Unknown Nuisance Parameters Research Articles

On procrustean mean shapes and the shape of the means

Two sets of k labelled points, or configurations, in ℝ m are defined to have the same shape if they differ only in translation, rotation and scaling. An important matter in practice is the estimation of the shape of the means; the shape determined by the means of data on the vertices of configurations. However, statistical models for vertices-based shapes always involve some unknown samplewise nuisance parameters associated with ambiguity of location, rotation and scaling. The use of procrustean mean shapes for a finite set of configurations, which are usually formulated directly in terms of their vertices, will enable one to eliminate these nuisance parameters.

On procrustean mean shapes and the shape of the means

Two sets of k labelled points, or configurations, in ℝm are defined to have the same shape if they differ only in translation, rotation and scaling. An important matter in practice is the estimation of the shape of the means; the shape determined by the means of data on the vertices of configurations. However, statistical models for vertices-based shapes always involve some unknown samplewise nuisance parameters associated with ambiguity of location, rotation and scaling. The use of procrustean mean shapes for a finite set of configurations, which are usually formulated directly in terms of their vertices, will enable one to eliminate these nuisance parameters.

A Weighted Least Squares Approach to Computer Simulation Factor Screening

In this paper, we develop a weighted least squares (WLS) approach to computer simulation factor screening. Like frequency domain methodology (FDM), the regression analysis is performed in the frequency domain. However, unlike FDM, where unknown nuisance parameters are eliminated by cancellation in a ratio statistic, the WLS procedure is designed to estimate these parameters (the WLS weights) and incorporate this information into the regression analysis. We propose and compare one- and two-run approaches for estimating the weights in the WLS approach.

P Values Maximized Over a Confidence Set for the Nuisance Parameter

Abstract For testing problems of the form H 0: v = v 0 with unknown nuisance parameter θ, various methods are used to deal with θ. The simplest approach is exemplified by the t test where the unknown variance is replaced by the sample variance and the t distribution accounts for estimation of the variance. In other problems, such as the 2 × 2 contingency table, one conditions on a sufficient statistic for 0 and proceeds as in Fisher's exact test. Because neither of these standard methods is appropriate for all situations, this article suggests a new method for handling the unknown θ. This new method is a simple modification of the formal definition of a p value that involves taking a maximum over the nuisance parameter space of a p value obtained for the case when θ is known. The suggested modification is to restrict the maximization to a confidence set for the nuisance parameter. After giving a brief justification, we give various examples to show how this new method gives improved results for 2 × 2 tabl...

Bahadur Efficiency of Tests of Separate Hypotheses and Adaptive Test Statistics

The notion of separability of hypotheses was first introduced by Cox (1961); two families of hypotheses are separate if no distribution in one family can be obtained as a limit of distributions from the other family. For testing separate hypotheses, Cox (1961, 1962) suggested the test based on the comparison of the logarithm of the likelihood ratio with its expected value (or estimate thereof) under each hypothesis. The tests of such hypotheses are required to have high power against the specified alternatives; when nuisance parameters are present, this concept leads to the notion of an adaptive test, which by definition must be asymptotically efficient for any value of the unknown nuisance parameter. A testing problem of separate hypotheses for an exponential family is studied from the standpoint of Bahadur asymptotic optimality. A formula for the Bahadur-exact slope of any smooth test statistic is obtained, and in the example of testing lognormality versus exponentiality, it is shown that Cox's test can...

Optimal estimation in random coefficient regression models

In linear regression models with random coefficients, the score function usually involves unknown nuisance parameters in the form of weights. Conditioning with respect to the sufficient statistics for the nuisance parameter, when the parameter of interest is held fixed, eliminates the nuisance parameters and is expected to give reasonably good estimating functions. The present paper adopts this approach to the problem of estimation of average slope in random coefficient regression models. Four sampling situations are discussed. Some asymptotic results are also obtained for a model where neither the regressors nor the random regression coefficients replicate. Simulation studies for normal as well as non-normal models show that the performance of the suggested estimating functions is quite satisfactory.

