Multivariate autoregressive extreme value process and its application for modeling the time series properties of the extreme daily asset prices

Abstract

In this article we suggest a new multivariate autoregressive process for modeling time-dependent extreme value distributed observations. The idea behind the approach is to transform the original observations to latent variables that are univariate normally distributed. Then the vector autoregressive DCC model is fitted to the multivariate latent process. The distributional properties of the suggested model are extensively studied. The process parameters are estimated by applying a two-stage estimation procedure. We derive a prediction interval for future values of the suggested process. The results are applied in an empirically study by modeling the behavior of extreme daily stock prices.     

On the Convergence of the Monte Carlo Maximum Likelihood Method for Latent Variable Models

While much used in practice, latent variable models raise challenging estimation problems due to the intractability of their likelihood. Monte Carlo maximum likelihood (MCML), as proposed by Geyer & Thompson (1992 ), is a simulation-based approach to maximum likelihood approximation applicable to general latent variable models. MCML can be described as an importance sampling method in which the likelihood ratio is approximated by Monte Carlo averages of importance ratios simulated from the complete data model corresponding to an arbitrary value of the unknown parameter. This paper studies the asymptotic (in the number of observations) performance of the MCML method in the case of latent variable models with independent observations. This is in contrast with previous works on the same topic which only considered conditional convergence to the maximum likelihood estimator, for a fixed set of observations. A first important result is that when is fixed, the MCML method can only be consistent if the number of simulations grows exponentially fast with the number of observations. If on the other hand, is obtained from a consistent sequence of estimates of the unknown parameter, then the requirements on the number of simulations are shown to be much weaker.     

Robust variable selection in finite mixture of regression models using the t distribution

Abstract

Variable selection in finite mixture of regression (FMR) models is frequently used in statistical modeling. The majority of applications of variable selection in FMR models use a normal distribution for regression error. Such assumptions are unsuitable for a set of data containing a group or groups of observations with heavy tails and outliers. In this paper, we introduce a robust variable selection procedure for FMR models using the t distribution. With appropriate selection of the tuning parameters, the consistency and the oracle property of the regularized estimators are established. To estimate the parameters of the model, we develop an EM algorithm for numerical computations and a method for selecting tuning parameters adaptively. The parameter estimation performance of the proposed model is evaluated through simulation studies. The application of the proposed model is illustrated by analyzing a real data set.     

Joint modeling of mixed skewed continuous and ordinal longitudinal responses: a Bayesian approach

In this paper, a joint model for analyzing multivariate mixed ordinal and continuous responses, where continuous outcomes may be skew, is presented. For modeling the discrete ordinal responses, a continuous latent variable approach is considered and for describing continuous responses, a skew-normal mixed effects model is used. A Bayesian approach using Markov Chain Monte Carlo (MCMC) is adopted for parameter estimation. Some simulation studies are performed for illustration of the proposed approach. The results of the simulation studies show that the use of the separate models or the normal distributional assumption for shared random effects and within-subject errors of continuous and ordinal variables, instead of the joint modeling under a skew-normal distribution, leads to biased parameter estimates. The approach is used for analyzing a part of the British Household Panel Survey (BHPS) data set. Annual income and life satisfaction are considered as the continuous and the ordinal longitudinal responses, respectively. The annual income variable is severely skewed, therefore, the use of the normality assumption for the continuous response does not yield acceptable results. The results of data analysis show that gender, marital status, educational levels and the amount of money spent on leisure have a significant effect on annual income, while marital status has the highest impact on life satisfaction.     

