Exact Goodness‐of‐Fit Testing for the Ising Model

The Ising model is one of the simplest and most famous models of interacting systems. It was originally proposed to model ferromagnetic interactions in statistical physics and is now widely used to model spatial processes in many areas such as ecology, sociology, and genetics, usually without testing its goodness of fit. Here, we propose various test statistics and an exact goodness‐of‐fit test for the finite‐lattice Ising model. The theory of Markov bases has been developed in algebraic statistics for exact goodness‐of‐fit testing using a Monte Carlo approach. However, finding a Markov basis is often computationally intractable. Thus, we develop a Monte Carlo method for exact goodness‐of‐fit testing for the Ising model that avoids computing a Markov basis and also leads to a better connectivity of the Markov chain and hence to a faster convergence. We show how this method can be applied to analyze the spatial organization of receptors on the cell membrane.     

Goodness‐of‐fit testing based on a weighted bootstrap: A fast large‐sample alternative to the parametric bootstrap

The process comparing the empirical cumulative distribution function of the sample with a parametric estimate of the cumulative distribution function is known as the empirical process with estimated parameters and has been extensively employed in the literature for goodness‐of‐fit testing. The simplest way to carry out such goodness‐of‐fit tests, especially in a multivariate setting, is to use a parametric bootstrap. Although very easy to implement, the parametric bootstrap can become very computationally expensive as the sample size, the number of parameters, or the dimension of the data increase. An alternative resampling technique based on a fast weighted bootstrap is proposed in this paper, and is studied both theoretically and empirically. The outcome of this work is a generic and computationally efficient multiplier goodness‐of‐fit procedure that can be used as a large‐sample alternative to the parametric bootstrap. In order to approximately determine how large the sample size needs to be for the parametric and weighted bootstraps to have roughly equivalent powers, extensive Monte Carlo experiments are carried out in dimension one, two and three, and for models containing up to nine parameters. The computational gains resulting from the use of the proposed multiplier goodness‐of‐fit procedure are illustrated on trivariate financial data. A by‐product of this work is a fast large‐sample goodness‐of‐fit procedure for the bivariate and trivariate t distribution whose degrees of freedom are fixed. The Canadian Journal of Statistics 40: 480–500; 2012 © 2012 Statistical Society of Canada     

Multivariate Kendall's tau for change‐point detection in copulas

Statistical procedures for the detection of a change in the dependence structure of a series of multivariate observations are studied in this work. The test statistics that are proposed are $L_1$ , $L_2$ , and $L_{\infty }$ distances computed from vectors of differences of Kendall's tau; two multivariate extensions of Kendall's measure of association are used. Since the distributions of these statistics under the null hypothesis of no change depend on the unknown underlying copula of the vectors, a procedure based on the multiplier central limit theorem is used for the computation of p‐values; the method is shown to be valid both asymptotically and for moderate sample sizes. Alternative versions of the tests that take into account possible breakpoints in the marginal distributions are also investigated. Monte Carlo simulations show that the tests are powerful under many scenarios of change‐point. In addition, two estimators of the time of change are proposed and their efficiency is carefully studied. The methodologies are illustrated on simulated series from the Canadian Regional Climate Model. The Canadian Journal of Statistics 41: 65–82; 2013 © 2012 Statistical Society of Canada     

Second‐order bias‐corrected AIC in multivariate normal linear models under non‐normality

This paper deals with a bias correction of Akaike's information criterion (AIC) for selecting variables in multivariate normal linear regression models when the true distribution of observation is an unknown non‐normal distribution. It is well known that the bias of AIC is $O(1)$ , and there are a number of the first‐order bias‐corrected AICs which improve the bias to $O(n^{-1})$ , where $n$ is the sample size. A new information criterion is proposed by slightly adjusting the first‐order bias‐corrected AIC. Although the adjustment is achieved by merely using constant coefficients, the bias of the new criterion is reduced to $O(n^{-2})$ . Then, a variance of the new criterion is also improved. Through numerical experiments, we verify that our criterion is superior to others. The Canadian Journal of Statistics 39: 126–146; 2011 © 2011 Statistical Society of Canada     

