Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semi-algebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions. We show that the assumptions of Chernoff's theorem hold for semi-algebraic sets such that the asymptotics are determined by the tangent cone at the true parameter point. At boundary points or singularities, the tangent cone need not be a linear space and limiting distributions other than chi-square distributions may arise. While boundary points often lead to mixtures of chi-square distributions, singularities give rise to nonstandard limits. We demonstrate that minima of chi-square random variables are important for locally identifiable models, and in a study of the factor analysis model with one factor, we reveal connections to eigenvalues of Wishart matrices.
The likelihood ratio statistic, with its asymptotic $χ^2$ distribution at regular model points, is often used for hypothesis testing. At model singularities and boundaries, however, the asymptotic distribution may not be $χ^2$, as highlighted by recent work of Drton. Indeed, poor behavior of a $χ^2$ for testing near singularities and boundaries is apparent in simulations, and can lead to conservative or anti-conservative tests. Here we develop a new distribution designed for use in hypothesis testing near singularities and boundaries, which asymptotically agrees with that of the likelihood rat…
We consider a likelihood ratio method for testing whether a monotone baseline hazard function in the Cox model has a particular value at a fixed point. The characterization of the estimators involved is provided both in the nondecreasing and the nonincreasing setting. These characterizations facilitate the derivation of the asymptotic distribution of the likelihood ratio test, which is identical in the nondecreasing and in the nonincreasing case. The asymptotic distribution of the likelihood ratio test enables, via inversion, the construction of pointwise confidence intervals. Simulations show…
Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In this paper we consider two such likelihood ratios. The first one is an exponential functional of a two-sided Poisson process driven by some parameter, while the second one is an exponential functional of a two-sided Brownian motion. We establish that for sufficiently small values of the parameter, the Poisson type likelihood ratio can be approximated by the Brownian type one. As a consequence, several statistically interesting quantitie…
This paper considers the asymptotic power of likelihood ratio test (LRT) for the identity test when the dimension p is large compared to the sample size n. The asymptotic distribution of LRT under alternatives is given and an explicit expression of the power is derived. A simulation study is carried out to compare LRT with other tests. All these studies show that LRT is a powerful test to detect eigenvalues around zero. Key words and phrases: Covariance matrix, High dimensional data, Identity test, Likelihood ratio test, Power
In this paper, we develop modified versions of the likelihood ratio test for multivariate heteroskedastic errors-in-variables regression models. The error terms are allowed to follow a multivariate distribution in the elliptical class of distributions, which has the normal distribution as a special case. We derive the Skovgaard adjusted likelihood ratio statistics, which follow a chi-squared distribution with a high degree of accuracy. We conduct a simulation study and show that the proposed tests display superior finite sample behavior as compared to the standard likelihood ratio test. We ill…
Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In a previous paper of one of the authors it was established that one of these likelihood ratios, which is an exponential functional of a two-sided Poisson process driven by some parameter, can be approximated (for sufficiently small values of the parameter) by another one, which is an exponential functional of a two-sided Brownian motion. In this paper we consider yet another likelihood ratio, which is the exponent of a two-sided compound…
The problem of testing the goodness of fit of an absolutely continuous distribution to a set of observations grouped into equal probability intervals, against to a family of sequences of alternatives approaching the hypothesis is considered. We focus our study to the asymptotic intermediate efficiency (AIE) due to Inglot (1999) of the chi-squared and log-likelihood ratio tests. It is assumed that the number of groups increases together with the number of observations.
Comparison of two univariate distributions based on independent samples from them is a fundamental problem in statistics, with applications in a wide variety of scientific disciplines. In many situations, we might hypothesize that the two distributions are stochastically ordered, meaning intuitively that samples from one distribution tend to be larger than those from the other. One type of stochastic order that arises in economics, biomedicine, and elsewhere is the likelihood ratio order, also known as the density ratio order, in which the ratio of the density functions of the two distribution…
This article focuses on the study of toric algebraic statistical models which correspond to toric Del Pezzo surfaces with Du Val singularities. A closed-form for the Maximum Likelihood Estimate of algebraic statistical models which correspond to cubic and quartic toric Del Pezzo surfaces with Du Val singular points is given. Also, we calculate the ML degrees of some toric Del Pezzo surfaces of degree less than or equal to six, which equals the degree of the surface in all the case but one, namely the quintic with two points of type $\mathbb{A}_1$.
In finite mixtures of location-scale distributions, if there is no constraint or penalty on the parameters, then the maximum likelihood estimator does not exist because the likelihood is unbounded. To avoid this problem, we consider a penalized likelihood, where the penalty is a function of the minimum of the ratios of the scale parameters and the sample size. It is shown that the penalized maximum likelihood estimator is strongly consistent. We also analyze the consistency of a penalized maximum likelihood estimator where the penalty is imposed on the scale parameters themselves.
