The universal validity of the possible triangle constraint for affected sib pairs

The authors consider affected‐sib‐pair analysis, in which genetic marker data are collected from families with at least two sibs affected by a disease under investigation. At any locus not linked to the disease gene, a sib pair shares 0, 1 and 2 alleles identical by descent (IBD) with probabilities of 1/4, 1/2 and 1/4, respectively. With linkage, the IBD value increases stochastically. Louis, Payami & Thomson (1987) and Holmans (1993) were the first ones who discovered that the IBD distribution satisfies the “possible triangle constraint” in some situations. Consequently, more powerful statistical procedures can be designed in detecting linkage. It is of statistical and genetical importance to investigate whether the possible triangle constraint remains true under general genetic models. In this paper, the authors introduce a new technique to prove the possible triangle constraint. Their proof is particularly simple for the single disease locus case. The general case is proved by linking IBD distributions between marker loci through a transition probability matrix.     

A modified likelihood ratio test for homogeneity in finite mixture models

Testing for homogeneity in finite mixture models has been investigated by many researchers. The asymptotic null distribution of the likelihood ratio test (LRT) is very complex and difficult to use in practice. We propose a modified LRT for homogeneity in finite mixture models with a general parametric kernel distribution family. The modified LRT has a χ-type of null limiting distribution and is asymptotically most powerful under local alternatives. Simulations show that it performs better than competing tests. They also reveal that the limiting distribution with some adjustment can satisfactorily approximate the quantiles of the test statistic, even for moderate sample sizes.     

Hypothesis testing for quantitative trait locus effects in both location and scale in genetic backcross studies

Testing the existence of a quantitative trait locus (QTL) effect is an important task in QTL mapping studies. Most studies concentrate on the case where the phenotype distributions of different QTL groups follow normal distributions with the same unknown variance. In this paper we make a more general assumption that the phenotype distributions come from a location-scale distribution family. We derive the limiting distribution of the likelihood ratio test (LRT) for the existence of the QTL effect in both location and scale in genetic backcross studies. We further identify an explicit representation for this limiting distribution. As a complement, we study the limiting distribution of the LRT and its explicit representation for the existence of the QTL effect in the location only. The asymptotic properties of the LRTs under a local alternative are also investigated. Simulation studies are used to evaluate the asymptotic results, and a real-data example is included for illustration.     

Inference of Seasonal Long‐memory Time Series with Measurement Error

We consider the Whittle likelihood estimation of seasonal autoregressive fractionally integrated moving‐average models in the presence of an additional measurement error and show that the spectral maximum Whittle likelihood estimator is asymptotically normal. We illustrate by simulation that ignoring measurement errors may result in incorrect inference. Hence, it is pertinent to test for the presence of measurement errors, which we do by developing a likelihood ratio (LR) test within the framework of Whittle likelihood. We derive the non‐standard asymptotic null distribution of this LR test and the limiting distribution of LR test under a sequence of local alternatives. Because in practice, we do not know the order of the seasonal autoregressive fractionally integrated moving‐average model, we consider three modifications of the LR test that takes model uncertainty into account. We study the finite sample properties of the size and the power of the LR test and its modifications. The efficacy of the proposed approach is illustrated by a real‐life example.     

A likelihood ratio test for goodness‐of‐fit of recessive and dominant models for case–control studies

Testing goodness‐of‐fit of commonly used genetic models is of critical importance in many applications including association studies and testing for departure from Hardy–Weinberg equilibrium. Case–control design has become widely used in population genetics and genetic epidemiology, thus it is of interest to develop powerful goodness‐of‐fit tests for genetic models using case–control data. This paper develops a likelihood ratio test (LRT) for testing recessive and dominant models for case–control studies. The LRT statistic has a closed‐form formula with a simple $\chi^{2}(1)$ null asymptotic distribution, thus its implementation is easy even for genome‐wide association studies. Moreover, it has the same power and optimality as when the disease prevalence is known in the population. The Canadian Journal of Statistics 41: 341–352; 2013 © 2013 Statistical Society of Canada     

Non‐Parametric Change‐Point Tests for Long‐Range Dependent Data

Abstract. We propose a non‐parametric change‐point test for long‐range dependent data, which is based on the Wilcoxon two‐sample test. We derive the asymptotic distribution of the test statistic under the null hypothesis that no change occurred. In a simulation study, we compare the power of our test with the power of a test which is based on differences of means. The results of the simulation study show that in the case of Gaussian data, our test has only slightly smaller power minus.3pt than the ‘difference‐of‐means’ test. For heavy‐tailed data, our test outperforms the ‘difference‐of‐means’ test.     