A Case Study for Modelling Cancer Incidence Using Bayesian Spatio‐Temporal Models

Researchers familiar with spatial models are aware of the challenge of choosing the level of spatial aggregation. Few studies have been published on the investigation of temporal aggregation and its impact on inferences regarding disease outcome in space–time analyses. We perform a case study for modelling individual disease outcomes using several Bayesian hierarchical spatio‐temporal models, while taking into account the possible impact of spatial and temporal aggregation. Using longitudinal breast cancer data from South East Queensland, Australia, we consider both parametric and non‐parametric formulations for temporal effects at various levels of aggregation. Two temporal smoothness priors are considered separately; each is modelled with fixed effects for the covariates and an intrinsic conditional autoregressive prior for the spatial random effects. Our case study reveals that different model formulations produce considerably different model performances. For this particular dataset, a classical parametric formulation that assumes a linear time trend produces the best fit among the five models considered. Different aggregation levels of temporal random effects were found to have little impact on model goodness‐of‐fit and estimation of fixed effects.     

Asymptotic properties of likelihood ratio test statistics in affected‐sib‐pair analysis

In an affected‐sib‐pair genetic linkage analysis, identical by descent data for affected sib pairs are routinely collected at a large number of markers along chromosomes. Under very general genetic assumptions, the IBD distribution at each marker satisfies the possible triangle constraint. Statistical analysis of IBD data should thus utilize this information to improve efficiency. At the same time, this constraint renders the usual regularity conditions for likelihood‐based statistical methods unsatisfied. In this paper, the authors study the asymptotic properties of the likelihood ratio test (LRT) under the possible triangle constraint. They derive the limiting distribution of the LRT statistic based on data from a single locus. They investigate the precision of the asymptotic distribution and the power of the test by simulation. They also study the test based on the supremum of the LRT statistics over the markers distributed throughout a chromosome. Instead of deriving a limiting distribution for this test, they use a mixture of chi‐squared distributions to approximate its true distribution. Their simulation results show that this approach has desirable simplicity and satisfactory precision.     

Goodness‐of‐Fit Test for Monotone Functions

Abstract. In this article, we develop a test for the null hypothesis that a real‐valued function belongs to a given parametric set against the non‐parametric alternative that it is monotone, say decreasing. The method is described in a general model that covers the monotone density model, the monotone regression and the right‐censoring model with monotone hazard rate. The criterion for testing is an ‐distance between a Grenander‐type non‐parametric estimator and a parametric estimator computed under the null hypothesis. A normalized version of this distance is shown to have an asymptotic normal distribution under the null, whence a test can be developed. Moreover, a bootstrap procedure is shown to be consistent to calibrate the test.     

On a new goodness‐of‐fit process for families of copulas

A goodness‐of‐fit procedure is proposed for parametric families of copulas. The new test statistics are functionals of an empirical process based on the theoretical and sample versions of Spearman's dependence function. Conditions under which this empirical process converges weakly are seen to hold for many families including the Gaussian, Frank, and generalized Farlie–Gumbel–Morgenstern systems of distributions, as well as the models with singular components described by Durante [Durante ( 2007 ) Comptes Rendus Mathématique. Académie des Sciences. Paris, 344, 195–198]. Thanks to a parametric bootstrap method that allows to compute valid P‐values, it is shown empirically that tests based on Cramér–von Mises distances keep their size under the null hypothesis. Simulations attesting the power of the newly proposed tests, comparisons with competing procedures and complete analyses of real hydrological and financial data sets are presented. The Canadian Journal of Statistics 37: 80‐101; 2009 © 2009 Statistical Society of Canada     

Three‐level regular designs with general minimum lower‐order confounding

In this paper, we extend the general minimum lower‐order confounding (GMC) criterion to the case of three‐level designs. First, we review the relationship between GMC and other criteria. Then we introduce an aliased component‐number pattern (ACNP) and a three‐level GMC criterion via the consideration of component effects, and obtain some results on the new criterion. All the 27‐run GMC designs, 81‐run GMC designs with factor numbers $n=5,\ldots,20$ and 243‐run GMC designs with resolution $IV$ or higher are tabulated. The Canadian Journal of Statistics 41: 192–210; 2013 © 2012 Statistical Society of Canada     

