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A kappa-like classification statistic is proposed for assessing the fit of GEE regression models with a categorical response. The proposed statistic is a summary measure depicting how well categorical responses are predicted from the fitted GEE model. The statistic takes on a value of 1 if prediction is perfect and a value of 0 if the fitted model fares no better than random chance, i.e., fitting the repeated categorical responses with an intercept-only model. To demonstrate the usefulness of the classification statistic, we present simulation results as well as two examples from biomedical studies.

We develop a likelihood ratio test statistic, based on the beta- binomial distribution, for comparing a single treated group with dichoto- mous data to dual control groups. This statistic is useful in cases where there is overdispersion or extra-binomial variation. We apply the statis- tic to data from a two year rodent carcinogenicity study with dual control groups. The test statistic we developed is similar to others that have been developed for incorporation of historical control groups with rodent carcino- genicity experiments. However, for the small sample case we considered, large sample theory used by the other test statistics did not apply. We de- termined the critical values of this statistic by enumerating its distribution. A small Monte Carlo study shows the new test statistic controls the signifi- cance level much better than Fisher's exact test when there is overdispersion and that it has adequate power.

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.

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 perceiv...

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.

The concept of pattern arises in many applications comprising experimental trials with two or more possible outcomes in each trial. A binary scan of type r / k is a special pattern referring to success–failure strings of fixed length k that contain at least r-successes, where r, k are positive integers with $$r\le k$$ . The multiple scan statistic $$W_{t,k,r}$$ is defined as the enumerating random variable for the overlapping moving windows occurring until trial t which include a scan of type r / k. In the present work, we consider a sequence of independent binary trials with not necessarily equal probabilities of success and develop upper bounds for the probability of the event that the multiple scan statistic will perform a jump from $$\ell $$ to $$\ell +1$$ (where $$\ell $$ is a nonnegative integer) in a finite time horizon.

Summary 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.

In this study, a simple diagnostic statistic for determining the size of random forest is proposed. This method is based on MV (margin of victory), a scaled difference in the votes at the infinite forest between the first and second most popular categories of the current random forest. We can note that if MV is negative then there is discrepancy between the current and infinite forests. More precisely, our method is based on the proportion of cases that -MV is greater than a fixed small positive number (say, 0.03). We derive an appropriate diagnostic statistic for our method and then calculate the distribution of the statistic. A simulation study is performed to compare our method with a recently proposed diagnostic statistic.

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.

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.

This article reexamines ANOVA problems for autocorrelated data. Using linear prediction techniques for stationary time series, a new test statistic that assesses a null hypothesis of equal means is proposed and investigated. Our test statistic mimics the classical F-type ratio form used with independent data, but substitutes estimated prediction residuals in for the errors. This simple tactic departs from past studies that adjust the quadratic forms in the numerator and denominator in the F ratio for autocorrelation. One of the advantages is that our statistic retains the classical null hypothesis F distribution (now as a limit) with the customary degrees of freedom. The statistic is shown to perform well in simulations. Asymptotic proofs are given in the case of autoregressive random errors; a sports application is supplied.[Received December 2014. Revised August 2015.]

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.

The C statistic is a commonly reported measure of screening test performance. Optimistic estimation of the C statistic is a frequent problem because of overfitting of statistical models in small data sets, and methods exist to correct for this issue. However, many studies do not use such methods, and those that do correct for optimism use diverse methods, some of which are known to be biased. We used clinical data sets (United Kingdom Down syndrome screening data from Glasgow (1991-2003), Edinburgh (1999-2003), and Cambridge (1990-2006), as well as Scottish national pregnancy discharge data (2004-2007)) to evaluate different approaches to adjustment for optimism. We found that sample splitting, cross-validation without replication, and leave-1-out cross-validation produced optimism-adjusted estimates of the C statistic that were biased and/or associated with greater absolute error than other available methods. Cross-validation with replication, bootstrapping, and a new method (leave-pair-out cross-validation) all generated unbiased optimism-adjusted estimates of the C statistic and had similar absolute errors in the clinical data set. Larger simulation studies confirmed that all 3 methods performed similarly with 10 or more events per variable, or when the C statistic was 0.9 or greater. However, with lower events per variable or lower C statistics, bootstrapping tended to be optimistic but with lower absolute and mean squared errors than both methods of cross-validation.

