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We consider inference about the parameter that determines the distribution of the data. In frequentist inference a very important and useful idea is that data reduction to a sufficient statistic does not lose any information about this parameter. We recall two justifications for this idea in frequentist inference. We then examine the extent to which these justifications carry over to conditional frequentist inference inference, which consists of carrying out frequentist inference conditional on an ancillary statistic. This examination shows that, in the context of conditional frequentist inference, first reducing data to a sufficient statistic is not always justified, so we should first condition on an ancillary statistic. Finally, we describe two types of practically important statistical models that illustrate this finding. [ABSTRACT FROM AUTHOR]

Data-driven most powerful tests are statistical hypothesis decision-making tools that deliver the greatest power against a fixed null hypothesis among all corresponding data-based tests of a given size. When the underlying data distributions are known, the likelihood ratio principle can be applied to conduct most powerful tests. Reversing this notion, we consider the following questions. (a) Assuming a test statistic, say T, is given, how can we transform T to improve the power of the test? (b) Can T be used to generate the most powerful test? (c) How does one compare test statistics with respect to an attribute of the desired most powerful decision-making procedure? To examine these questions, we propose one-to-one mapping of the term "most powerful" to the distribution properties of a given test statistic via matching characterization. This form of characterization has practical applicability and aligns well with the general principle of sufficiency. Findings indicate that to improve a given test, we can employ relevant ancillary statistics that do not have changes in their distributions with respect to tested hypotheses. As an example, the present method is illustrated by modifying the usual t-test under nonparametric settings. Numerical studies based on generated data and a real-data set confirm that the proposed approach can be useful in practice. [ABSTRACT FROM AUTHOR]

For the analysis of clinical trials, the study participants are usually assumed to be representative sample of a target population. This assumption is rarely fulfilled in clinical trials, and particularly not if the sample size is small. In addition, covariate imbalances may affect the trial. Randomization tests provide a nonparametric analysis method of the treatment effect that does not rely on population‐based assumptions. We propose a nonparametric statistical model that yields a formal basis for randomization tests. We adapt the model for the presence of covariate imbalance in the form of selection bias and investigate the effects of bias on the rejection probability of the randomization test using Monte Carlo simulations. Finally, we show that ancillary statistics can be used to control for the influence of bias. We show that covariate imbalance leads to an inflation of the type I error probability. The proposed nonparametric model allows for the use of ancillary statistics that yield an unbiased adjusted randomization test. [ABSTRACT FROM AUTHOR]

In comparing two treatments via a randomized clinical trial, the analysis of covariance (ANCOVA) technique is often utilized to estimate an overall treatment effect. The ANCOVA is generally perceived as a more efficient procedure than its simple two sample estimation counterpart. Unfortunately, when the ANCOVA model is nonlinear, the resulting estimator is generally not consistent. Recently, various nonparametric alternatives to the ANCOVA, such as the augmentation methods, have been proposed to estimate the treatment effect by adjusting the covariates. However, the properties of these alternatives have not been studied in the presence of treatment allocation imbalance. In this article, we take a different approach to explore how to improve the precision of the naive two-sample estimate even when the observed distributions of baseline covariates between two groups are dissimilar. Specifically, we derive a bias-adjusted estimation procedure constructed from a conditional inference principle via relevant ancillary statistics from the observed covariates. This estimator is shown to be asymptotically equivalent to an augmentation estimator under the unconditional setting. We utilize the data from a clinical trial for evaluating a combination treatment of cardiovascular diseases to illustrate our findings. [ABSTRACT FROM AUTHOR]

Consider a high-dimensional linear regression problem, where the number of covariates is larger than the number of observations and the interest is in estimating the conditional variance of the response variable given the covariates. A conditional and an unconditional framework are considered, where conditioning is with respect to the covariates, which are ancillary to the parameter of interest. In recent papers, a consistent estimator was developed in the unconditional framework when the marginal distribution of the covariates is normal with known mean and variance. In the present work, a certain Bayesian hypothesis test is formulated under the conditional framework, and it is shown that the Bayes risk is a constant. This implies that no consistent estimator exists in the conditional framework. However, when the marginal distribution of the covariates is normal, the conditional error of the above consistent estimator converges to zero, with probability converging to one. It follows that even in the conditional setting, information about the marginal distribution of an ancillary statistic may have a significant impact on statistical inference. The practical implication in the context of high-dimensional regression models is that additional observations where only the covariates are given are potentially very useful and should not be ignored. This finding is most relevant to semi-supervised learning problems where covariate information is easy to obtain. [ABSTRACT FROM AUTHOR]