Residual Analysis for Inhomogeneous Neyman–Scott Processes

Abstract. In this article, we introduce a residual analysis for inhomogeneous Neyman–Scott models based on Laplace functionals. Our simulation study shows that this residual analysis method has a good performance in assessing goodness‐of‐fit and revealing inadequacy of the fitted model. The method is employed in fitting a Thomas model to California redwood trees data and a Matérn model to the locations of hickory trees in Lansing woods, Michigan.     

Central Limit Theorems for the Non‐Parametric Estimation of Time‐Changed Lévy Models

Abstract. Let {Z_t}_{t 0} be a Lévy process with Lévy measure ν and let be a random clock, where g is a non‐negative function and is an ergodic diffusion independent of Z. Time‐changed Lévy models of the form are known to incorporate several important stylized features of asset prices, such as leptokurtic distributions and volatility clustering. In this article, we prove central limit theorems for a type of estimators of the integral parameter β(?):=∫?(x)ν(dx), valid when both the sampling frequency and the observation time‐horizon of the process get larger. Our results combine the long‐run ergodic properties of the diffusion process with the short‐term ergodic properties of the Lévy process Z via central limit theorems for martingale differences. The performance of the estimators are illustrated numerically for Normal Inverse Gaussian process Z and a Cox–Ingersoll–Ross process .     

A modified Hosmer–Lemeshow test for large data sets

The Hosmer–Lemeshow test is a widely used method for evaluating the goodness of fit of logistic regression models. But its power is much influenced by the sample size, like other chi-square tests. Paul, Pennell, and Lemeshow (2013 Paul, P., M. L. Pennell, and S. Lemeshow. 2013. Standardizing the power of the Hosmer–Lemeshow goodness of fit test in large data sets. Statistics in Medicine 32:67–80.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) considered using a large number of groups for large data sets to standardize the power. But simulations show that their method performs poorly for some models. In addition, it does not work when the sample size is larger than 25,000. In the present paper, we propose a modified Hosmer–Lemeshow test that is based on estimation and standardization of the distribution parameter of the Hosmer–Lemeshow statistic. We provide a mathematical derivation for obtaining the critical value and power of our test. Through simulations, we can see that our method satisfactorily standardizes the power of the Hosmer–Lemeshow test. It is especially recommendable for enough large data sets, as the power is rather stable. A bank marketing data set is also analyzed for comparison with existing methods.     

New estimation and feature selection methods in mixture‐of‐experts models

We study estimation and feature selection problems in mixture‐of‐experts models. An $l_2$ ‐penalized maximum likelihood estimator is proposed as an alternative to the ordinary maximum likelihood estimator. The estimator is particularly advantageous when fitting a mixture‐of‐experts model to data with many correlated features. It is shown that the proposed estimator is root‐$n$ consistent, and simulations show its superior finite sample behaviour compared to that of the maximum likelihood estimator. For feature selection, two extra penalty functions are applied to the $l_2$ ‐penalized log‐likelihood function. The proposed feature selection method is computationally much more efficient than the popular all‐subset selection methods. Theoretically it is shown that the method is consistent in feature selection, and simulations support our theoretical results. A real‐data example is presented to demonstrate the method. The Canadian Journal of Statistics 38: 519–539; 2010 © 2010 Statistical Society of Canada     

A note on goodness-of-fit test of continuation ratio logistic regression models under case–control data

We extend the discussion of Qin and Zhang's [1997. A goodness of fit test for logistic regression models base on case–control data. Biometrika 84, 609–618] goodness-of-fit test of logistic regression under case–control data to continuation ratio logistic regression (CRLR) models. We first showed that the retrospective CRLR model, which is valid for case–control data (the null hypothesis _H0)${\mathrm{H}}_0)$, is equivalent to an I  -sample semiparametric model. Then under _H0${\mathrm{H}}_0$, we find the semiparametric profile empirical likelihood estimators of distributions of the covariate conditioning on each response category and use them to define a Kolmogorov–Smirnov type test for assessing the global fit of CRLR models under case–control data. Unlike prospective CRLR models, retrospective CRLR models cannot be partitioned to a series of retrospective binary logistic regression models studied by Qin and Zhang [1997. A goodness of fit test for logistic regression models base on case–control data. Biometrika 84, 609–618].     