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.

As an important extension of the regular kappa statistic, the weighted kappa statistic has been widely used to assess the agreement between two procedures for independent matched-pair ordinal data. For clustered matched-pair ordinal data, based on the delta method and sampling techniques, a non-parametric variance estimator for the weighted kappa statistic is proposed without within-cluster correlation structure or distributional assumptions. The results of an extensive Monte Carlo simulation study demonstrate that the proposed weighted kappa statistic provides consistent estimation, and the proposed variance estimator behaves reasonably well for at least a moderately large number of clusters (e.g., K ≥ 50 ). Compared with the variance estimator ignoring dependence within a cluster, the proposed variance estimator performs better in maintaining the nominal coverage probability when the intra-cluster correlation is fair ( ρ ≥ 0.3 ), with more pronounced improvement when ρ is further increased. Moreover, under the general analysis of variance setting with systematic variability between procedures and clusters being included as a component of total variation, the equivalence between weighted kappa statistic and intraclass correlation coefficient is established. To illustrate the practical application of the proposed estimator, two real medical research data examples of clustered matched-pair ordinal data are analyzed, including an agreement study to compare two methods for assessing cervical ectopy, and a physician–patients data example from the Enhancing Communication and HIV Outcomes study.

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.

A U-statistic for the tail index of a multivariate stable random vector is given as an extension of the univariate case introduced by Fan (2006). Asymptotic normality and consistency of the proposed U-statistic for the tail index are proved theoretically. The proposed estimator is used to estimate the spectral measure. The performance of both introduced tail index and spectral measure estimators is compared with the known estimators by comprehensive simulations and real datasets.

Single moments of order statistics from the modified Makeham distribution (MMD) are derived, an identity about the single moments of order statistics is given, and the specific expected value and variance of the single moments of order statistics from the MMD are calculated. In this study, the order statistic from the MMD was applied to the rank sum test in a two-sample problem. The exact critical values of the designated statistics were evaluated. Simulations were used to investigate the power of these statistics for the two-sided alternative with several population distributions. The powers of the statistics were compared with the Wilcoxon rank sum statistic, the Lepage statistic, the modified Baumgartner statistic, the Savage test and the normal score test. The Edgeworth expansion was used to evaluate the upper tail probability for the preferred statistic, given finite sample sizes.

The scan statistic is widely used in spatial cluster detection applications of inhomogeneous Poisson processes. However, real data may present substantial departure from the underlying Poisson process. One of the possible departures has to do with zero excess. Some studies point out that when applied to data with excess zeros, the spatial scan statistic may produce biased inferences. In this work, we develop a closed-form scan statistic for cluster detection of spatial zero-inflated count data. We apply our methodology to simulated and real data. Our simulations revealed that the Scan-Poisson statistic steadily deteriorates as the number of zeros increases, producing biased inferences. On the other hand, our proposed Scan-ZIP and Scan-ZIP+EM statistics are, most of the time, either superior or comparable to the Scan-Poisson statistic.

A wide literature is available on the asymptotic behavior of the Durbin-Watson statistic for autoregressive models. However, it is impossible to find results on the Durbin-Watson statistic for autoregressive models with adaptive control. Our purpose is to fill the gap by establishing the asymptotic behavior of the Durbin Watson statistic for ARX models in adaptive tracking. On the one hand, we show the almost sure convergence as well as the asymptotic normality of the least squares estimators of the unknown parameters of the ARX models. On the other hand, we establish the almost sure convergence of the Durbin-Watson statistic and its asymptotic normality. Finally, we propose a bilateral statistical test for residual autocorrelation in adaptive tracking.