Testing equality of mean vectors in a one-way MANOVA with monotone missing data

In this study, testing the equality of mean vectors in a one-way multivariate analysis of variance (MANOVA) is considered when each dataset has a monotone pattern of missing observations. The likelihood ratio test (LRT) statistic in a one-way MANOVA with monotone missing data is given. Furthermore, the modified test (MT) statistic based on likelihood ratio (LR) and the modified LRT (MLRT) statistic with monotone missing data are proposed using the decomposition of the LR and an asymptotic expansion for each decomposed LR. The accuracy of the approximation for the Chi-square distribution is investigated using a Monte Carlo simulation. Finally, an example is given to illustrate the methods.     

A class of partially linear single‐index survival models

The authors define a class of “partially linear single‐index” survival models that are more flexible than the classical proportional hazards regression models in their treatment of covariates. The latter enter the proposed model either via a parametric linear form or a nonparametric single‐index form. It is then possible to model both linear and functional effects of covariates on the logarithm of the hazard function and if necessary, to reduce the dimensionality of multiple covariates via the single‐index component. The partially linear hazards model and the single‐index hazards model are special cases of the proposed model. The authors develop a likelihood‐based inference to estimate the model components via an iterative algorithm. They establish an asymptotic distribution theory for the proposed estimators, examine their finite‐sample behaviour through simulation, and use a set of real data to illustrate their approach.     

Empirical likelihood for the varying‐coefficient single‐index model

In this article the author investigates the application of the empirical‐likelihood‐based inference for the parameters of varying‐coefficient single‐index model (VCSIM). Unlike the usual cases, if there is no bias correction the asymptotic distribution of the empirical likelihood ratio cannot achieve the standard chi‐squared distribution. To this end, a bias‐corrected empirical likelihood method is employed to construct the confidence regions (intervals) of regression parameters, which have two advantages, compared with those based on normal approximation, that is, (1) they do not impose prior constraints on the shape of the regions; (2) they do not require the construction of a pivotal quantity and the regions are range preserving and transformation respecting. A simulation study is undertaken to compare the empirical likelihood with the normal approximation in terms of coverage accuracies and average areas/lengths of confidence regions/intervals. A real data example is given to illustrate the proposed approach. The Canadian Journal of Statistics 38: 434–452; 2010 © 2010 Statistical Society of Canada     

A modified likelihood ratio test for the mean direction in the von mises distribution

The likelihood ratio test (LRT) for the mean direction in the von Mises distribution is modified for possessing a common asymptotic distribution both for large sample size and for large concentration parameter. The test statistic of the modified LRT is compared with the F distribution but not with the chi-square distribution usually employed, Good performances of the modified LRT are shown by analytical studies and Monte Carlo simulation studies, A notable advantage of the test is that it takes part in the unified likelihood inference procedures including both the marginal MLE and the marginal LRT for the concentration parameter.     

Box‐Cox transformations in linear models: Large sample theory and tests of normality

The authors provide a rigorous large sample theory for linear models whose response variable has been subjected to the Box‐Cox transformation. They provide a continuous asymptotic approximation to the distribution of estimators of natural parameters of the model. They show, in particular, that the maximum likelihood estimator of the ratio of slope to residual standard deviation is consistent and relatively stable. The authors further show the importance for inference of normality of the errors and give tests for normality based on the estimated residuals. For non‐normal errors, they give adjustments to the log‐likelihood and to asymptotic standard errors.     

The equivalence between likelihood ratio test and F-test for testing variance component in a balanced one-way random effects model

In mixed linear models, it is frequently of interest to test hypotheses on the variance components. F-test and likelihood ratio test (LRT) are commonly used for such purposes. Current LRTs available in literature are based on limiting distribution theory. With the development of finite sample distribution theory, it becomes possible to derive the exact test for likelihood ratio statistic. In this paper, we consider the problem of testing null hypotheses on the variance component in a one-way balanced random effects model. We use the exact test for the likelihood ratio statistic and compare the performance of F-test and LRT. Simulations provide strong support of the equivalence between these two tests. Furthermore, we prove the equivalence between these two tests mathematically.     

Empirical likelihood inference for the regression model of mean quality‐adjusted lifetime with censored data

The authors consider the empirical likelihood method for the regression model of mean quality‐adjusted lifetime with right censoring. They show that an empirical log‐likelihood ratio for the vector of the regression parameters is asymptotically a weighted sum of independent chi‐squared random variables. They adjust this empirical log‐likelihood ratio so that the limiting distribution is a standard chi‐square and construct corresponding confidence regions. Simulation studies lead them to conclude that empirical likelihood methods outperform the normal approximation methods in terms of coverage probability. They illustrate their methods with a data example from a breast cancer clinical trial study.     