Nonparametric Estimation of Distribution Functions of Nonstandard Mixtures

ABSTRACT

Nonstandard mixtures are those that result from a mixture of a discrete and a continuous random variable. They arise in practice, for example, in medical studies of exposure. Here, a random variable that models exposure might have a discrete mass point at no exposure, but otherwise may be continuous. In this article we explore estimating the distribution function associated with such a random variable from a nonparametric viewpoint. We assume that the locations of the discrete mass points are known so that we will be able to apply a classical nonparametric smoothing approach to the problem. The proposed estimator is a mixture of an empirical distribution function and a kernel estimate of a distribution function. A simple theoretical argument reveals that existing bandwidth selection algorithms can be applied to the smooth component of this estimator as well. The proposed approach is applied to two example sets of data.     

Latent Variable Models for Mixed Discrete and Continuous Outcomes

We propose a latent variable model for mixed discrete and continuous outcomes. The model accommodates any mixture of outcomes from an exponential family and allows for arbitrary covariate effects, as well as direct modelling of covariates on the latent variable. An EM algorithm is proposed for parameter estimation and estimates of the latent variables are produced as a by-product of the analysis. A generalized likelihood ratio test can be used to test the significance of covariates affecting the latent outcomes. This method is applied to birth defects data, where the outcomes of interest are continuous measures of size and binary indicators of minor physical anomalies. Infants who were exposed in utero to anticonvulsant medications are compared with controls.     

Joint modeling of longitudinal data with a dependent terminal event

ABSTRACT

Longitudinal data often arise in longitudinal follow-up studies, and there may exist a dependent terminal event such as death that stops the follow-up. In this article, we propose a new joint modeling for the analysis of longitudinal data with informative observation times via a dependent terminal event and two latent variables. Estimating equations are developed for parameter estimation, and asymptotic properties of the resulting estimators are established. In addition, a generalization of the joint model with time-varying coefficients for the longitudinal response variable is considered, and goodness-of-fit methods for assessing the adequacy of the model are also provided. The proposed method works well in our simulation studies, and is applied to a data set from a bladder cancer study.     

On minimum Lp-distance estimation for inhomogeneous Poisson processes

ABSTRACT

We consider the problem of parameter estimation by the observations of the inhomogeneous Poisson processes. We suppose that the intensity function of these processes is a smooth function of the unknown parameter and as a method of estimation we take the minimum distance approach. We are interested by the behavior of estimators in non Hilbertian situation and we define the minimum distance estimation (MDE) with the help of the L^p metrics. We show that (under regularity conditions) the MDE is consistent and we describe its limit distribution.     

The effect of parameter estimation on phase II monitoring of poisson regression profiles

ABSTRACT

The effect of parameters estimation on profile monitoring methods has only been studied by a few researchers and only the assumption of a normal response variable has been tackled. However, in some practical situation, the normality assumption is violated and the response variable follows a discrete distribution such as Poisson. In this paper, we evaluate the effect of parameters estimation on the Phase II monitoring of Poisson regression profiles by considering two control charts, namely the Hotelling’s T² and the multivariate exponentially weighted moving average (MEWMA) charts. Simulation studies in terms of the average run length (ARL) and the standard deviation of the run length (SDRL) are carried out to assess the effect of estimated parameters on the performance of Phase II monitoring approaches. The results reveal that both in-control and out-of-control performances of these charts are adversely affected when the regression parameters are estimated.     

Selection of Latent Variables for Multiple Mixed‐outcome Models

Latent variable models have been widely used for modelling the dependence structure of multiple outcomes data. However, the formulation of a latent variable model is often unknown a priori, the misspecification will distort the dependence structure and lead to unreliable model inference. Moreover, multiple outcomes with varying types present enormous analytical challenges. In this paper, we present a class of general latent variable models that can accommodate mixed types of outcomes. We propose a novel selection approach that simultaneously selects latent variables and estimates parameters. We show that the proposed estimator is consistent, asymptotically normal and has the oracle property. The practical utility of the methods is confirmed via simulations as well as an application to the analysis of the World Values Survey, a global research project that explores peoples’ values and beliefs and the social and personal characteristics that might influence them.     

