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Adaptive two-stage designs allow a data-driven change of design characteristics during the ongoing trial. One of the available options is an adaptive choice of the test statistic for the second stage of the trial based on the results of the interim analysis. Since there is often only a vague knowledge of the distribution shape of the primary endpoint in the planning phase of a study, a change of the test statistic may then be considered if the data indicate that the assumptions underlying the initial choice of the test are not correct. Collings and Hamilton proposed a bootstrap method for the estimation of the power of the two-sample Wilcoxon test for shift alternatives. We use this approach for the selection of the test statistic. By means of a simulation study, we show that the gain in terms of power may be considerable when the initial assumption about the underlying distribution was wrong, whereas the loss is relatively small when in the first instance the optimal test statistic was chosen. The results also hold true for comparison with a one-stage design. Application of the method is illustrated by a clinical trial example.

Motivation: In high-dimensional testing problems π0, the proportion of null hypotheses that are true is an important parameter. For discrete test statistics, the P values come from a discrete distribution with finite support and the null distribution may depend on an ancillary statistic such as a table margin that varies among the test statistics. Methods for estimating π0 developed for continuous test statistics, which depend on a uniform or identical null distribution of P values, may not perform well when applied to discrete testing problems. Results: This article introduces a number of π0 estimators, the regression and ‘T’ methods that perform well with discrete test statistics and also assesses how well methods developed for or adapted from continuous tests perform with discrete tests. We demonstrate the usefulness of these estimators in the analysis of high-throughput biological RNA-seq and single-nucleotide polymorphism data. Availability and implementation: implemented in R Contact: nsa1@psu.edu or naomi@psu.edu Supplementary information: Supplementary data are available at Bioinformatics online.

The score statistic S2 is commonly used for general likelihood-based inference. Pearson’s Chi-squared statistic X2 = ∑(O − E)2/E is ubiquitous in contingency table inference. Because tests and confidence intervals based on S2 have been shown to work well in practice and theory and because X2 has such a simple and intuitively appealing form, it is of interest to know when S2 is identical to X2 and when X2 has an approximate Chi-squared distribution. Toward these ends, this paper gives a simple proof that S2 = X2 for the broad class of multinomial-Poisson distributions when the alternative hypothesis is unrestricted in a certain sense. This paper also gives a sufficient condition under which the null distribution of the Pearson score statistic is approximately Chi-squared. Several examples illustrate the utility of the results and counter-examples highlight the importance of the sufficient conditions of the results.

The purpose of this paper is to assess goodness of fit properties of the conditional distribution of the Pearson statistic, for noncanonical generalized linear models with data that are extensive but sparse, by Edgeworth approximation of p-values using higher order moment corrections. In this paper we obtain approximations to the fourth moments of the unconditional and conditional distribution of the modified Pearson statistic with noncanonical links. This extends previous results where approximations to the first three moments are available and completes all usual higher order moment calculations of the modified Pearson statistic. We consider the asymptotic limit in which the data are extensive but sparse and a supplementary estimating equation for the dispersion parameter. Specific results for binomial and Poisson data are obtained separately. The methods for assessing goodness of fit using higher order moments are discussed. For testing goodness of fit of generalized linear models to sparse data some simulations are conducted to compare, in terms of empirical size and power, the performance of the classical Pearson statistic (X 2), a standardized modified Pearson statistic ( $X_{\ast}^2$ ), a standardized modified deviance statistic (D ∗ ), a modified Pearson statistic based on Edgeworth approximation with the first three conditional moments (Z 1), and a modified Pearson statistic based on Edgeworth approximation with the first four conditional moments (Z 2). The statistic Z 2 holds level most effectively in most situations and has some power advantage.

The formula of the limit distribution of the maximum value test statistic is represented in the general case in this paper. The formula provides a precise calculation of a p-value in hypothesis testing. Also, the proof is represented in this paper.

In the paper of Akahira (Ann Inst Statist Math 48:349–364, 1996), it was shown that the second order asymptotic loss of information in reducing to a statistic consisting of extreme values and an asymptotically ancillary statistic vanished for a family of non-regular distributions whose densities have the same values and the sum of differential coefficients at the endpoints of the bounded support is equal to zero. In this paper, the result can be shown to be extended to the case of a family of non-regular distributions without the above restriction.

A test statistic for the detection of spatial clusters is developed by generalizing the common chi-square goodness-of-fit test. The paper includes a discussion of the relationship between the statistic and other associated statistics, and provides an analysis of both its null distribution and power. The paper concludes with the development of a local version of the statistic and an application to leukemia clustering in central New York.

