Testing Hypotheses in the Presence of Nuisance Parameters

Binary-discrimination tasks useful to arms-control-treaty verification were performed by applying mathematical observer models used by the medical-imaging community. It is a difficult task, as the monitor needs to verify a measured object is a warhead while the host wants to prevent dissemination of sensitive information on their weapons. Inspection objects were classified by their projection data, without reconstructing an image, which often contains sensitive information. Furthermore, the models process data event-by-event, with only a scalar test statistic being updated before the observed data is purged from memory, preventing the aggregation of information which would be sensitive. Template matching models were developed from a set of calibration data on these objects, with the ultimate goal being to develop an observer model that stores only non-sensitive information sufficient for confirmation. These models were also analyzed in the presence of nuisance parameters — unknowns in the objects or imaging system that affect the projection data but aren't of interest to the discrimination task. Examples of these include object location and orientation. The Hotelling observer and channelized Hotelling observer were modeled and their benefits to information security analyzed. The Hotelling observer-the ideal linear observer when the statistics of the data are Gaussian-stores only a set of weights which are the product of the inverse average covariance matrix and difference in mean data between the two objects in the discrimination task. When testing a source, an inner product of the Hotelling weights and binned testing data is taken, resulting in a scalar that is thresholded to make a decision. If nuisance parameters are present, the mean and covariance matrix are found by averaging not only over the always present Poisson noise, but the nuisance parameter distributions as well. Hence, even if detector data is gathered from multiple realizations of an object, the Hotelling weights will be a single smeared out data set the size of the measured data. The Hotelling weights also sifts out all information other than the differences between the two objects. Hence, if the monitor was to gain access to the weights and apply the inverse to their test statistic, they could only back out a scaled version of the template, not the image. The channelized Hotelling observer-a common tool used in medical image quality assessment-was also investigated. This method drastically reduces the size of the data by applying a channeiizing matrix to the binned testing data set. An optimal set of weights for the channelized data can then be found, and the inner product between these weights and the channelized vector results in the scalar test statistic. Optimal performance is retained by optimizing the matrix to maximize the SNR2 between the test-statistic distributions for the two sources in the task. The channelized Hotelling observer gives the monitor access to multiple non-sensitive test statistics. In practice, the channelizing matrix could be implemented in hardware or software behind an information barrier. The addition of penalty terms to the channelized Hotelling observer offers further potential for this method. Individual channel performance could be penalized, leading to a large number of non-sensitive channels that the monitor can use to verify the channelization routine is working as described without gaining access to sensitive information. Noise could be added to the resulting channels, causing additional reduction of the total stored information. Finally, if the host can define what information in its object geometries is sensitive in advance, they could optimize the differences between the two objects in the task while penalizing out the sensitive information, creating a non-sensitive channeiizing matrix that could be shared with the monitor. To test these models, Monte Carlo simulations were performed with the GEANT4 toolkit. Photons were tracked from plutonium inspection objects developed by Idaho National Laboratory. We simulated the Fast-Neutron Imaging system designed by Oak Ridge National Laboratory and Sandia National Laboratories, which consists of 40×40 1cm2 liquid scintillator pixels with a plastic coded aperture. Observer models were evaluated using the area under the ROC curve. This work is supported by the Office of Defense Nuclear Nonproliferation Research and Development, Nuclear Weapon and Material Security Team. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000. (SAND2015-10391A).

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

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.

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 generalizedP‐value) for a variety of one‐sided testing problems where the nuisance parameters are involved.

An algorithm to determine the uniformly most powerful unbiased test (UMPUT) in 2×2 tables is introduced and shown to provide a useful tool in power computations. Sample sizes based on exact power functions are obtained and compared with existing sample size estimations.