Estimates for cell counts and common odds ratio in three-way contingency tables by homogeneous log-linear models with missing data

Missing observations often occur in cross-classified data collected during observational, clinical, and public health studies. Inappropriate treatment of missing data can reduce statistical power and give biased results. This work extends the Baker, Rosenberger and Dersimonian modeling approach to compute maximum likelihood estimates for cell counts in three-way tables with missing data, and studies the association between two dichotomous variables while controlling for a third variable in \( 2\times 2 \times K \) tables. This approach is applied to the Behavioral Risk Factor Surveillance System data. Simulation studies are used to investigate the efficiency of estimation of the common odds ratio.     

Composite quantile regression for varying-coefficient single-index models

ABSTRACT

The varying-coefficient single-index model (VCSIM) is a very general and flexible tool for exploring the relationship between a response variable and a set of predictors. Popular special cases include single-index models and varying-coefficient models. In order to estimate the index-coefficient and the non parametric varying-coefficients in the VCSIM, we propose a two-stage composite quantile regression estimation procedure, which integrates the local linear smoothing method and the information of quantile regressions at a number of conditional quantiles of the response variable. We establish the asymptotic properties of the proposed estimators for the index-coefficient and varying-coefficients when the error is heterogeneous. When compared with the existing mean-regression-based estimation method, our simulation results indicate that our proposed method has comparable performance for normal error and is more robust for error with outliers or heavy tail. We illustrate our methodologies with a real example.     

Cross-validated mixed-datatype bandwidth selection for nonparametric cumulative distribution/survivor functions

ABSTRACT

We propose a computationally efficient data-driven least square cross-validation method to optimally select smoothing parameters for the nonparametric estimation of cumulative distribution/survivor functions. We allow for general multivariate covariates that can be continuous, discrete/ordered categorical or a mix of either. We provide asymptotic analysis, examine finite-sample properties through Monte Carlo simulation, and consider an illustration involving nonparametric copula modeling. We also demonstrate how the approach can also be used to construct a smooth Kolmogorov–Smirnov test that has a slightly better power profile than its nonsmooth counterpart.     

On the estimation of normal copula discrete regression models using the continuous extension and simulated likelihood

The continuous extension of a discrete random variable is amongst the computational methods used for estimation of multivariate normal copula-based models with discrete margins. Its advantage is that the likelihood can be derived conveniently under the theory for copula models with continuous margins, but there has not been a clear analysis of the adequacy of this method. We investigate the asymptotic and small-sample efficiency of two variants of the method for estimating the multivariate normal copula with univariate binary, Poisson, and negative binomial regressions, and show that they lead to biased estimates for the latent correlations, and the univariate marginal parameters that are not regression coefficients. We implement a maximum simulated likelihood method, which is based on evaluating the multidimensional integrals of the likelihood with randomized quasi-Monte Carlo methods. Asymptotic and small-sample efficiency calculations show that our method is nearly as efficient as maximum likelihood for fully specified multivariate normal copula-based models. An illustrative example is given to show the use of our simulated likelihood method.     

Bayesian inference for zero-and-one-inflated geometric distribution regression model using Pólya-Gamma latent variables

Abstract

In the fields of internet financial transactions and reliability engineering, there could be more zero and one observations simultaneously. In this paper, considering that it is beyond the range where the conventional model can fit, zero-and-one-inflated geometric distribution regression model is proposed. Ingeniously introducing Pólya-Gamma latent variables in the Bayesian inference, posterior sampling with high-dimensional parameters is converted to latent variables sampling and posterior sampling with lower-dimensional parameters, respectively. Circumventing the need for Metropolis-Hastings sampling, the sample with higher sampling efficiency is obtained. A simulation study is conducted to assess the performance of the proposed estimation for various sample sizes. Finally, a doctoral dissertation data set is analyzed to illustrate the practicability of the proposed method, research shows that zero-and-one-inflated geometric distribution regression model using Pólya-Gamma latent variables can achieve better fitting results.     