The G statistic and its local version have been used extensively in spatial data analysis. The paper proves the asymptotic normality of the G statistic. Theorems in this paper imply that the regular permutation test for the G statistic is valid.

This paper is an investigation on the sufficient statistic for the parameters of the vector-valued (multivariate) ARMA models, when a finite sample is available. In the simplest case ARMA(1,1), by using the factorization theorem, we present a sufficient statistic whose dimension depends on the sample size and this dimension is even larger than the sample size. In this case and under some restrictions, we have solved this problem and have presented a sufficient statistic whose dimension does not depend on the sample size. In the general case, due to the complexity of the problem, we will use the modified versions of the likelihood function to find an approximate sufficient statistic in terms of the periodogram. The dimension of this sufficient statistic depends on the sample size; however, this dimension is much lower than the sample size.

Xie and Priebe [2002. ''Generalizing the Mann-Whitney-Wilcoxon Statistic''. J. Nonparametric Statist. 12, 661-682] introduced the class of weighted generalized Mann-Whitney-Wilcoxon (WGMWW) statistics which contained as special cases the classical Mann-Whitney test statistic and many other nonparametric distribution-free test statistics commonly used for the two-sample testing problem. The two-sample test that they proposed was based on any statistic within the class of WGMWW statistics optimal in the Pitman asymptotic efficacy (PAE) sense. In this paper, among other things, we show via simulation studies that for finite samples the PAE-optimal WGMWW test has substantially higher empirical power compared to the classical Mann-Whitney test for various underlying densities (especially for those densities for which Mann-Whitney test is considered a better alternative to parametric tests such as t-tests). The PAE-optimal WGMWW test is not a candidate for the practitioner's toolbox since the corresponding test statistic contains parameters which are functions of the underlying null distribution function of the samples. The main thrust of this paper is in introducing a data-adaptive alternative to the PAE-optimal WGMWW test, which has efficacy and power as good as the latter. We provide an estimate @[email protected]^ for the PAE function @j of a WGMWW statistic, and our test is based on a @[email protected]^-optimal WGMWW statistic. We prove strong consistency of @[email protected]^, thereby showing that our test has approximately the same efficacy as the @j-optimal WGMWW test for large sample sizes. Via simulation studies we show that for finite samples the empirical power of @[email protected]^-optimal WGMWW test is almost the same as @j-optimal WGMWW test for various underlying densities. We also analyze magnetic imaging data related to subjects with and without Alzheimer's disease to illustrate our methodology. In summary, we present a strong competitor for the classical Mann-Whitney-Wilcoxon test and many other existing nonparametric distribution-free tests, especially for moderate and large samples.

Often neither the exact density nor the exact cumulative distribution function (c.d.f.) of a statistic of interest is available in the statistics and econometrics literature (e.g., the maximum likelihood estimator of the autocorrelation coefficient in a simple Gaussian AR(1) model with zero start-up value). In other cases the exact c.d.f. of a statistic of interest is very complicated despite the statistic being “simple” (e.g., the circular serial correlation coefficient, or a quadratic form of a vector uniformly distributed over the unit n-sphere). The first part of the paper tries to explain why this is the case by studying the analytic properties of the c.d.f. of a statistic under very general assumptions. Differential geometric considerations show that there can be points where the c.d.f. of a given statistic is not analytic, and such points do not depend on the parameters of the model but only on the properties of the statistic itself. The second part of the paper derives the exact c.d.f. of a ratio of quadratic forms in normal variables, and for the first time a closed form solution is found. These results are then specialized to the maximum likelihood estimator of the autoregressive parameter in a Gaussian AR(1) model with zero start-up value, which is shown to have precisely those properties highlighted in the first part of the paper.

In this paper, we consider a simple risk model and study the occurrences of clusters of threshold exceedances by the individual claims. The statistic used to study the model is the discrete multiple scan statistic. A compound Poisson approximation is established and certain asymptotic results are obtained for both the risk model and a similar in nature financial problem. Finally, we review two typical examples from areas of applied science where the outcomes of this paper may have beneficial impact.

This paper provides a general method of modifying a statistic of interest in such a way that the distribution of the modified statistic can be approximated by an arbitrary reference distribution to an order of accuracy of O(n -1/2) or even O(n -1). The reference distribution is usually the asymptotic distribution of the original statistic. We prove that the multiplication of the statistic by a suitable stochastic correction improves the asymptotic approximation to its distribution. This paper extends the results of the closely related paper by Cordeiro and Ferrari (1991) to cope with several other statistical tests. The resulting expression for the adjustment factor requires knowledge of the Edgeworth-type expansion to order O(n-1) for the distribution of the unmodified statistic. In practice its functional form involves some derivatives of the reference distribution. Certain difference between the cumulants of appropriate order in n of the unmodified statistic and those of its first-order approximation, ...

