Adaptive likelihood estimator of conditional variance function

Modelling volatility in the form of conditional variance function has been a popular method mainly due to its application in financial risk management. Among others, we distinguish the parametric GARCH models and the nonparametric local polynomial approximation using weighted least squares or gaussian likelihood function. We introduce an alternative likelihood estimate of conditional variance and we show that substitution of the error density with its estimate yields similar asymptotic properties, that is, the proposed estimate is adaptive to the error distribution. Theoretical comparison with existing estimates reveals substantial gains in efficiency, especially if error distribution has fatter tails than Gaussian distribution. Simulated data confirm the theoretical findings while an empirical example demonstrates the gains of the proposed estimate.     

Tree-structured generalized autoregressive conditional heteroscedastic models

We propose a new generalized autoregressive conditional heteroscedastic (GARCH) model with tree-structured multiple thresholds for the estimation of volatility in financial time series. The approach relies on the idea of a binary tree where every terminal node parameterizes a (local) GARCH model for a partition cell of the predictor space. The fitting of such trees is constructed within the likelihood framework for non-Gaussian observations: it is very different from the well-known regression tree procedure which is based on residual sums of squares. Our strategy includes the classical GARCH model as a special case and allows us to increase model complexity in a systematic and flexible way. We derive a consistency result and conclude from simulation and real data analysis that the new method has better predictive potential than other approaches.     

The conditional level of confidence intervals for the normal variance and applications

Suppose we observe two independent random vectors each having a multivariate normal distribution with covariance matrix known up to an unknown scale factor σ . The first random vector has a known mean vector while the second has an unknown mean vector. Interest centers around finding confidence intervals for σ² with confidence coefficient 1 ? α. Standard results show that, when we only observe the first random vector, an optimal (i.e., smallest length) confidence interval C, based on the well-known chi- squared statistic, can be constructed for σ² . When we additionally observe the second random vector, the confidence interval C is no longer optimal for estimating σ². One criterion useful for detecting the non-optimality of a confidence interval C concerns whether C admits positively or negatively biased relevant subsets. This criterion has recently received a good deal of attention. It is shown here that under some conditions the confidence interval C admits positively biased relevant subsets.

Applications of this result to the construction of ‘better‘ unconditional confidence intervals for σ² are presented. Some simulation results are given to indicate the typical extent of improvement attained.     

Jackknife empirical likelihood for the error variance in linear models

Variance estimation is a fundamental yet important problem in statistical modelling. In this paper, we propose jackknife empirical likelihood (JEL) methods for the error variance in a linear regression model. We prove that the JEL ratio converges to the standard chi-squared distribution. The asymptotic chi-squared properties for the adjusted JEL and extended JEL estimators are also established. Extensive simulation studies to compare the new JEL methods with the standard method in terms of coverage probability and interval length are conducted, and the simulation results show that our proposed JEL methods perform better than the standard method. We also illustrate the proposed methods using two real data sets.     

Inferences of variance function - a parametric robust way

Tsou (2003a) proposed a parametric procedure for making robust inference for mean regression parameters in the context of generalized linear models. This robust procedure is extended to model variance heterogeneity. The normal working model is adjusted to become asymptotically robust for inference about regression parameters of the variance function for practically all continuous response variables. The connection between the novel robust variance regression model and the estimating equations approach is also provided.     

On estimation of conditional density models with two-phase sampling

Suppose that the conditional density of a response variable given a vector of explanatory variables is parametrically modelled, and that data are collected by a two-phase sampling design. First, a simple random sample is drawn from the population. The stratum membership in a finite number of strata of the response and explanatory variables is recorded for each unit. Second, a subsample is drawn from the phase-one sample such that the selection probability is determined by the stratum membership. The response and explanatory variables are fully measured at this phase. We synthesize existing results on nonparametric likelihood estimation and present a streamlined approach for the computation and the large sample theory of profile likelihood in four different situations. The amount of information in terms of data and assumptions varies depending on whether the phase-one data are retained, the selection probabilities are known, and/or the stratum probabilities are known. We establish and illustrate numerically the order of efficiency among the maximum likelihood estimators, according to the amount of information utilized, in the four situations.     