This study deals with the small sample likelihood based inference for the ratio of twonormal variances. The small sample likelihood inference is an approximation method.The signed log-likelihood ratio statistic and the modi ed signed log-likelihood ratiostatistic, which converge to standard normal distribution, are proposed for the normalvariance ratio. Through the simulation study, the coverage probabilities of con denceinterval and power of the exact, the signed log-likelihood and the modi ed signed log-likelihood ratio statistic will be compared. A real data example will be provided.Keywords: Likelihood based inference, modi ed signed log-likelihood ratio statistic,normal variance ratio, signed log-likelihood ratio statistic. 1. Introduction The normal distribution plays an important role in statistical inference, and there are somany studies about this distribution. The statistical inference for the ratio of two normalvariances arises in the areas for comparing the precision of two independent normal popula-tions. About the ratio of two normal variances, there also exists an exact statistical inferencewith Fstatistic. This problem is simple, obvious and important.When comparing the dispersion of two normal population, one can use the distributiontable of F statistic. Since F distribution depends on degrees of freedom, the distributiontable of Fdistribution is more than several pages. For the degrees of freedom exceed 30, thetable does not have the value. In this case, one must use an interpolation to obtaining thequantile.An approximation of a statistic to standard normal distribution has been developed inmany statistical models. Since the percentile of standard normal distribution is well known,a statistic, which distributes as standard normal distribution asymptotically, may be veryuseful. But these statistics have the error rate depending on sample size. When the samplesize is small, these approximation to standard normal distribution is quite inaccurate. Forexample, a signed log-likelihood ratio statistic converges to standard normal distributionwith an error of O(n

This paper focuses on inferential tools in the logistic regression model fitted by the Firth penalized likelihood. In this context, the Likelihood Ratio statistic is often reported to be the preferred choice as compared to the ‘traditional’ Wald statistic. In this work, we consider and discuss a wider range of test statistics, including the robust Wald, the Score, and the recently proposed Gradient statistic. We compare all these asymptotically equivalent statistics in terms of interval estimation and hypothesis testing via simulation experiments and analyses of two real datasets. We find out that the Likelihood Ratio statistic does not appear the best inferential device in the Firth penalized logistic regression.

Under standard regularity assumptions, we provide simple approximations for specific classes of fiducial and confidence distributions and discuss their connections with objective Bayesian posteriors. For a real parameter the approximations are accurate at least to order O(1/n). For the mean parameter of a multivariate exponential family, our fiducial distribution is asymptotically normal and invariant to the importance ordering of the components of the mean parameter., Comment: New examples are added to compare our results with traditional MLE inference, with emphasis on coverage probabilities. New references are added

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.

Under model correctness, highly accurate inference on a scalar interest parameter in the presence of a nuisance parameter can be achieved by several routes, among them considering the bootstrap distribution of the signed root likelihood ratio statistic. The context of model mis-specification is considered and inference based on a robust form of the signed root statistic is discussed in detail. Stability of the distribution of the statistic allows accurate inference, outperforming that based on first-order asymptotic approximation, by considering the bootstrap distribution of the statistic under the incorrectly assumed distribution. Comparisons of this simple approach with alternative analytic and non-parametric inference schemes are discussed.

Goodness of fit testing in factor analysis is based on the assumption that the test statistic is asymptotically chi-square; but this property may not hold in small samples even when the factors and errors are normally distributed in the population. Robust methods such as Browne’s asymptotically distribution-free method and Satorra Bentler’s mean scaling statistic were developed under the presumption of non-normality in the factors and errors. This paper finds new application to the case where factors and errors are normally distributed in the population but the skewness of the obtained test statistic is still high due to sampling error in the observed indicators. An extension of Satorra Bentler’s statistic is proposed that not only scales the mean but also adjusts the degrees of freedom based on the skewness of the obtained test statistic in order to improve its robustness under small samples. A simple simulation study shows that this third moment adjusted statistic asymptotically performs on par with previously proposed methods, and at a very small sample size offers superior Type I error rates under a properly specified model. Data from Mardia, Kent and Bibby’s study of students tested for their ability in five content areas that were either open or closed book were used to illustrate the real-world performance of this statistic.

In this paper we study some basic statistics in quantile regression. In particular, we investigate the residual, goodness-of-fit statistic and the effect of one or few observations on estimates of regression coefficients. In addition, we compare the proposed goodness-of-fit statistic with the statistic considered by Koenker and Machado (1999). An illustrative example based on real data sets is given to see the numerical performance of the proposed basic statistics.

