Generalized t-tests are constructed under weaker than normal conditions. In the first part of this paper we assume only the symmetry (around zero) of the error distribution (i). In the second part we assume that the error distribution is a Gaussian scale mixture (ii). The optimal (smallest) critical values can be computed from generalizations of Student's cumulative distribution function (cdf), $t_n(x)$. The cdf's of the generalized $t$-test statistics are denoted by (i) $t_n^S(x)$ and (ii) $t_n^G(x)$, resp. As the sample size $n\to \infty$ we get the counterparts of the standard normal cdf $Φ(x)$: (i) $Φ^S(x):=\operatorname {lim}_{n\to \infty}t_n^S(x)$, and (ii) $Φ^G(x):=\operatorname {lim}_{n\to \infty}t_n^G(x)$. Explicit formulae are given for the underlying new cdf's. For example $Φ^G(x)=Φ(x)$ iff $|x|\ge \sqrt{3}$. Thus the classical 95% confidence interval for the unknown expected value of Gaussian distributions covers the center of symmetry with at least 95% probability for Gaussian scale mixture distributions. On the other hand, the 90% quantile of $Φ^G$ is $4\sqrt{3}/5=1.385... >Φ^{-1}(0.9)=1.282...$.
In the analysis of microarray data, and in some other contemporary statistical problems, it is not uncommon to apply hypothesis tests in a highly simultaneous way. The number, $ν$ say, of tests used can be much larger than the sample sizes, $n$, to which the tests are applied, yet we wish to calibrate the tests so that the overall level of the simultaneous test is accurate. Often the sampling distribution is quite different for each test, so there may not be an opportunity for combining data across samples. In this setting, how large can $ν$ be, as a function of $n$, before level accuracy beco…
We propose a new testing procedure about the tail weight parameter of multivariate Student $t$ distributions by having recourse to the Le Cam methodology. Our test is asymptotically as efficient as the classical likelihood ratio test, but outperforms the latter by its flexibility and simplicity: indeed, our approach allows to estimate the location and scatter nuisance parameters by any root-$n$ consistent estimators, hereby avoiding numerically complex maximum likelihood estimation. The finite-sample properties of our test are analyzed in a Monte Carlo simulation study, and we apply our method…
We propose the finite mixture of skewed sub-Gaussian stable distributions. The maximum likelihood estimator for the parameters of proposed finite mixture model is computed through the expectation-maximization algorithm. The proposed model contains the finite mixture of normal and skewed normal distributions. Since the tails of proposed model is heavier than even the Student's t distribution, it can be used as a powerful model for robust model-based clustering. Performance of the proposed model is demonstrated by clustering simulation data and two sets of real data.
The testing problem for the order of finite mixture models has a long history and remains an active research topic. Since Ghosh and Sen (1985) revealed the hard-to-manage asymptotic properties of the likelihood ratio test, there has been marked progress. The most successful attempts include the modified likelihood ratio test and the EM-test, which lead to neat solutions for finite mixtures of univariate normal distributions, finite mixtures of single-parameter distributions, and several mixture-like models. The problem remains challenging, and there is still no generic solution for location-sc…
Fitting mixture distributions is needed in applications where data belongs to inhomogeneous populations comprising homogeneous sub-populations. The mixing proportions of the sub populations are in general unknown and need to be estimated as well. A goodness of fit test based on the empirical distribution function is proposed for assessing the goodness of fit in model fits comprising two components, each distributed as two parameter Weibull. The applicability of the proposed test procedure was empirically established using a Monte Carlo simulation study. The proposed test procedure can be easil…
A new mixture autoregressive model based on Student's $t$-distribution is proposed. A key feature of our model is that the conditional $t$-distributions of the component models are based on autoregressions that have multivariate $t$-distributions as their (low-dimensional) stationary distributions. That autoregressions with such stationary distributions exist is not immediate. Our formulation implies that the conditional mean of each component model is a linear function of past observations and the conditional variance is also time varying. Compared to previous mixture autoregressive models ou…
Gaussian mixture models with eigen-decomposed covariance structures make up the most popular family of mixture models for clustering and classification, i.e., the Gaussian parsimonious clustering models (GPCM). Although the GPCM family has been used for almost 20 years, selecting the best member of the family in a given situation remains a troublesome problem. Likelihood ratio tests are developed to tackle this problems. These likelihood ratio tests use the heteroscedastic model under the alternative hypothesis but provide much more flexibility and real-world applicability than previous approa…
We present general results on the univariate tail conditional expectation (TCE) and multivariate tail conditional expectation for location-scale mixture of elliptical distributions. Examples include the location-scale mixture of normal distributions, location-scale mixture of Student-$t$ distributions, location-scale mixture of Logistic distributions and location-scale mixture of Laplace distributions. We also consider portfolio risk decomposition with TCE for location-scale mixture of elliptical distributions.