Although the log-likelihood is widely used in model selection, the log-likelihood ratio has had few applications in this area. We develop a log-likelihood ratio based method for selecting regression models by focusing on the set of models deemed plausible by the likelihood ratio test. We show that when the sample size is large and the significance level of the test is small, there is a high probability that the smallest model in the set is the true model; thus, we select this smallest model. The significance level of the test serves as a parameter for this method. We consider three levels of t…
We compute the loss of power in likelihood ratio tests when we test the original parameter of a probability density extended by the first Lehmann alternative.
The present manuscript studies signal detection by likelihood ratio tests in a number of spiked random matrix models, including but not limited to Gaussian mixtures and spiked Wishart covariance matrices. We work directly with multi-spiked cases in these models and with flexible priors on the signal component that allow dependence across spikes. We derive asymptotic normality for the log-likelihood ratios when the signal-to- noise ratios are below certain thresholds. In addition, we show that the variances of the log-likelihood ratios can be asymptotically decomposed as the sums of those of a …
A profile likelihood ratio test is proposed for inferences on the index coefficients in generalized single-index models. Key features include its simplicity in implementation, invariance against parametrization, and exhibiting substantially less bias than standard Wald-tests in finite-sample settings. Moreover, the R routine to carry out the profile likelihood ratio test is demonstrated to be over two orders of magnitude faster than the recently proposed generalized likelihood ratio test based on kernel regression. The advantages of the method are demonstrated on various simulations and a data…
This work is concern with testing the low-dimensional parameters of interest with divergent dimensional data and variable selection for the rest under the sparse case. A consistent test via the partial penalized likelihood approach, called the partial penalized likelihood ratio test statistic is derived, and its asymptotic distributions under the null hypothesis and the local alternatives of order $n^{-1/2}$ are obtained under some regularity conditions. Meanwhile, the oracle property of the partial penalized likelihood estimator also holds. The proposed partial penalized likelihood ratio test…
Mixtures of factor analyzers are becoming more and more popular in the area of model based clustering of high-dimensional data. According to the likelihood approach in data modeling, it is well known that the unconstrained log-likelihood function may present spurious maxima and singularities and this is due to specific patterns of the estimated covariance structure, when their determinant approaches 0. To reduce such drawbacks, in this paper we introduce a procedure for the parameter estimation of mixtures of factor analyzers, which maximizes the likelihood function in a constrained parameter …
In many fields of science, generalized likelihood ratio tests are established tools for statistical inference. At the same time, it has become increasingly common that a simulator (or generative model) is used to describe complex processes that tie parameters $θ$ of an underlying theory and measurement apparatus to high-dimensional observations $\mathbf{x}\in \mathbb{R}^p$. However, simulator often do not provide a way to evaluate the likelihood function for a given observation $\mathbf{x}$, which motivates a new class of likelihood-free inference algorithms. In this paper, we show that likeli…
Centre Borelli, ENS Paris-Saclay, Université Paris-Saclay（巴黎萨克雷大学巴黎萨克雷高等师范学院博雷利中心）
Consider each node of a graph to be generating a data stream that is synchronized and observed at near real-time. At a change-point $τ$, a change occurs at a subset of nodes $C$, which affects the probability distribution of their associated node streams. In this paper, we propose a novel kernel-based method to both detect $τ$ and localize $C$, based on the direct estimation of the likelihood-ratio between the post-change and the pre-change distributions of the node streams. Our main working hypothesis is the smoothness of the likelihood-ratio estimates over the graph, i.e connected nodes are …
This paper provides a generalization of a classical result obtained by Wilks about the asymptotic behavior of the likelihood ratio. The new results deal with the asymptotic behavior of the joint distribution of a vector of likelihood ratios which turn out to be stochastically dependent.
This paper studies the least upper bounds on coverage probabilities of the empirical likelihood ratio confidence regions based on estimating equations. The implications of the bounds on empirical likelihood inference are also discussed.
We explore the applications of our previously established likelihood-ratio method for deriving concentration inequalities for a wide variety of univariate and multivariate distributions. New concentration inequalities for various distributions are developed without the idea of minimizing moment generating functions.
A new large deviation results for the Pearson chi-square and Log-likelihood ratio statistics are obtained. Here attention is focused on the case when the number of groups increases to infinity and the probabilities of groups decreases to zero, as the sample size tends to infinity.
Louisiana State University（路易斯安那州立大学）；Southern University and A&M College（南方大学及农工学院）
We propose a new approach for deriving probabilistic inequalities based on bounding likelihood ratios. We demonstrate that this approach is more general and powerful than the classical method frequently used for deriving concentration inequalities such as Chernoff bounds. We discover that the proposed approach is inherently related to statistical concepts such as monotone likelihood ratio, maximum likelihood, and the method of moments for parameter estimation. A connection between the proposed approach and the large deviation theory is also established. We show that, without using moment gener…