On the use of priors in goodness‐of‐fit tests

Priors are introduced into goodness‐of‐fit tests, both for unknown parameters in the tested distribution and on the alternative density. Neyman–Pearson theory leads to the test with the highest expected power. To make the test practical, we seek priors that make it likely a priori that the power will be larger than the level of the test but not too close to one. As a result, priors are sample size dependent. We explore this procedure in particular for priors that are defined via a Gaussian process approximation for the logarithm of the alternative density. In the case of testing for the uniform distribution, we show that the optimal test is of the U‐statistic type and establish limiting distributions for the optimal test statistic, both under the null hypothesis and averaged over the alternative hypotheses. The optimal test statistic is shown to be of the Cramér–von Mises type for specific choices of the Gaussian process involved. The methodology when parameters in the tested distribution are unknown is discussed and illustrated in the case of testing for the von Mises distribution. The Canadian Journal of Statistics 47: 560–579; 2019 © 2019 Statistical Society of Canada     

Testing parametric models in linear‐directional regression

This paper presents a goodness‐of‐fit test for parametric regression models with scalar response and directional predictor, that is, a vector on a sphere of arbitrary dimension. The testing procedure is based on the weighted squared distance between a smooth and a parametric regression estimator, where the smooth regression estimator is obtained by a projected local approach. Asymptotic behaviour of the test statistic under the null hypothesis and local alternatives is provided, jointly with a consistent bootstrap algorithm for application in practice. A simulation study illustrates the performance of the test in finite samples. The procedure is applied to test a linear model in text mining.     

Central Limit Theorems for Reduced U‐Statistics Under Dependence and Their Usefulness

A reduced ‐statistic is a ‐statistic with its summands drawn from a restricted but balanced set of pairs. In this article, central limit theorems are derived for reduced ‐statistics under ‐mixing, which significantly extends the work of Brown & Kildea in various aspects. It will be shown and illustrated that reduced ‐statistics are quite useful in deriving test statistics in various nonparametric testing problems.     

Analysis of band‐recovery data in a multistate capture‐recapture framework

Dead recoveries of marked animals are commonly used to estimate survival probabilities. Band‐recovery models can be parameterized either by r (the probability of recovering a band conditional on death of the animal) or by f (the probability that an animal will be killed, retrieved, and have its band reported). The T parametrization can be implemented in a capture‐recapture framework with two states (alive and newly dead), mortality being the transition probability between the two states. The authors show here that the f parametrization can also be implemented in a multistate framework by imposing simple constraints on some parameters. They illustrate it using data on the mallard and the snow goose. However, they mention that because it does not entirely separate the individual survival and encounter processes, the f parametrization must be used with care on reduced models, or in the presence of estimates at the boundary of the parameter space. As they show, a multistate framework allows the use of powerful software for model fitting or testing the goodness‐of‐fit of models; it also affords the implementation of complex models such as those based on mixture of information or uncertain states     

Generalised Smooth Tests of Goodness of Fit Utilising L‐moments

In this paper we present a semiparametric test of goodness of fit which is based on the method of L‐moments for the estimation of the nuisance parameters. This test is particularly useful for any distribution that has a convenient expression for its quantile function. The test proceeds by investigating equality of the first few L‐moments of the true and the hypothesised distributions. We provide details and undertake simulation studies for the logistic and the generalised Pareto distributions. Although for some distributions the method of L‐moments estimator is less efficient than the maximum likelihood estimator, the former method has the advantage that it may be used in semiparametric settings and that it requires weaker existence conditions. The new test is often more powerful than competitor tests for goodness of fit of the logistic and generalised Pareto distributions.