GENERALIZED EXTREME VALUE ADDITIVE MODEL ANALYSIS VIA MEAN FIELD VARIATIONAL BAYES

We develop Mean Field Variational Bayes methodology for fast approximate inference in Bayesian Generalized Extreme Value additive model analysis. Such models are useful for flexibly assessing the impact of continuous predictor variables on sample extremes. The new methodology allows large Bayesian models to be fitted and assessed without the significant computing costs of Markov Chain Monte Carlo methods. We illustrate our new methodology with maximum rainfall data from the Sydney, Australia, hinterland. Comparisons are made between the Mean Field Variational Bayes and Markov Chain Monte Carlo approaches.     

SEMIPARAMETRIC REGRESSION AND GRAPHICAL MODELS

Semiparametric regression models that use spline basis functions with penalization have graphical model representations. This link is more powerful than previously established mixed model representations of semiparametric regression, as a larger class of models can be accommodated. Complications such as missingness and measurement error are more naturally handled within the graphical model architecture. Directed acyclic graphs, also known as Bayesian networks, play a prominent role. Graphical model-based Bayesian 'inference engines', such as bugs and vibes , facilitate fitting and inference. Underlying these are Markov chain Monte Carlo schemes and recent developments in variational approximation theory and methodology.     

Mixture of Distributions in the Biparametric Exponential Family: A Bayesian Approach

In this article we propose mixture of distributions belonging to the biparametric exponential family, considering joint modeling of the mean and variance (or dispersion) parameters. As special cases we consider mixtures of normal and gamma distributions. A novel Bayesian methodology, using Markov Chain Monte Carlo (MCMC) methods, is proposed to obtain the posterior summaries of interest. We include simulations and real data examples to illustrate de performance of the proposal.     

The Stochastic Volatility in Mean Model With Time-Varying Parameters: An Application to Inflation Modeling

This article generalizes the popular stochastic volatility in mean model to allow for time-varying parameters in the conditional mean. The estimation of this extension is nontrival since the volatility appears in both the conditional mean and the conditional variance, and its coefficient in the former is time-varying. We develop an efficient Markov chain Monte Carlo algorithm based on band and sparse matrix algorithms instead of the Kalman filter to estimate this more general variant. The methodology is illustrated with an application that involves U.S., U.K., and Germany inflation. The estimation results show substantial time-variation in the coefficient associated with the volatility, highlighting the empirical relevance of the proposed extension. Moreover, in a pseudo out-of-sample forecasting exercise, the proposed variant also forecasts better than various standard benchmarks.     

Survival estimation and the effects of dependency among animals

Survival models assume that fates of individuals are independent, yet the robustness of this assumption has been poorly quantified. We examine how empirically derived estimates of the variance of survival rates are affected by dependency in survival probability among individuals. We used Monte Carlo simulations to generate known amounts of dependency among pairs of individuals and analyzed these data with Kaplan-Meier and Cormack-Jolly-Seber models. Dependency significantly increased these empirical variances as compared to theoretically derived estimates of variance from the same populations. Using resighting data from 168 pairs of black brant ( Branta bernicla nigricans ), we used a resampling procedure and program RELEASE to estimate empirical and mean theoretical variances. We estimated that the relationship between paired individuals caused the empirical variance of the survival rate to be 155% larger than the empirical variance for unpaired individuals. Monte Carlo simulations and use of this resampling strategy can provide investigators with information on how robust their data are to this common assumption of independent survival probabilities.     

Survival estimation and the effects of dependency among animals

Survival models assume that fates of individuals are independent, yet the robustness of this assumption has been poorly quantified. We examine how empirically derived estimates of the variance of survival rates are affected by dependency in survival probability among individuals. We used Monte Carlo simulations to generate known amounts of dependency among pairs of individuals and analyzed these data with Kaplan-Meier and Cormack-Jolly-Seber models. Dependency significantly increased these empirical variances as compared to theoretically derived estimates of variance from the same populations. Using resighting data from 168 pairs of black brant ( Branta bernicla nigricans ), we used a resampling procedure and program RELEASE to estimate empirical and mean theoretical variances. We estimated that the relationship between paired individuals caused the empirical variance of the survival rate to be 155% larger than the empirical variance for unpaired individuals. Monte Carlo simulations and use of this resampling strategy can provide investigators with information on how robust their data are to this common assumption of independent survival probabilities.     