In this paper, we present an accurate procedure, called “within-sample prediction of order statistics,” to obtain prediction limits for the number of failures that will be observed in a future inspection of a sample of units, based only on the results of the first in-service inspection of the same sample. The failure-time of such units is modeled with a two-parameter Weibull distribution indexed by scale and shape parameters β and δ, respectively. It will be noted that in the literature only the case is considered when the scale parameter β is unknown, but the shape parameter δ is known. As a rule, in practice the Weibull shape parameter δ is not known. Instead it is estimated subjectively or from relevant data. Thus, its value is uncertain. This δ uncertainty may contribute greater uncertainty to the construction of prediction limits for a future number of failures. In this paper, we consider the case when both parameters β and δ are unknown. The technique proposed here for constructing prediction limits emphasizes pivotal quantities relevant for obtaining ancillary statistics and represents a special case of the method of invariant embedding of sample statistics into a performance index applicable whenever the statistical problem is invariant under a group of transformations, which acts transitively on the parameter space. Application to other distributions could follow directly.

The paper considers the K-statistic, Kleibergen’s (2000) adaptation ofthe Anderson-Rubin (AR) statistic in instrumental variables regression.Compared to the AR-statistic this K-statistic shows improvedasymptotic efficiency in terms of degrees of freedom in overidentifiedmodels and yet it shares, asymptotically, the pivotal property of theAR statistic. That is, asymptotically it has a chi-square distributionwhether or not the model is identified. This pivotal property is veryrelevant for size distortions in finite-sample tests. Whereas Kleibergen(2000) focuses especially on the asymptotic behavior of the statistic,the present paper concentrates on finite-sample properties in a Gaussianframework. In that case the AR statistic has an F-distribution.However, the K-statistic is not exactly pivotal. Its finite-sample distributionis affected by nuisance parameters. Here we consider the twoextreme cases, which provide tight bounds for the exact distribution.The first case amounts to perfect identification—which is similar tothe asymptotic case—where the statistic has an F-distribution. Inthe other extreme case there is total underidentification. For the lattercase we show how to compute the exact distribution. Thus weprovide tight bounds for exact confidence sets based on the efficientK-statistic. Asymptotically the two bounds converge, except whenthere is a large number of redundant instruments.This paper has resulted in a publication in Econometric Theory, 2003, 19, 744-53.

A widely held notion of classical conditional theory is that statistical inference in the presence of ancillary statistics should be independent of the distribution of those ancillary statistcs. In this paper, ancillary paradoxes which contradict this notion are presented for two scenarios involving confidence estimation. These results are related to Brown’s ancillary paradox in point estimation. Moreover, the confidence coefficient, the usual constant coverage probability estimator, is shown to be inadmissible for confidence estimation in the multiple regression model with random predictor variables if the dimension of the slope parameters is greater than five. Some estimators better than the confidence coefficient are provided in this paper. These new estimators are constructed based on empirical Bayes estimators.

In a recent paper Kwiatkowski et al. (1992) propose the so-called KPSS statistic for testing the null hypothesis of stationarity against the alternative of a unit root process. The statistic employs a spectral estimator which can be shown to diverge with increasing sample size, given the alternative is true. Here, we suggest a modified spectral estimator which is shown to stabilize for moving average models. It is shown that this test statistic uniformly outperforms the KPSS statistic in an MA(1) model. Furthermore, a two-step nonparametric correction procedure is suggested, giving a test statistic with similar asymptotic properties as the original KPSS statistic. However, in small samples this correction performs better especially in detecting large random walk components.

Summary 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.

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.

Detection of multiple outliers or subset of influential points has been rarely considered in the linear measurement error models. In this paper a new influence statistic for one or a set of observations is generalized and characterized based on the corrected likelihood in the linear measurement error models. This influence statistic can be expressed in terms of the residuals and the leverages of linear measurement error regression. Unlike Cook’s statistic, this new measure of influence has asymptotically normal distribution and is able to detect a subset of high leverage outliers which is not identified by Cook’s statistic. As an illustrative example, simulation studies and a real data set are analysed.

