Fixed-width output analysis for Markov chain Monte Carlo

Markov chain Monte Carlo is a method of producing a correlated sample in order to estimate features of a target distribution via ergodic averages. A fundamental question is when should sampling stop? That is, when are the ergodic averages good estimates of the desired quantities? We consider a method that stops the simulation when the width of a confidence interval based on an ergodic average is less than a user-specified value. Hence calculating a Monte Carlo standard error is a critical step in assessing the simulation output. We consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution. We give sufficient conditions for the strong consistency of both methods and investigate their finite sample properties in a variety of examples

Markov Chain Monte Carlo confidence intervals

For a reversible and ergodic Markov chain $\{X_n,n\geq0\}$ with invariant distribution $\pi$, we show that a valid confidence interval for $\pi(h)$ can be constructed whenever the asymptotic variance $\sigma^2_P(h)$ is finite and positive. We do not impose any additional condition on the convergence rate of the Markov chain. The confidence interval is derived using the so-called fixed-b lag-window estimator of $\sigma_P^2(h)$. We also derive a result that suggests that the proposed confidence interval procedure converges faster than classical confidence interval procedures based on the Gaussian distribution and standard central limit theorems for Markov chains.Comment: Published at http://dx.doi.org/10.3150/15-BEJ712 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Change-Point Testing and Estimation for Risk Measures in Time Series

We investigate methods of change-point testing and confidence interval construction for nonparametric estimators of expected shortfall and related risk measures in weakly dependent time series. A key aspect of our work is the ability to detect general multiple structural changes in the tails of time series marginal distributions. Unlike extant approaches for detecting tail structural changes using quantities such as tail index, our approach does not require parametric modeling of the tail and detects more general changes in the tail. Additionally, our methods are based on the recently introduced self-normalization technique for time series, allowing for statistical analysis without the issues of consistent standard error estimation. The theoretical foundation for our methods are functional central limit theorems, which we develop under weak assumptions. An empirical study of S&P 500 returns and US 30-Year Treasury bonds illustrates the practical use of our methods in detecting and quantifying market instability via the tails of financial time series during times of financial crisis

BAYESIAN FORECASTING USING STOCHASTIC SIMULATION

In this article, we present a general framework to construct forecasts using simulation. This framework allows us to incorporate available data into a forecasting model in order to assess parameter uncertainty through a posterior distribution, which is then used to estimate a point forecast in the form of a conditional (given the data) expectation. The uncertainty on the point forecast is assessed through the estimation of a conditional variance and a prediction interval. We discuss how to construct asymptotic confidence intervals to assess the estimation error for the estimators obtained using simulation. We illustrate how this approach is consistent with Bayesian forecasting by presenting two examples, and experimental results that confirm our analytical results are discussed.Forecasting; simulation output analysis; Bayesian estimation; quantile estimation.

Pronósticos bayesianos usando simulación estocástica*

In this article, we present a general framework to construct forecasts using simulation. This framework allows us to incorporate available data into a forecasting model in order to assess parameter uncertainty through a posterior distribution, which is then used to estimate a point forecast in the form of a conditional (given the data) expectation. The uncertainty on the point forecast is assessed through the estimation of a conditional variance and a prediction interval. We discuss how to construct asymptotic confidence intervals to assess the estimation error for the estimators obtained using simulation. We illustrate how this approach is consistent with Bayesian forecasting by presenting two examples, and experimental results that confirm our analytical results are discussed.En este artículo se presenta un marco general para la construcción de pronósticos usando simulación. El marco permite la incor-poración de datos disponibles en un modelo de pronóstico, con la finalidad de cuantificar la incertidumbre en los parámetros del modelo a través de una distribución posterior, la que es utilizada para estimar un pronóstico puntual en la forma de una es-peranza condicional (dados los datos). La incertidumbre en el pronóstico puntual se mide a través de una varianza condicional y un intervalo de predicción. Se discute cómo construir intervalos de confianza asintóticos para evaluar la precisión de los esti-madores obtenidos por simulación, y se presentan dos ejemplos para ilustrar cómo este enfoque es consistente con las técnicas bayesianas para pronóstico. Se discuten resultados experimentales que confirman la validez de las metodologías propuestas

Applications of stochastic simulation in two-stage multiple comparisons with the best problem and time average variance constant estimation

In this dissertation, we study two problems. In the first part, we consider the two-stage methods for comparing alternatives using simulation. Suppose there are a finite number of alternatives to compare, with each alternative having an unknown parameter that is the basis for comparison. The parameters are to be estimated using simulation, where the alternatives are simulated independently. We develop two-stage selection and multiple-comparison procedures for simulations under a general framework. The assumptions are that each alternative has a parameter estimation process that satisfies a random- time-change central limit theorem (CLT), and there is a weakly consistent variance estimator (WCVE) for the variance constant appearing in the CLT. The framework encompasses comparing means of independent populations, functions of means, and steady-state means. One problem we consider of considerable practical interest and not handled in previous work on two-stage multiple-comparison procedures is comparing quantiles of alternative populations. We establish the asymptotic validity of our procedures as the prescribed width of the confidence intervals or indifference-zone parameter shrinks to zero. Also, for the steady-state simulation context, we compare our procedures based on WCVEs with techniques that instead use standardized time series methods. In the second part, we propose a new technique of estimating the variance parameter of a wide variety of stochastic processes. This new technique is better than the existing techniques for some standard stochastic processes in terms of bias and variance properties, since it reduces bias at the cost of no significant increase in variance