Testing variance components in balanced linear growth curve models

It is well known that the testing of zero variance components is a non-standard problem since the null hypothesis is on the boundary of the parameter space. The usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold because of this null hypothesis. To circumvent this difficulty in balanced linear growth curve models, we introduce an appropriate test statistic and suggest a permutation procedure to approximate its finite-sample distribution. The proposed test alleviates the necessity of any distributional assumptions for the random effects and errors and can easily be applied for testing multiple variance components. Our simulation studies show that the proposed test has Type I error rate close to the nominal level. The power of the proposed test is also compared with the likelihood ratio test in the simulations. An application on data from an orthodontic study is presented and discussed.     

Likelihood analysis of joint marginal and conditional models for longitudinal categorical data

The authors develop a Markov model for the analysis of longitudinal categorical data which facilitates modelling both marginal and conditional structures. A likelihood formulation is employed for inference, so the resulting estimators enjoy the optimal properties such as efficiency and consistency, and remain consistent when data are missing at random. Simulation studies demonstrate that the proposed method performs well under a variety of situations. Application to data from a smoking prevention study illustrates the utility of the model and interpretation of covariate effects. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

A test for the parametric form of the variance function in a partial linear regression model

We consider the problem of testing for a parametric form of the variance function in a partial linear regression model. A new test is derived, which can detect local alternatives converging to the null hypothesis at a rate n^-1/2$n^{-1/2}$ and is based on a stochastic process of the integrated variance function. We establish weak convergence to a Gaussian process under the null hypothesis, fixed and local alternatives. In the special case of testing for homoscedasticity the limiting process is a scaled Brownian bridge. We also compare the finite sample properties with a test based on an L²$L^2$-distance, which was recently proposed by You and Chen [2005. Testing heteroscedasticity in partially linear regression models. Statist. Probab. Lett. 73, 61–70].     

The exact distribution and density functions of the stein-type estimator for normal variance

In this paper, we derive the exact distribution and density functions of the Stein-type estimator for the normal variance. It is shown by numerical evaluation that the density function of the Stein-type estimator is unimodal and concentrates around the mode more than that of the usual estimator.     

Bootstrap tests for variance components in generalized linear mixed models

In many applications of generalized linear mixed models to clustered correlated or longitudinal data, often we are interested in testing whether a random effects variance component is zero. The usual asymptotic mixture of chi‐square distributions of the score statistic for testing constrained variance components does not necessarily hold. In this article, the author proposes and explores a parametric bootstrap test that appears to be valid based on its estimated level of significance under the null hypothesis. Results from a simulation study indicate that the bootstrap test has a level much closer to the nominal one while the asymptotic test is conservative, and is more powerful than the usual asymptotic score test based on a mixture of chi‐squares. The proposed bootstrap test is illustrated using two sets of real‐life data obtained from clinical trials. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Choice of link and variance function for generalized linear mixed models: a case study with binomial response in proteomics

Abstract

Non-normality is a common phenomenon in data from agricultural and biological research, especially in molecular data (for example; -omics, RNAseq, flow cytometric data, etc.). For over half a century, the leading paradigm called for using analysis of variance (ANOVA) after applying a data transformation. The introduction of generalized linear mixed models (GLMM) provides a new way of analyzing non-normal data. Selecting an apt link function in GLMM can be quite influential, however, and is as critical as selecting an appropriate transformation for ANOVA. In this paper, we assess the performance of different parametric link families available in literature. Then, we propose a new estimation method for selecting an appropriate link function with a suitable variance function in a quasi-likelihood framework. We apply these methods to a proteomics data set, showing that GLMMs provide a very flexible framework for analyzing these kinds of data.     

Asymptotic normality of a conditional hazard function estimate in the single index for quasi-associated data

Abstract

The main goal of this paper is to study the estimation of the conditional hazard function of a scalar response variable Y given a hilbertian random variable X in functional single-index model. We construct an estimator of this nonparametric function and we study its asymptotic properties, under quasi-associated structure. Precisely, we establish the asymptotic normality of the constructed estimator. We carried out simulation experiments to examine the behavior of this asymptotic property over finite sample data.     