Boundary Crossing Probabilities for Locally Poisson Processes

We derive large deviation approximations to boundary crossing probabilities for a class of point processes which can be approximated locally as Poisson processes. In the special case of empirical processes, we are able to obtain second order correction terms. The methods are applied to Kolmogorov-Smirnov testing, where we are able to obtain accurate approximations to the significance level when the null hypothesis is an exponential family with unknown nuisance parameters.

On the Information Content of Partial Observations: A Bayesian Approach

On the Information Content of Partial Observations: A Bayesian Approach

Construction of Permutation Tests

Abstract A general method for constructing permutation tests for various experimental designs follows from invariance and sufficiency. In this framework, randomization (or rerandomization) tests are just special cases of permutation tests. The methodology extends the applicability of permutation tests: An example demonstrates a test for an interaction effect in a factorial experiment, a problem thought to be unapproachable by permutation or randomization methods. Tests are constructed as follows. The modeling of block and treatment effects closely parallels classical analysis of variance (ANOVA). The ANOVA assumptions of iid normal random errors are replaced, however, by the much weaker assumption of exchangeability of subsets of random errors. Distributional dependences on the unknown nuisance parameters for blocks, covariates, and untested treatments and on the unknown error distribution are eliminated by invariance and sufficiency, respectively. Initial reduction of the data by invariance appears to extend considerably the applicability of permutation tests. Because of the assumed exchangeability of the errors, the sufficiency step leads to data permutations when computing the test. The method also addresses some philosophical problems associated with randomization tests. For instance, in this article's framework, it is the design, and not the data, which is always regarded as fixed, so ancillarity of the design causes no difficulty.

Construction of Permutation Tests

Abstract A general method for constructing permutation tests for various experimental designs follows from invariance and sufficiency. In this framework, randomization (or rerandomization) tests are just special cases of permutation tests. The methodology extends the applicability of permutation tests: An example demonstrates a test for an interaction effect in a factorial experiment, a problem thought to be unapproachable by permutation or randomization methods. Tests are constructed as follows. The modeling of block and treatment effects closely parallels classical analysis of variance (ANOVA). The ANOVA assumptions of iid normal random errors are replaced, however, by the much weaker assumption of exchangeability of subsets of random errors. Distributional dependences on the unknown nuisance parameters for blocks, covariates, and untested treatments and on the unknown error distribution are eliminated by invariance and sufficiency, respectively. Initial reduction of the data by invariance appears to ex...

Deconvolution with partially known kernel of nonnegative signals

This paper extends the Bayesian direct deconvolution method to the case in which the convolution kernel depends on few unknown nuisance parameters. Moreover, an acceleration procedure is proposed that drastically reduces the computational burden. Finally, the implementation of the method by means of the fast Fourier transform is fully discussed in the multidimensional case.

Efficient dynamic programming in presence of nuisance parameters

The dynamic programming approach for maximum a posteriori (MAP) estimation of Markov sequences is frequently proposed for problems in control theory, communications, and signal processing. It is usually assumed that the observation sequence is a perfectly known function of the Markov sequence of interest, except for some additive noise with known statistics. However, often the observation is not only a function of the Markov sequence but also of a vector of unknown nuisance parameters. It is shown how the dynamic programming methodology can be extended to estimate both the nuisance parameters and the Markov sequence, using a combined maximum-likelihood and MAP framework. The technique is efficient relative to other possible solutions. The problem of detecting and tracking moving targets observed by imaging sensors is used to demonstrate the efficiency of the procedure.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Joint asymptotic distribution of marginal quantiles and quantile functions in samples from a multivariate population

The joint asymptotic distributions of the marginal quantiles and quantile functions in samples from a p-variate population are derived. Of particular interest is the joint asymptotic distribution of the marginal sample medians, on the basis of which tests of significance for population medians are developed. Methods of estimating unknown nuisance parameters are discussed. The approach is completely nonparametric.

Improved estimation for the parameters of an inverse gaussian distribution

James-Stein estimators are proposed for the #-parameter of an inverse Gaussian #G# distribution. The estimators of this class have smaller expected quadratic loss than the maximum likelihood estimator usually employed when analysing real sets of data. This problem is also studied for the case of an unknown nuisance parameter. Finally, improved estimators are considered for # in the two sample problem.