A test for ordered categorical variables is of considerable importance, because they are frequently encountered in biomedical studies. This paper introduces a simple ordering test approach for the two-way r × c contingency tables with incomplete counts by developing six test statistics, i.e., the likelihood ratio test statistic, score test statistic, global score test statistic, Hausman-Wald test statistic, Wald test statistic and distance-based test statistic. Bootstrap resampling methods are also presented. The performance of the proposed tests is evaluated with respect to their empirical type I error rates and empirical powers. The results show that the likelihood ratio test statistic based on the bootstrap resampling methods perform satisfactorily for small to large sample sizes. A real example from a wheeze study in six cities is used to illustrate the proposed methodologies.

In practice, it is often not possible to find an appropriate family of distributions which can be used for fitting the sample distribution with high precision. In these cases, it seems to be opportune to search for the best approximation by a family of distributions instead of an exact fit. In this paper, we consider the Anderson–Darling statistic with plugged-in minimum distance estimator for the parameter vector. We prove asymptotic normality of the Anderson–Darling statistic which is used for a test of goodness of approximation. Moreover, we introduce a measure of discrepancy between the sample distribution and the model class.

The statistician often needs to test whether or not a time series has a seasonal first moment. The problem often arises in environmental series, where most time-ordered data display some type of periodic structure. This paper reviews the problem, proposing new statistics in both the time and frequency domains. Our new time domain statistic has an analysis of variance form that is based on the one-step-ahead prediction errors of the series. This statistic inherits the classic traits of the F-distribution arising in one-way analysis of variance tests, is easy to use, and is asymptotically equivalent to the likelihood ratio test. The statistics asymptotic distribution is quantified when time series parameters are estimated. In the frequency domain, a statistic modifying Fisher's classical test for a sinusoidal mean superimposed on independent and identically distributed Gaussian noise is devised. The performance and comparison of these statistics are studied via simulation. Implementation of the methods merely requires sample means, autocovariances, and periodograms of the series. Application to a data set of monthly temperatures from Tuscaloosa, Alabama, is given. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider the consistency of Cp-type statistics for selecting variables in a normality-assumed linear regression with multiple responses when the dimension of the vector of the response variables may be large. We propose a new consistent Cp-type statistic for which consistency can be achieved whenever the dimension of the response variables vector is fixed or goes to infinity. A high probability of selecting the true subset of explanatory variables can be expected under a moderate sample size when the proposed Cp-type statistic is used to select variables, even when there is a high-dimensional response variables vector.

Barndorff-Nielsen’s celebrated p*-formula and variations thereof have amongst their various attractions the ability to approximate bimodal distributions. In this paper we show that in general this requires a crucial adjustment to the basic formula. The adjustment is based on a simple idea and straightforward to implement, yet delivers important improvements. It is based on recognizing that certain outcomes are theoretically impossible and the density of the MLE should then equal zero, rather than the positive density that a straight application of p* would suggest. This has implications for inference and we show how to use the new p**-formula to construct improved confidence regions. These can be disjoint as a consequence of the bimodality. The degree of bimodality depends heavily on the value of an approximate ancillary statistic and conditioning on the observed value of this statistic is therefore desirable. The p**-formula naturally delivers the relevant conditional distribution. We illustrate these results in small and large samples using a simple nonlinear regression model and errors in variables model where the measurement errors in dependent and explanatory variables are correlated and allow for weak proxies.

Summary The general nonnull distribution of Wilks statistic, the likelihood ratio statistic in MANOVA, can be expressed as a product of conditional beta variables [3]. Making use of this result, in the present paper, an upper bound for the nonnull distribution of Wilks statistic is obtained, which provides a conservative evaluation of the power of the likelihood ratio test for the cases when the alternative hypothesis is of rank 1, 2 or 3. For p=2, where p is the number of variables, and large f2, the degrees of freedom, it has been shown that the results of this paper give a much better approximation to the power of Wilks statistic than Mikhail's approximation [10]. A few percentage points have also been computed for p=3 and selected values of the degrees of freedom and the noncentrality parameters, which in the linear case have been compared with the exact values obtained by the author [7].