Use of likelihood ratio tests to detect outliers under the variance shift outlier model

In this paper, we revisit the alternative outlier model of Thompson [A note on restricted maximum likelihood estimation with an alternative outlier model, J. Roy. Stat. Soc. Ser. B 47 (1985), pp. 53–55] for detecting outliers in the linear model. Gumedze et al. [A variance shift model for detection of outliers in the linear mixed model, Comput. Statist. Data Anal. 54 (2010), pp. 2128–2144] called this model the variance shift outlier model (VSOM). The basic idea behind the VSOM is to detect observations with inflated variance and isolate them for further investigation. The VSOM is appealing because it downweights an outlier in the analysis, with the weighting determined automatically as part of the estimation procedure. We set up the VSOM as a linear mixed model and then use the likelihood ratio test (LRT) statistic as an objective measure for determining whether the weighting is required, i.e. whether the observation is an outlier. We also derived one-step updates of the variance parameter estimates based on observed, expected and average information matrices to obtain one-step LRT statistics which usually require less computation. Both the fully iterated and one-step LRTs are functions of the squared standard residuals from the null model and therefore can be computed directly without the need to fit the VSOM. We investigated the properties of the likelihood ratio tests and compare them. An extension of the model to detect a group of outliers is also given. We illustrate the proposed methodology using simulated datasets and a real dataset.     

Proportional hazards estimate of the conditional survival function

We introduce a new estimator of the conditional survival function given some subset of the covariate values under a proportional hazards regression. The new estimate does not require estimating the base-line cumulative hazard function. An estimate of the variance is given and is easy to compute, involving only those quantities that are routinely calculated in a Cox model analysis. The asymptotic normality of the new estimate is shown by using a central limit theorem for Kaplan–Meier integrals. We indicate the straightforward extension of the estimation procedure under models with multiplicative relative risks, including non-proportional hazards, and to stratified and frailty models. The estimator is applied to a gastric cancer study where it is of interest to predict patients' survival based only on measurements obtained before surgery, the time at which the most important prognostic variable, stage, becomes known.     

Conservative prior distributions for variance parameters in hierarchical models

Bayesian hierarchical models typically involve specifying prior distributions for one or more variance components. This is rather removed from the observed data, so specification based on expert knowledge can be difficult. While there are suggestions for “default” priors in the literature, often a conditionally conjugate inverse‐gamma specification is used, despite documented drawbacks of this choice. The authors suggest “conservative” prior distributions for variance components, which deliberately give more weight to smaller values. These are appropriate for investigators who are skeptical about the presence of variability in the second‐stage parameters (random effects) and want to particularly guard against inferring more structure than is really present. The suggested priors readily adapt to various hierarchical modelling settings, such as fitting smooth curves, modelling spatial variation and combining data from multiple sites.     

Regularization and variable selection for infinite variance autoregressive models

Autoregressive models with infinite variance are of great importance in modeling heavy-tailed time series and have been well studied. In this paper, we propose a penalized method to conduct model selection for autoregressive models with innovations having Pareto-like distributions with index α∈(0,2)$\alpha \in (\mathrm{0,2})$. By combining the least absolute deviation loss function and the adaptive lasso penalty, the proposed method is able to consistently identify the true model and at the same time produce efficient estimators with a convergence rate of n^−1/α$n^{-1/\alpha }$. In addition, our approach provides a unified way to conduct variable selection for autoregressive models with finite or infinite variance. A simulation study and a real data analysis are conducted to illustrate the effectiveness of our method.     

Functional analysis of variance for Hilbert-valued multivariate fixed effect models

This paper presents new results on functional analysis of variance for fixed effect models with correlated Hilbert-valued Gaussian error components. The geometry of the reproducing kernel Hilbert space of the error term is considered in the computation of the total sum of squares, the residual sum of squares, and the sum of squares due to the regression. Under suitable linear transformation of the correlated functional data, the distributional characteristics of these statistics, their moment generating and characteristic functions, are derived. Fixed effect linear hypothesis testing is finally formulated in the Hilbert-valued multivariate Gaussian context considered.     

A test for the variance residual life

In this paper we develop a test based on the empirical distribution function for the alternative representing 'decreasing variance residual life1 property. The test is consistent with asymptotically normal test statistic and is shown to perform well in the Pitman's asymptotic relative efficiency sense.