The simplest and the most important distribution in survival analysis is exponential distribution. Koziol and Green (1976) derived Cramr-von Mises statistic's randomly censored version based on the Kaplan-Meier product limit estimate of the distribution function; however, it could not be practical for a real data set since the statistic is for testing a simple goodness of fit hypothesis. We generalized it to the composite hypothesis for exponentiality with an unknown scale parameter. We also considered the classical Kolmogorov-Smirnov statistic and generalized it by the exact same way. The two statistics are compared through a simulation study. As a result, we can see that the generalized Koziol-Green statistic has better power in most of the alternative distributions considered.

In this paper, the ratio of parameters in two independent Maxwell distributions is parameter of interest. We proposed test statistics, which converge to standard normal distribution, based on likelihood function. The exact distribution for testing the ratio is hard to obtain. We proposed the signed log-likelihood ratio statistic and the modified signed log-likelihood ratio statistic for testing the ratio. Through simulation, we show that the modified signed log-likelihood ratio statistic converges faster than signed log-likelihood ratio statistic to standard normal distribution. We compare two statistics in terms of type I error and power. We give an example using real data.

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.

The nonparametric location–scale statistic is introduced for a change-point problem. A combination of the Wilcoxon and Mood statistics is extended to the change-point setting. The limiting distribution of proposed statistic is derived for testing the null hypothesis of no change. The finite sample critical value of the suggested statistic is estimated by simulation studies. The accuracy of detecting a change point is investigated by simulation studies for various distributions. The method is illustrated by the analysis of real data.

For survival data we sometimes want to test a log normality hypothesis that can be changed into normality by transforming the survival data. Hence the Shapiro-Wilk type statistic for normality is generalized to randomly censored data based on the Kaplan-Meier product limit estimate of the distribution function. Koziol and Green (1976) derived Cramr-von Mises statistic's randomly censored version under the simpl hypothesis. These two test statistics are compared through a simulation study. As for the distribution of censoring variables, we consider Koziol and Green (1976)'s model and other similar models. Through the simulation results, we can see that the power of the proposed statistic is higher than that of Koziol-Green statistic and that the proportion of the censored observations (rather than the distribution of censoring variables) has a strong influence on the power of the proposed statistic.

In logistic regression models, we consider the deviance statistic (the log likelihood ratio statistic) D as a goodness-of-fit test statistic. In this paper, we show the derivation of an expression of asymptotic expansion for the distribution of D under a null hypothesis. Using the continuous term of the expression, we obtain a Bartlett-type transformed statistic D̃ that improves the speed of convergence to the chi-square limiting distribution of D. By numerical comparison, we find that the transformed statistic D̃ performs much better than D. We also give a real data example of D̃ being more reliable than D for testing a hypothesis.

We propose a new omnibus test statistic for normality based on the Jarque–Bera test statistic. We give the exact first four moments of the null distribution for the statistic using a computer algebra system. Our proposed statistic is an improvement of Jarque–Bera test statistic. Then the cumulants of the standardized statistic satisfy the Cornish–Fisher assumption. We give a normalizing transformation of the statistic based on the Wilson–Hilferty transformation.

This paper investigates the equality test of risk ratios in multiple 2x2 tables with structural zero, which is formed by a confounding factor such as age, gender, severity of disease, region or other variables of interest. Score statistic, Wald statistic and likelihood ratio statistic for testing equality of risk ratios in multiple 2x2 tables with structural zero are proposed. Three procedures for calculating p-values of the above three statistics based on the MaxT, MinP and Bonferroni methods are developed. An algorithm for determining sample size required for attaining the pre-specified power is presented. Empirical results show that (1) likelihood ratio statistic and score statistic via the MaxT and MinP methods behave satisfactorily in the sense that their type I error rates are close to the pre-given significance level; (2) the Wald-type statistic with logarithmic transformation behaves better than the Wald-type statistic. Simulation studies and a real example are used to illustrate the proposed methodologies.

Let T n be a Studentized U-statistic. It is proved that a Cramer type moderate deviation P (T n ≥ x )/(1 − Φ(x )) → 1 holds uniformly in x ∈ [0, o (n 1/6 )) when the kernel satisfies some regular conditions.