For a two-group comparative study, a stratified inference procedure is routinely used to estimate an overall group contrast to increase the precision of the simple two-sample estimator. Unfortunately, most commonly used methods including the Cochran-Mantel-Haenszel statistic for a binary outcome and the stratified Cox procedure for the event time endpoint do not serve this purpose well. In fact, these procedures may be worse than their two-sample counterparts even when the observed treatment allocations are imbalanced across strata. Various procedures beyond the conventional stratified methods have been proposed to increase the precision of estimation when the naive estimator is consistent. In this paper, we are interested in the case when the treatment allocation proportions vary markedly across strata. We study the stochastic properties of the two-sample naive estimator conditional on the ancillary statistics, the observed treatment allocation proportions and/or the stratum sizes, and present a biased-adjusted estimator. This adjusted estimator is asymptotically equivalent to the augmentation estimators proposed under the unconditional setting. Moreover, this consistent estimation procedure is also equivalent to a rather simple procedure, which estimates the mean response of each treatment group first via a stratum-size weighted average and then constructs the group contrast estimate. This simple procedure is flexible and readily applicable to any target patient population by choosing appropriate stratum weights. All the proposals are illustrated with the data from a cardiovascular clinical trial, whose treatment allocations are imbalanced. [ABSTRACT FROM AUTHOR]

Summary: N‐mixture models describe count data replicated in time and across sites in terms of abundance N and detectability p. They are popular because they allow inference about N while controlling for factors that influence p without the need for marking animals. Using a capture–recapture perspective, we show that the loss of information that results from not marking animals is critical, making reliable statistical modeling of N and p problematic using just count data. One cannot reliably fit a model in which the detection probabilities are distinct among repeat visits as this model is overspecified. This makes uncontrolled variation in p problematic. By counter example, we show that even if p is constant after adjusting for covariate effects (the “constant p” assumption) scientifically plausible alternative models in which N (or its expectation) is non‐identifiable or does not even exist as a parameter, lead to data that are practically indistinguishable from data generated under an N‐mixture model. This is particularly the case for sparse data as is commonly seen in applications. We conclude that under the constant p assumption reliable inference is only possible for relative abundance in the absence of questionable and/or untestable assumptions or with better quality data than seen in typical applications. Relative abundance models for counts can be readily fitted using Poisson regression in standard software such as R and are sufficiently flexible to allow controlling for p through the use covariates while simultaneously modeling variation in relative abundance. If users require estimates of absolute abundance, they should collect auxiliary data that help with estimation of p. [ABSTRACT FROM AUTHOR]

In this paper, tests for the skewness parameter of the two-piece double exponential distribution are derived when the location parameter is unknown. Classical tests like Neyman structure test and likelihood ratio test (LRT), that are generally used to test hypotheses in the presence of nuisance parameters, are not feasible for this distribution since the exact distributions of the test statistics become very complicated. As an alternative, we identify a set of statistics that are ancillary for the location parameter. When the scale parameter is known, Neyman-Pearson's lemma is used, and when the scale parameter is unknown, the LRT is applied to the joint density function of ancillary statistics, in order to obtain a test for the skewness parameter of the distribution. Test for symmetry of the distribution can be deduced as a special case. It is found that power of the proposed tests for symmetry is only marginally less than the power of corresponding classical optimum tests when the location parameter is known, especially for moderate and large sample sizes. [ABSTRACT FROM AUTHOR]