The likelihood ratio test for homogeneity in finite mixture models

The authors study the asymptotic behaviour of the likelihood ratio statistic for testing homogeneity in the finite mixture models of a general parametric distribution family. They prove that the limiting distribution of this statistic is the squared supremum of a truncated standard Gaussian process. The autocorrelation function of the Gaussian process is explicitly presented. A re‐sampling procedure is recommended to obtain the asymptotic p‐value. Three kernel functions, normal, binomial and Poisson, are used in a simulation study which illustrates the procedure.     

Bayesian analysis of mark‐recapture data with travel time‐dependent survival probabilities

The authors extend the classical Cormack‐Jolly‐Seber mark‐recapture model to account for both temporal and spatial movement through a series of markers (e.g., dams). Survival rates are modeled as a function of (possibly) unobserved travel times. Because of the complex nature of the likelihood, they use a Bayesian approach based on the complete data likelihood, and integrate the posterior through Markov chain Monte Carlo methods. They test the model through simulations and apply it also to actual salmon data arising from the Columbia river system. The methodology was developed for use by the Pacific Ocean Shelf Tracking (POST) project.     

An improved test of equality of mean directions for the Langevin‐von Mises‐Fisher distribution

A multi‐sample test for equality of mean directions is developed for populations having Langevin‐von Mises‐Fisher distributions with a common unknown concentration. The proposed test statistic is a monotone transformation of the likelihood ratio. The high‐concentration asymptotic null distribution of the test statistic is derived. In contrast to previously suggested high‐concentration tests, the high‐concentration asymptotic approximation to the null distribution of the proposed test statistic is also valid for large sample sizes with any fixed nonzero concentration parameter. Simulations of size and power show that the proposed test outperforms competing tests. An example with three‐dimensional data from an anthropological study illustrates the practical application of the testing procedure.     

Likelihood inference for small variance components

The authors explore likelihood‐based methods for making inferences about the components of variance in a general normal mixed linear model. In particular, they use local asymptotic approximations to construct confidence intervals for the components of variance when the components are close to the boundary of the parameter space. In the process, they explore the question of how to profile the restricted likelihood (REML). Also, they show that general REML estimates are less likely to fall on the boundary of the parameter space than maximum‐likelihood estimates and that the likelihood‐ratio test based on the local asymptotic approximation has higher power than the likelihood‐ratio test based on the usual chi‐squared approximation. They examine the finite‐sample properties of the proposed intervals by means of a simulation study.     

Asymptotic test of mixture model and its applications to QTL interval mapping

Quantitative trait loci (QTL) mapping has been a standard means in identifying genetic regions harboring potential genes underlying complex traits. Likelihood ratio test (LRT) has been commonly applied to assess the significance of a genetic locus in a mixture model content. Given the time constraint in commonly used permutation tests to assess the significance of LRT in QTL mapping, we study the behavior of the LRT statistic in mixture model when the proportions of the distributions are unknown. We found that the asymptotic null distribution is stationary Gaussian process after suitable transformation. The result can be applied to one-parameter exponential family mixture model. Under certain condition, such as in a backcross mapping model, the tail probability of the supremum of the process is calculated and the threshold values can be determined by solving the distribution function. Simulation studies were performed to evaluate the asymptotic results.     

Location‐invariant Multi‐sample U‐tests for Covariance Matrices with Large Dimension

For two or more multivariate distributions with common covariance matrix, test statistics for certain special structures of the common covariance matrix are presented when the dimension of the multivariate vectors may exceed the number of such vectors. The test statistics are constructed as functions of location‐invariant estimators defined as U‐statistics, and the corresponding asymptotic theory is used to derive the limiting distributions of the proposed tests. The properties of the test statistics are established under mild and practical assumptions, and the same are numerically demonstrated using simulation results with small or moderate sample sizes and large dimensions.     

One‐Way anova for Functional Data via Globalizing the Pointwise F‐test

In this paper, we propose and study a new global test, namely, GPF test, for the one‐way anova problem for functional data, obtained via globalizing the usual pointwise F‐test. The asymptotic random expressions of the test statistic are derived, and its asymptotic power is investigated. The GPF test is shown to be root‐n consistent. It is much less computationally intensive than a parametric bootstrap test proposed in the literature for the one‐way anova for functional data. Via some simulation studies, it is found that in terms of size‐controlling and power, the GPF test is comparable with two existing tests adopted for the one‐way anova problem for functional data. A real data example illustrates the GPF test.