C337. Misspecified location and scale. 11. independence of and s2

A hierarchical Bayesian approach to the problem of estimating the largest normal mean is considered. Calculation of the posterior mean and the posterior variance involves, at worst, 3-dimensional numerical integration, for which an efficient Monte Carlo method of evaluation is given. An example is presented to illustrate the methodology. In the two populations case, computation of the posterior estimates can be substantially simplified and in special cases can actually be performed using closed form solutions. A simulation study has been done to compare mean square errors of some hierarchical Bayesian estimators that are expressed in closed forms and several existing estimators of the larger mean.     

Variational Bayes with synthetic likelihood

Synthetic likelihood is an attractive approach to likelihood-free inference when an approximately Gaussian summary statistic for the data, informative for inference about the parameters, is available. The synthetic likelihood method derives an approximate likelihood function from a plug-in normal density estimate for the summary statistic, with plug-in mean and covariance matrix obtained by Monte Carlo simulation from the model. In this article, we develop alternatives to Markov chain Monte Carlo implementations of Bayesian synthetic likelihoods with reduced computational overheads. Our approach uses stochastic gradient variational inference methods for posterior approximation in the synthetic likelihood context, employing unbiased estimates of the log likelihood. We compare the new method with a related likelihood-free variational inference technique in the literature, while at the same time improving the implementation of that approach in a number of ways. These new algorithms are feasible to implement in situations which are challenging for conventional approximate Bayesian computation methods, in terms of the dimensionality of the parameter and summary statistic.     

Likelihood ratio test for simultaneous testing of the mean and the variance of a normal distribution

In this paper, we consider the simultaneous testing of the mean and the variance of a normal distribution. The exact distribution of the likelihood ratio test statistic is obtained, which is not available in the literature. The critical points of the exact test are reported. We also consider some of the other exact and asymptotic tests. The powers of these tests are compared using the Monte Carlo simulations.     

Inference for Lévy‐Driven Stochastic Volatility Models via Adaptive Sequential Monte Carlo

Abstract. We investigate simulation methodology for Bayesian inference in Lévy‐driven stochastic volatility (SV) models. Typically, Bayesian inference from such models is performed using Markov chain Monte Carlo (MCMC); this is often a challenging task. Sequential Monte Carlo (SMC) samplers are methods that can improve over MCMC; however, there are many user‐set parameters to specify. We develop a fully automated SMC algorithm, which substantially improves over the standard MCMC methods in the literature. To illustrate our methodology, we look at a model comprised of a Heston model with an independent, additive, variance gamma process in the returns equation. The driving gamma process can capture the stylized behaviour of many financial time series and a discretized version, fit in a Bayesian manner, has been found to be very useful for modelling equity data. We demonstrate that it is possible to draw exact inference, in the sense of no time‐discretization error, from the Bayesian SV model.     

A Survey of Sequential Monte Carlo Methods for Economics and Finance

This article serves as an introduction and survey for economists to the field of sequential Monte Carlo methods which are also known as particle filters. Sequential Monte Carlo methods are simulation-based algorithms used to compute the high-dimensional and/or complex integrals that arise regularly in applied work. These methods are becoming increasingly popular in economics and finance; from dynamic stochastic general equilibrium models in macro-economics to option pricing. The objective of this article is to explain the basics of the methodology, provide references to the literature, and cover some of the theoretical results that justify the methods in practice.     

Bias-corrected realized variance

We propose a novel “bias-corrected realized variance” (BCRV) estimator based upon the appropriate re-weighting of two realized variances calculated at different sampling frequencies. Our bias-correction methodology is found to be extremely accurate, with the finite sample variance being significantly minimized. In our Monte Carlo experiments and a finite sample MSE comparison of alternative estimators, the performance of our straightforward BCRV estimator is shown to be comparable to other widely-used integrated variance estimators. Given its simplicity, our BCRV estimator is likely to appeal to researchers and practitioners alike for the estimation of integrated variance.     

The mixed model for survey regression estimation

Estimation of the population mean under the regression model with random components is considered. Conditions under which the random components regression estimator is design consistent are given. It is shown that consistency holds when incorrect values are used for the variance components. The regression estimator constructed with model parameters that differ considerably from the true parameters performed well in a Monte Carlo study. Variance estimators for the regression predictor are suggested. A variance estimator appropriate for estimators constructed with a biased estimator for the between-group variance component performed well in the Monte Carlo study.     