We offer an exposition of modern higher order likelihood inference and introduce software to implement this in a quite general setting. The aim is to make more accessible an important development in statistical theory and practice. The software, implemented in an R package, requires only that the user provide code to compute the likelihood function and to specify extra-likelihood aspects of the model, such as stopping rule or censoring model, through a function generating a dataset under the model. The exposition charts a narrow course through the developments, intending thereby to make these more widely accessible. It includes the likelihood ratio approximation to the distribution of the maximum likelihood estimator, that is the p∗ formula, and the transformation of this yielding a second-order approximation to the distribution of the signed likelihood ratio test statistic, based on a modified signed likelihood ratio statistic r∗. This follows developments of Barndorff-Nielsen and others. The software utilises the approximation to required Jacobians as developed by Skovgaard, which is included in the exposition. Several examples of using the software are provided. [ABSTRACT FROM AUTHOR]

Accurate inference on a scalar interest parameter in the presence of a nuisance parameter may be obtained using an adjusted version of the signed root likelihood ratio statistic, in particular Barndorff-Nielsen’s R ∗ statistic. The adjustment made by this statistic may be decomposed into a sum of two terms, interpreted as correcting respectively for the possible effect of nuisance parameters and the deviation from standard normality of the signed root likelihood ratio statistic itself. We show that the adjustment terms are determined to second-order in the sample size by their means. Explicit expressions are obtained for the leading terms in asymptotic expansions of these means. These are easily calculated, allowing a simple way of quantifying and interpreting the respective effects of the two adjustments, in particular of the effect of a high dimensional nuisance parameter. Illustrations are given for a number of examples, which provide theoretical insight to the effect of nuisance parameters on parametric inference. The analysis provides a decomposition of the mean of the signed root statistic involving two terms: the first has the property of taking the same value whether there are no nuisance parameters or whether there is an orthogonal nuisance parameter, while the second is zero when there are no nuisance parameters. Similar decompositions are discussed for the Bartlett correction factor of the likelihood ratio statistic, and for other asymptotically standard normal pivots. [ABSTRACT FROM AUTHOR]

Developments in the theory of frequentist parametric inference in recent decades have been driven largely by the desire to achieve higher-order accuracy, in particular distributional approximations that improve on first-order asymptotic theory by one or two orders of magnitude. At the same time, much methodology is specifically designed to respect key principles of parametric inference, in particular conditionality principles. Two main routes to higher-order accuracy have emerged: analytic methods based on ‘small-sample asymptotics’, and simulation, or ‘bootstrap’, approaches. It is argued here that, of these, the simulation methodology provides a simple and effective approach, which nevertheless retains finer inferential components of theory. The paper seeks to track likely developments of parametric inference, in an era dominated by the emergence of methodological problems involving complex dependences and/or high-dimensional parameters that typically exceed available data sample sizes. [ABSTRACT FROM AUTHOR]

In 1922 R. A. Fisher introduced the modern regression model, synthesizing the regression theory of Pearson and Yule and the least squares theory of Gauss. The innovation was based on Fisher's realization that the distribution associated with the regression coefficient was unaffected by the distribution of X. Subsequently Fisher interpreted the fixed X assumption in terms of his notion of ancillarity. This paper considers these developments against the background of the development of statistical theory in the early twentieth century. [ABSTRACT FROM AUTHOR]

Sample size calculations are important and difficult in clinical trails because they depend on the nuisance parameter and treatment effect. Recently, much attention has been focused on two-stage methods whereby the first stage constitutes an internal pilot study used to estimate parameters and revise the final sample size. This paper reviews two-stage methods based on estimation of nuisance parameters in either a continuous or dichotomous outcome setting. [ABSTRACT FROM AUTHOR]

In this note we consider von Mises–Fisher families of probability densities on spheres and more generally on Stiefel manifolds, which include the orthogonal groups. It addresses the estimation of the mean direction or the mean location by empirical mean location, which for the von Mises–Fisher family coincides with the maximum likelihood estimator. It is shown that (with a few exceptions) the empirical mean location of a sample is almost surely uniquely defined and that it is unbiased in the sense that its mean location coincides with the mean location of the von Mises–Fisher distribution. The main goal, however, is to show that empirical mean location is an admissible estimator for the mean location of the von Mises–Fisher distribution. [Copyright &y& Elsevier]