An optimality criterion for vector unbiased statistical estimation functions

SummaryLet (X, A, P)be a given statistical model and letg be the class of all one-to-one, bimeasurable maps g of ( X, A) onto itself such that g is measure-preserving for each P ε P, i.e. Pg−1= P for all P. Let us suppose that there exists a least (minimal) sufficient sub-fieldL. Then, for each LεL, it is true that g−1L is P-equivalent to L for each gεg , i.e., the least sufficient sub-field is almost g -invariant. It is demonstrated that, in many familiar statistical models, the least sufficient sub-field and the sub-field of all almost g -invariant sets are indeed P-equivalent. The problem of data reduction in the presence of nuisance parameters has been discussed very briefly. It is shown that in many situations the principle of invariance is strong enough to lead us to the standard reductions. For instance, given n independent observations on a normal variable with unknown mean (the nuisance parameter) and unknown variance, it is shown how the principle of invariance alone can reduce the data to the sample variance.

The Fisher information about parameters of interest (P‐information) is invariant with respect to nuisance parameters, and induces an information inequality associated with likelihood factorization. This information inequality provides a natural basis for measuring information loss due to using only a sublikelihood function for inference. In contrast with the global reparametrization of some previous concepts in the literature, the concepts of P‐ancillarity and P‐sufficiency proposed in this article are characterized by the notion of no pointwise information loss with respect to the parameters of interest. A conditional version of P‐sufficiency is also proposed. The asymptotic efficiency of likelihood inference under P‐ancillarity or P‐sufficiency is outlined.

Simple robust tests for the specification of high-frequency predictors of a low-frequency series

: Certain best population problems adopting a Bayesian approach were investigated. One main advantage of such an approach is that satisfactory decision procedures can be obtained in the presence of nuisance parameters. The use of Bayes' theorem allows one to analyze best population problems from many points of view. A general description of best population problems is outlined. The criterion for 'bestness' is regarded as a 'utility' function of the statistician. The decision procedure then adopted is based upon the principle of maximizing posterior expected utility. The application of this procedure when sampling from normal populations and exponential populations was discussed. The criterion defining the best population is taken to be the coverage of the population considered in a certain given interval. The procedures are shown to be consistent. The decision analysis by considering the pos terior distribution of the criterion is presented. This extension enables other decision procedures to be proposed which may be more appropriate in certain situations than that resulting from the principle of maximizing posterior expected utility.

A locally most powerful unbiased test for the proportion of mixture

: The report contains the derivation and properties of sequential tests of composite hypotheses for families of distributions satisfying certain conditions. The following topics are discussed: Sequential test 'S'; Discrimination among k > 2 hypotheses in the presence of nuisance parameters; Some tests with the usual decision boundaries; Sequential tests for analysis of variance under random and mixed models; Some exact results.

A method is given for obtaining sequential tests in the presence of nuisance parameters. It is assumed that a jointly sufficient set of estimators exists for the unknown parameters. A number of special tests are described and their properties discussed.

SUMMARY Series expansions for confidence limits based on consistent estimators are obtained in the presence of nuisance parameters. The maximum likelihood estimator is treated as a special case and the necessary characteristics of its sampling distribution are obtained in terms of basic quantities derivable from the likelihood function. The theory is applied to the problem of setting separate confidence limits for the scale and shape parameters of the Weibull distribution. Comparisons are made with earlier simulation studies.

The U.S. Food and Drug Administration (FDA) has proposed new regulations that address the 'prescribability' and 'switchability' of new formulations of already-approved drugs. These new criteria are known, respectively, as population and individual bioequivalence. Two methods have been proposed in the bioequivalence literature for assessing population and individual bioequivalence that calculate an upper 95 per cent confidence bound for the bioequivalence criterion in question, and then test bio-equivalence by comparing this bound to the limit established by the FDA. In this paper we propose applying the generalized test function (GTF) methodology of Tsui and Weerahandi (Journal of the American Statistical Association 1989; 602-607) to this problem to produce tests based on a generalized p-value (GPV). This methodology allows us to construct hypothesis tests in the presence of nuisance parameters. Using simulation we show that these tests perform well in comparison to the confidence interval methods and have superior power for assessing population bioequivalence.