SUMMARY A consistency criterion is proposed for fiducial probability distributions: if a fiducial density function is used to modulate a prior density function, the resulting frequency function should be that obtained by a Bayesian analysis. For a real variable and a real parameter, the fulfilment of this criterion is equivalent to a condition examined by Lindley (1958) and it requires the parameter to be essentially a location parameter. For several variables and the same number of parameters, the criterion if applied to fiducial probability derived from a pivotal quantity produces a simple differential equation condition on the pivotal quantity. If one-dimensional fiducial distributions can be generated in an arbitrary direction from any point in the parameter space, then the fulfilment of the criterion requires the model to be of transformation-parameter form; for such models a variety of properties concerned with consistency and interpretation are available (Fraser, 1961). The fiducial method of inference was introduced by Fisher (1930) and has been discussed and developed in many of his subsequent papers. The development is primarily in terms of examples, with but little attention to precise governing principles. In this paper the method will be interpreted in terms of the pivotal quantities that are a common feature in the examples. A pivotal quantity is a function of a sufficient statistic and a parameter that has a fixed frequency distribution. For a single real variable and a single real parameter the distribution function is a pivotal quantity and it has the uniform distribution on (0, 1) regardless of the parameter value. In some statistical problems there may be a natural choice for the pivotal quantity based, perhaps on the physical origin of the problem. The quantity may then express a position, a probability or error position for an outcome relative to the parameter value. In other problems there may be a natural choice based on mathematical properties of the specification. In this paper a pivotal quantity will be assumed given as part of the statistical problem: the distribution of the pivotal variable will be interpreted as describing the error distribution or error pattern, and the pivotal equation as describing the way that the probability mass of the error pattern is applied to the sample space. The fiducial method as presented by Fisher involves a preliminary reduction to an exhaustive statistic. If the exhaustive statistic is of the same dimension as the parameter it is used as the basic variable in the pivotal quantity. If the exhaustive statistic is of higher dimension, then an ancillary statistic is needed such that conditionally the exhaustive statistic has the same dimension as the parameter and can be used as basic variable in the pivotal quantity.

The Durbin-Watson statistic is used for testing the first order serial correlation among residuals in a linear model. It is based on the residuals from a corresponding regression analysis. In this paper a generalization of the statistic which tests for higher order dependence among residuals is proposed. The paper gives a brief review of the Durbin-Watson theory and the construction of the corresponding significance tables based on Jacobi orthogonal polynomials and the beta density. The distribution of the Durbin-Watson statistic based on Tmhof's distribution of Quadratic forms is indicated as an alternative method of directly computing the distribution function of the Durbin-Watson statistic, as well as its generalization, Significance values for 5 percent level of significance for lags 2 to 4 are given in Table II.

Pearson's Chi-square statistic for frequency tables depends on what is hypothesized as the expected frequencies. Its partitions also depend on the hypothesis. Lancaster (J R Stat Soc B 13:242---249, 1951) proposed ANOVA-like partitions of Pearson's statistic under several representative hypotheses about the expected frequencies. His expositions were, however, not entirely clear. In this paper, we clarify his method of derivations, and extend it to more general situations. A comparison is made with analogous decompositions of the log likelihood ratio statistic associated with log-linear analysis of contingency tables.

This paper proposes a new simple panel unit-root test by extending the cross-sectionally augmented panel unit-root test (CIPS) developed by Pesaran et al. (2013) to allow for smoothing structural changes in deterministic terms, approximated by a Fourier series. The proposed statistic is the simple average of the individual statistics constructed from the breaks and cross-sectional dependence augmented Dickey-Fuller (BCADF) regression and is called the BCIPS statistic. We initially develop the tests by assuming that the number of factors in the model is known and show that the limiting distribution of the BCADF statistic is free of nuisance parameters. The nonstandard limiting distribution of the (truncated) BCIPS statistic is also shown to exist and its critical values are tabulated. Monte-Carlo experiments point out that the sizes and powers of the BCIPS statistic are generally satisfactory as long as T is greater than or equal to fifty and a hundred, respectively. By using two different methods to determine the number of factors, both the BCIPS and CIPS tests are applied to examine the validity of long-run purchasing power parity. The proposed test complements the panel unit-root tests with breaks using dummy variables.

In normal theory exploratory factor analysis, likelihood ratio (LR) statistic plays an important role in evaluating the goodness-of-fit of the model. In this paper, we derive an approximation of the LR statistic. The approximation is then used to show explicitly that the expectation of the LR statistic agrees with the degrees of freedom of the asymptotic chi-square distribution.

This paper proposes an overlapping-based test statistic for testing the equality of two exponential distributions with different scale and location parameters. The test statistic is defined as the maximum likelihood estimate of the Weitzman's overlapping coefficient, which estimates the agreement of two densities. The proposed test statistic is derived in closed form. Simulated critical points are generated for the proposed test statistic for various sample sizes and significance levels via Monte Carlo Simulations. Statistical powers of the proposed test are computed via simulation studies and compared to those of the existing Log likelihood ratio test.