In applied statistics, the coefficient of variation is widely calculated and interpreted even when the sample size of the data set is very small. However, confidence intervals for the coefficient of variation are rarely reported. One of the reasons is the exact confidence interval for the coefficient of variation, which is given in Lehmann (Testing Statistical Hypotheses, 2nd Edition, Wiley, New York, 1996), is very difficult to calculate. Various asymptotic methods have been proposed in literature. These methods, in general, require the sample size to be large. In this article, we will apply a recently developed small sample asymptotic method to obtain approximate confidence intervals for the coefficient of variation for both normal and nonnormal models. These small sample asymptotic methods are very accurate even for very small sample size. Numerical examples are given to illustrate the accuracy of the proposed method. [Copyright &y& Elsevier]

The likelihood ratio approximation, also called Barndorff‐Nielsen's approximation and often denoted by p*, provides a highly accurate approximation to the conditional density of a maximum likelihood estimator Θ given an ancillary statistic. In this paper, the properties of p* are considered for the case in which the underlying random variables have a lattice distribution and Θ has a discrete, but not necessarily lattice, distribution. If Θ has a lattice distribution, then p* provides a valid approximation to the density of Θ with respect to counting measure. If the distribution of Θ is non‐lattice, then p* is still a valid approximation to the conditional density of Θ however, the dominating measure is no longer counting measure. [ABSTRACT FROM PUBLISHER]

For estimation of functions involving only parameters of interest, in the presence of nuisance parameters, some optimality properties are established for partially sufficient (i.e. p-sufficient) statistics in two classes of regular probability models. The results are based on a characterization of regular unbiased estimating functions for parameters of interest in probability models for which a statistic exists such that its marginal distribution depends on unknown parameters only through the parameters of interest. [ABSTRACT FROM AUTHOR]

A structured model is essentially a family of random vectors Xθ defined on a probability space with values in a sample space. If, for a given sample value x and for each ω in the probability space, there is at most one parameter value θ for which Xθ(ω) is equal to x, then the model is called additive at x. When a certain conditional distribution exists, a frequency interpretation specific to additive structured models holds, and is summarized in a unique structured distribution for the parameter. Many of the techniques used by Fisher in deriving and handling his fiducial probability distribution are shown to be valid when dealing with a structured distribution. [ABSTRACT FROM AUTHOR]

Lei A* be a multinomial (n. p) variable, where n is an ancillary statistic. In Section 2. it is shown that the minimax estimator of p for fixed sample size n is not minimax for squared error loss. In Section 3, it is shown that the minimax estimator of p for fixed sample size n is still minimax for relative squared error loss. [ABSTRACT FROM AUTHOR]

Two lists are given. The first is a list of properties of ancillary statistics; the second is a list of examples of ancillary statistics. It is then indicated which of the properties are satisfied by each example. Many of the models have the property that the parameter θ is a location parameter for the maximum likelihood estimator (MLE) θ in every conditional distribution determined by a fixed value of the ancillary statistic. In certain other models the same state of affairs is achieved by parameter transformation. In either of these cases we call the ancillary an "exact precision index." There exist irregular models in which the precision of estimation depends not only on the ancillary but on θ as well. Some comments are offered on approximate ancillarity and on the conditionality principle. [ABSTRACT FROM AUTHOR]

In order to obtain a statistical interpretation of completeness of a sufficient statistic T, an attempt is made to characterize this property as equivalent to the condition that all ancillary statistics are independent of T. For technical reasons, this characterization is not correct, but two correct versions are obtained in Section 3 by modifying either the definition of ancillarity or of completeness. The light this throws on the meaning of completeness is discussed in Section 4. Finally, Section 5 reviews some of the difficulties encountered in models that lack completeness. [ABSTRACT FROM AUTHOR]

Conditional and unconditional confidence intervals have been compared by Grice, Bain, and Engelhardt (Commun. Statist. B7 (1978), 515-524) in terms of the location-scale model with double-exponential distribution form. Preference was found for the conditional intervals based on mean length and coverage probability for untrue parameters values. These two criteria for a location-scale system are shown to be inappropriate criteria for assessing the conditional versus unconditional approaches to inference. The usual ancillarity concept is also noted to be inappropriate. Support for many conditional analyses, however, is found in a more careful formulation of the statistical model. [ABSTRACT FROM AUTHOR]