An explicit representation of an arbitrary zero-mean distribution as the mixture of (at-most-)two-point zero-mean distributions is given. Based in this representation, tests for (i) asymmetry patterns and (ii) for location without symmetry conditions can be constructed. Exact inequalities implying conservative properties of such tests are presented. These developments extend results established earlier by Efron, Eaton, and Pinelis under a symmetry condition.
Normal variance-mean mixtures encompass a large family of useful distributions such as the generalized hyperbolic distribution, which itself includes the Student t, Laplace, hyperbolic, normal inverse Gaussian, and variance gamma distributions as special cases. We study shape properties of normal variance-mean mixtures, in both the univariate and multivariate cases, and determine conditions for unimodality and log-concavity of the density functions. This leads to a short proof of the unimodality of all generalized hyperbolic densities. We also interpret such results in practical terms and disc…
We consider the problem of detecting sparse heterogeneous mixtures from a nonparametric perspective, and develop distribution-free tests when all effects have the same sign. Specifically, we assume that the null distribution is symmetric about zero, while the true effects have positive median. We evaluate the precise performance of classical tests for the median (t-test, sign test) and classical tests for symmetry (signed-rank, Smirnov, total number of runs, longest run tests) showing that none of them is asymptotically optimal for the normal mixture model in all sparsity regimes. We then sugg…
This paper deals with statistical tests on the components of mixture densities. We propose to test whether the densities of two independent samples of independent random variables $Y_1, ..., Y_n$ and $Z_1, ..., Z_n$ result from the same mixture of $M$ components or not. We provide a test procedure which is proved to be asymptotically optimal according to the minimax setting. We extensively discuss the connection between the mixing weights and the performance of the testing procedure and illustrate it with numerical examples.
Testing hypotheses of goodness-of-fit about mixture distributions on the basis of independent but not necessarily identically distributed random vectors is considered. The hypotheses are given by a specific distribution or by a family of distributions. Moreover, testing hypotheses formulated by Hadamard differentiable functionals is discussed in this situation, in particular the hypothesis of central symmetry, homogeneity and independence. Kolmogorov-Smirnov or Cramér-von-Mises type statistics are suggested as well as methods to determine critical values. The focus of the investigation is on a…
Finite mixtures of multivariate normal distributions have been widely used in empirical applications in diverse fields such as statistical genetics and statistical finance. Testing the number of components in multivariate normal mixture models is a long-standing challenge even in the most important case of testing homogeneity. This paper develops likelihood-based tests of the null hypothesis of $M_0$ components against the alternative hypothesis of $M_0 + 1$ components for a general $M_0 \geq 1$. For heteroscedastic normal mixtures, we propose an EM test and derive the asymptotic distribution …
We focus on one-sided, mixture-based stopping rules for the problem of sequential testing a simple null hypothesis against a composite alternative. For the latter, we consider two cases---either a discrete alternative or a continuous alternative that can be embedded into an exponential family. For each case, we find a mixture-based stopping rule that is nearly minimax in the sense of minimizing the maximal Kullback-Leibler information. The proof of this result is based on finding an almost Bayes rule for an appropriate sequential decision problem and on high-order asymptotic approximations for…
Max-mixture processes are defined as Z = max(aX, (1 -- a)Y) with X an asymptotic dependent (AD) process, Y an asymptotic independent (AI) process and a $\in$ [0, 1]. So that, the mixing coefficient a may reveal the strength of the AD part present in the max-mixture process. In this paper we focus on two tests based on censored pairwise likelihood estimates. We compare their performance through an extensive simulation study. Monte Carlo simulation plays a fundamental tool for asymptotic variance calculations. We apply our tests to daily precipitations from the East of Australia. Drawbacks and p…
The test of homogeneity for normal mixtures has been conducted in diverse research areas, but constructing a theory of the test of homogeneity is challenging because the parameter set for the null hypothesis corresponds to singular points in the parameter space. In this paper, we examine this problem from a new perspective and offer a theory of hypothesis testing for homogeneity based on a variational Bayes framework. In the conventional theory, the constant order term of the free energy has remained unknown, however, we clarify its asymptotic behavior because it is necessary for constructing …
Testing for regime switching when the regime switching probabilities are specified either as constants (`mixture models') or are governed by a finite-state Markov chain (`Markov switching models') are long-standing problems that have also attracted recent interest. This paper considers testing for regime switching when the regime switching probabilities are time-varying and depend on observed data (`observation-dependent regime switching'). Specifically, we consider the likelihood ratio test for observation-dependent regime switching in mixture autoregressive models. The testing problem is hig…
