (2000b) generalized this result to m nuisance parameters. Mukhopadhyay (20002002ab) obtained lower bounds to the variance of regular estimating functions in the presence of nuisance parameters. Taking cues from these results we propose a method of finding optimal estimating function for θ1 by taking the multiple regression equation on θ1 score and Bhattacharyya's (1946) scores with respect to θ2. The result is extended to the case of m nuisance parameters.

Simultaneous testing of multiple hypotheses using generalized p-values

In the context of simultaneously testing many hypotheses, Benjamini and Hochberg [1995. Controlling the false discovery rate: a practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B (Methodological) 57, 289–300] propose a procedure that guarantees that the false discovery rate ( FDR) will be less than or equal to a specified value. Here, the FDR is the expected value of the ratio of the number of incorrectly rejected hypotheses and the total number of rejected hypotheses. To our knowledge, all of the existing research assumes that a usual p-value is available for each hypothesis. However, in circumstances when nuisance parameters are present, the usual p-values may not be free of the nuisance parameters. We develop a simultaneous testing procedure to control the FDR for the problem of simultaneously testing many Behrens–Fisher problems. Our multiple testing procedure, based on generalized p-values [Tsui, K.-W., Weerahandi, S., 1989. Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters. J. Amer. Statist. Assoc. 84(406), 602–607], is then illustrated with an application to data from a microarray experiment.

Simple and accurate one-sided inference based on a class of M-estimators

SUMMARY This paper presents a simple method of calculating one-sided p-values which yield accurate inferences for a simple null hypothesis or, equivalently, accurate one-sided con fidence limits regarding a scalar parameter in the presence of nuisance parameters. The method, which extends the work of DiCiccio et al. (2001) and Lee & Young (2005) in the context of adjusted likelihood estimation, is based on a test statistic analogous to the signed root of the loglikelihood, but derived from the objective function of an M-estimator from a certain class. Monte Carlo simulation is used to avoid the need for onerous analytical calculations typical of competing procedures based on Edgeworth or saddle point approximations. The specific class of M-estimators under consideration is character ised by the requirement that the associated test statistic has constant variance to second order. This class contains, among others, the maximum likelihood estimator as well as variants of commonly used M-estimators of location, such as Huber's (1964) Proposal 2.

Distributional properties for the generalized p-value for the Behrens–Fisher problem

The generalized p-value method introduced by Tsui and Weerahandi [1989. Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters. J. Amer. Statist. Assoc. 84 (406), 602–607] has been successfully used to provide small sample solutions for many hypothesis testing problems when nuisance parameters are present. Simulation studies show that generalized p-values have similar distributional properties as ordinary p-values. It is desirable to study theoretical properties of generalized p-values. Given a sample d, let p ( d ) be the generalized p-value for the Behrens–Fisher problem of testing the difference of two independent normal distribution means with possibly unequal distributional variances, as given in Tsui and Weerahandi [1989. Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters. J. Amer. Statist. Assoc. 84 (406), 602–607]. We derive a closed form expression to show that, for small samples, the probability P ( p ( d ) ⩽ r ) is approximately less than or equal to r, for 0 ⩽ r ⩽ 0.5 .

An exact test for testing the equality of parameter matrices in two multivariate linear models

A method of testing the hypothesis of the kind H 0 : C ′ Φ 1 M = C ′ Φ 2 M in two independent multivariate linear models is presented using the concept of generalized p-values introduced by Tsui and Weerahandi [K.W. Tsui, S. Weerahandi, Generalized p-values in significance testing of hypothesis in the presence of nuisance parameters, J. Amer. Statist. Assoc. 84 (1989) 602–607] under the assumption of error matrix variate normality and heteroscedasticity. This method calculates the exact p-value in the generalized sense. Nel [D.G. Nel, Tests for equality of parameter matrices in two multivariate linear models, J. Multivariate Anal. 61 (1997) 29– 37] provided an approximate degrees of freedom test to test this hypothesis.