The use of conditional cutoffs in a forward selection procedure

Using a forward selection procedure for selecting the best subset of regression variables involves the calculation of critical values (cutoffs) for an F-ratio at each step of a multistep search process. On dropping the restrictive (unrealistic) assumptions used in previous works, the null distribution of the F-ratio depends on unknown regression parameters for the variables already included in the subset. For the case of known σ, by conditioning the F-ratio on the set of regressors included so far and also on the observed (estimated) values of their regression coefficients, we obtain a forward selection procedure whose stepwise type I error does not depend on the unknown (nuisance) parameters. A numerical example with an orthogonal design matrix illustrates the difference between conditional cutoffs, cutoffs for the centralF-distribution, and cutoffs suggested by Pope and Webster.

Simulated Power Functions

Tests for a null hypothesis whose specification involves an unknown nuisance parameter may be obtained by inverting a bootstrap confidence region for the parameter being tested or by constructing a simulated null distribution for the test statistic. The power of either test against certain alternatives involving the same unknown nuisance parameter can itself be estimated by simulation.

Statistical analysis of a compound exponential failure model

This paper discusses the statistical analysis of a two parameter mixed(or compound)exponential failure model. This model is also a Pareto distribution of the second kind. Some computational aspects of estimating both parameters by the method of maximum likelihood are discussed and small-sample means and variances are derived and studied. Tests of hypotheses on each parameter with the other parameter an unknown nuisance parameter are provided, and the construction of confidence limits are discussed.

Survival Models for Fertility Evaluation

Abstract Two proportional hazards models for cohort fertility evaluation are constructed. Time-dependent covariates describe sources of heterogeneity between and within women regarding fertility characteristics. In the first model, U.S. birth rates specific to maternal age, race, parity, and birth cohort are used as underlying hazard rates. Covariate effects are estimated by maximizing the full likelihood. In the second model, covariate effects are estimated via Cox regression with stratified underlying hazard rates regarded as unknown nuisance parameters. The two models are illustrated with an evaluation of the fertility histories of the wives of workers at a manufacturing plant with potential for hazardous exposure. Adjustments to the U.S. birth rates for maternal age and parity zero experience are required with the first approach. Then, despite differences in the model-specific estimation procedures, the point estimates of the exposure effect and the estimated standard errors from the two models are practically equivalent.

Survival models for fertility evaluation.

"Two proportional hazards models for cohort fertility evaluation are constructed. Time-dependent covariates describe sources of heterogeneity between and within women regarding fertility characteristics. In the first model, U.S. birth rates specific to maternal age, race, parity, and birth cohort are used as underlying hazard rates. Covariate effects are estimated by maximizing the full likelihood. In the second model, covariate effects are estimated via Cox regression with stratified underlying hazard rates regarded as unknown nuisance parameters." The authors illustrate the models "with an evaluation of the fertility histories of the wives of workers at a manufacturing plant with potential for hazardous exposure. Adjustments to the U.S. birth rates for maternal age and parity zero experience are required with the first approach. Then, despite differences in the model-specific estimation procedures, the point estimates of the exposure effect and the estimated standard errors from the two models are practically equivalent."

Multiplicative Models and Cohort Analysis

Abstract Three methods of cohort analysis are presented for a statistical model wherein the explanatory or exposure variables act multiplicatively on age × calendar year specific death rates. The first method, which assumes that the baseline rates are known from national vital statistics, is a multiple regression analysis of the standardized mortality ratio. The second method is a variant of Cox's proportional hazards analysis in which the baseline rates are treated as unknown nuisance parameters. The third method consists of case-control sampling from the risk sets formed in the course of applying Cox's model. It requires substantially less computation than do the other two. In illustrative analysis of respiratory cancer deaths among a cohort of smelter workers, all three approaches yield roughly equivalent estimates of the relative risk associated with arsenic exposure. The discussion centers on the tradeoff between efficiency and bias in the selection of a particular method of analysis, and on practica...