New modified moment estimators for the two-parameter weighted Lindley distribution

We propose a modification of the moment estimators for the two-parameter weighted Lindley distribution. The modification replaces the second sample moment (or equivalently the sample variance) by a certain sample average which is bounded on the unit interval for all values in the sample space. In this method, the estimates always exist uniquely over the entire parameter space and have consistency and asymptotic normality over the entire parameter space. The bias and mean squared error of the estimators are also examined by means of a Monte Carlo simulation study, and the empirical results show the small-sample superiority in addition to the desirable large sample properties. Monte Carlo simulation study showed that the proposed modified moment estimators have smaller biases and smaller mean-square errors than the existing moment estimators and are compared favourably with the maximum likelihood estimators in terms of bias and mean-square error. Three illustrative examples are finally presented.     

Maximum likelihood estimation of variance components of heteroscedastic random anova model

the estimation of variance components of heteroscedastic random model is discussed in this paper. Maximum Likelihood (ML) is described for one-way heteroscedastic random models. The proportionality condition that cell variance is proportional to the cell sample size, is used to eliminate the efffect of heteroscedasticity. The algebraic expressions of the estimators are obtained for the model. It is seen that the algebraic expressions of the estimators depend mainly on the inverse of the variance-covariance matrix of the observation vector. So, the variance-covariance matrix is obtained and the formulae for the inversions are given. A Monte Carlo study is conducted. Five different variance patterns with different numbers of cells are considered in this study. For each variance pattern, 1000 Monte Carlo samples are drawn. Then the Monte Carlo biases and Monte Carlo MSE’s of the estimators of variance components are calculated. In respect of both bias and MSE, the Maximum Likelihood (ML) estimators of variance components are found to be sufficiently good.     

On maximum likelihood estimation of the long-memory parameter in fractional Gaussian noise

Approximate normality and unbiasedness of the maximum likelihood estimate (MLE) of the long-memory parameter H of a fractional Brownian motion hold reasonably well for sample sizes as small as 20 if the mean and scale parameter are known. We show in a Monte Carlo study that if the latter two parameters are unknown the bias and variance of the MLE of H both increase substantially. We also show that the bias can be reduced by using a parametric bootstrap procedure. In very large samples, maximum likelihood estimation becomes problematic because of the large dimension of the covariance matrix that must be inverted. To overcome this difficulty, we propose a maximum likelihood method based upon first differences of the data. These first differences form a short-memory process. We split the data into a number of contiguous blocks consisting of a relatively small number of observations. Computation of the likelihood function in a block then presents no computational problem. We form a pseudo-likelihood function consisting of the product of the likelihood functions in each of the blocks and provide a formula for the standard error of the resulting estimator of H. This formula is shown in a Monte Carlo study to provide a good approximation to the true standard error. The computation time required to obtain the estimate and its standard error from large data sets is an order of magnitude less than that required to obtain the widely used Whittle estimator. Application of the methodology is illustrated on two data sets.     

Double Generalized Spatial Econometric Models

The authors offer a unified method extending traditional spatial dependence with normally distributed error terms to a new class of spatial models based on the biparametric exponential family of distributions. Joint modeling of the mean and variance (or precision) parameters is proposed in this family of distributions, including spatial correlation. The proposed models are applied for analyzing Colombian land concentration, assuming that the variable of interest follows normal, gamma, and beta distributions. In all cases, the models were fitted using Bayesian methodology with the Markov Chain Monte Carlo (MCMC) algorithm for sampling from joint posterior distribution of the model parameters.     

Bayesian optimal sequential design for nonparametric regression via inhomogeneous evolutionary MCMC

We develop a novel computational methodology for Bayesian optimal sequential design for nonparametric regression. This computational methodology, that we call inhomogeneous evolutionary Markov chain Monte Carlo, combines ideas of simulated annealing, genetic or evolutionary algorithms, and Markov chain Monte Carlo. Our framework allows optimality criteria with general utility functions and general classes of priors for the underlying regression function. We illustrate the usefulness of our novel methodology with applications to experimental design for nonparametric function estimation using Gaussian process priors and free-knot cubic splines priors.     

A quadratic approximation for Jackknife estimators of the variance of sample mean functions

Jackknife estimators of the variance of estimators which are functions of the sample mean are considered. A quadratic approximation of them is proposed and compared with a linear approximation by Monte Carlo experiments carried out by statistical software Minitab.