A continuous-time HMM approach to modeling the magnitude-frequency distribution of earthquakes

The magnitude-frequency distribution (MFD) of earthquake is a fundamental statistic in seismology. The so-called b-value in the MFD is of particular interest in geophysics. A continuous time hidden Markov model (HMM) is proposed for characterizing the variability of b-values. The HMM-based approach to modeling the MFD has some appealing properties over the widely used sliding-window approach. Often, large variability appears in the estimation of b-value due to window size tuning, which may cause difficulties in interpretation of b-value heterogeneities. Continuous-time hidden Markov models (CT-HMMs) are widely applied in various fields. It bears some advantages over its discrete time counterpart in that it can characterize heterogeneities appearing in time series in a finer time scale, particularly for highly irregularly-spaced time series, such as earthquake occurrences. We demonstrate an expectation–maximization algorithm for the estimation of general exponential family CT-HMM. In parallel with discrete-time hidden Markov models, we develop a continuous time version of Viterbi algorithm to retrieve the overall optimal path of the latent Markov chain. The methods are applied to New Zealand deep earthquakes. Before the analysis, we first assess the completeness of catalogue events to assure the analysis is not biased by missing data. The estimation of b-value is stable over the selection of magnitude thresholds, which is ideal for the interpretation of b-value variability.     

SAVE: Robust or not?

Abstract

Sliced average variance estimation (SAVE) is one of the best methods for estimating central dimension-reduction subspace in semi parametric regression models when covariates are normal. In recent days SAVE is being used to analyze DNA microarray data especially in tumor classification but most important drawback is normality of covariates. In this article, the asymptotic behavior of estimates of CDR space under varying slice size is studied through simulation studies when covariates are non normal but follows linearity condition as well as when covariates slightly perturbed from normal distribution and we observed that serious error may occur under violation normality assumption.     

Variable selection for semiparametric varying coefficient partially linear model based on modal regression with missing data

Abstract

In this article, we focus on the variable selection for semiparametric varying coefficient partially linear model with response missing at random. Variable selection is proposed based on modal regression, where the non parametric functions are approximated by B-spline basis. The proposed procedure uses SCAD penalty to realize variable selection of parametric and nonparametric components simultaneously. Furthermore, we establish the consistency, the sparse property and asymptotic normality of the resulting estimators. The penalty estimation parameters value of the proposed method is calculated by EM algorithm. Simulation studies are carried out to assess the finite sample performance of the proposed variable selection procedure.     

Joint modeling of multivariate longitudinal mixed measurements and time to event data using a Bayesian approach

In many longitudinal studies multiple characteristics of each individual, along with time to occurrence of an event of interest, are often collected. In such data set, some of the correlated characteristics may be discrete and some of them may be continuous. In this paper, a joint model for analysing multivariate longitudinal data comprising mixed continuous and ordinal responses and a time to event variable is proposed. We model the association structure between longitudinal mixed data and time to event data using a multivariate zero-mean Gaussian process. For modeling discrete ordinal data we assume a continuous latent variable follows the logistic distribution and for continuous data a Gaussian mixed effects model is used. For the event time variable, an accelerated failure time model is considered under different distributional assumptions. For parameter estimation, a Bayesian approach using Markov Chain Monte Carlo is adopted. The performance of the proposed methods is illustrated using some simulation studies. A real data set is also analyzed, where different model structures are used. Model comparison is performed using a variety of statistical criteria.     

Mimicking Cox-Regression

Abstract

A class of objective functions, related to the Cox partial likelihood, that generates unbiased estimating equations is proposed. These equations allow for estimation of interest parameters when nuisance parameters are proportional to expectations. Examples of the objective functions are applied to binary data with a log-link in three situations: independent observations, independent groups of observations with common random intercept and discrete survival data. It is pointed out that the Peto–Breslow approximation to the partial likelihood with discrete failure times fits a conditional model with a log-link.