SHIFTED HILL'S ESTIMATOR FOR HEAVY TAILS

Hill's estimator is a popular method for estimating the thickness of heavy tails. In this paper we modify Hill's estimator to make it shift-invariant as well as scale-invariant. The resulting shifted Hill's estimator is a more robust method of estimating tail thickness.

     

Efficiency and Validity Analyses of Two-Stage Estimation Procedures and Derived Testing Procedures in Quantitative Linear Models with AR(1) Errors

Abstract

In a quantitative linear model with errors following a stationary Gaussian, first-order autoregressive or AR(1) process, Generalized Least Squares (GLS) on raw data and Ordinary Least Squares (OLS) on prewhitened data are efficient methods of estimation of the slope parameters when the autocorrelation parameter of the error AR(1) process, ρ, is known. In practice, ρ is generally unknown. In the so-called two-stage estimation procedures, ρ is then estimated first before using the estimate of ρ to transform the data and estimate the slope parameters by OLS on the transformed data. Different estimators of ρ have been considered in previous studies. In this article, we study nine two-stage estimation procedures for their efficiency in estimating the slope parameters. Six of them (i.e., three noniterative, three iterative) are based on three estimators of ρ that have been considered previously. Two more (i.e., one noniterative, one iterative) are based on a new estimator of ρ that we propose: it is provided by the sample autocorrelation coefficient of the OLS residuals at lag 1, denoted r(1). Lastly, REstricted Maximum Likelihood (REML) represents a different type of two-stage estimation procedure whose efficiency has not been compared to the others yet. We also study the validity of the testing procedures derived from GLS and the nine two-stage estimation procedures. Efficiency and validity are analyzed in a Monte Carlo study. Three types of explanatory variable x in a simple quantitative linear model with AR(1) errors are considered in the time domain: Case 1, x is fixed; Case 2, x is purely random; and Case 3, x follows an AR(1) process with the same autocorrelation parameter value as the error AR(1) process. In a preliminary step, the number of inadmissible estimates and the efficiency of the different estimators of ρ are compared empirically, whereas their approximate expected value in finite samples and their asymptotic variance are derived theoretically. Thereafter, the efficiency of the estimation procedures and the validity of the derived testing procedures are discussed in terms of the sample size and the magnitude and sign of ρ. The noniterative two-stage estimation procedure based on the new estimator of ρ is shown to be more efficient for moderate values of ρ at small sample sizes. With the exception of small sample sizes, REML and its derived F-test perform the best overall. The asymptotic equivalence of two-stage estimation procedures, besides REML, is observed empirically. Differences related to the nature, fixed or random (uncorrelated or autocorrelated), of the explanatory variable are also discussed.     

PERFORMANCE OF THE LOCATION LINEAR DISCRIMINANT FUNCTION UNDER ACROSS-LOCATION HETEROSCEDASTICITY

ABSTRACT

Classification of data consisting of both categorical and continuous variables between two groups is often handled by the sample location linear discriminant function confined to each of the locations specified by the observed values of the categorical variables. Homoscedasticity of across-location conditional dispersion matrices of the continuous variables is often assumed. Quite often, interactions between continuous and categorical variables cause across-location heteroscedasticity. In this article, we examine the effect of heterogeneous across-location conditional dispersion matrices on the overall expected and actual error rates associated with the sample location linear discriminant function. Performance of the sample location linear discriminant function is evaluated against the results for the restrictive classifier adjusted for across-location heteroscedasticity. Conclusions based on a Monte Carlo study are reported.     

Multiple Linear Regression Model Under Nonnormality

Abstract

We consider multiple linear regression models under nonnormality. We derive modified maximum likelihood estimators (MMLEs) of the parameters and show that they are efficient and robust. We show that the least squares esimators are considerably less efficient. We compare the efficiencies of the MMLEs and the M estimators for symmetric distributions and show that, for plausible alternatives to an assumed distribution, the former are more efficient. We provide real-life examples.     

Out-of-Sample Performance of Discrete-Time Spot Interest Rate Models

We provide a comprehensive analysis of the out-of-sample performance of a wide variety of spot rate models in forecasting the probability density of future interest rates. Although the most parsimonious models perform best in forecasting the conditional mean of many financial time series, we find that the spot rate models that incorporate conditional heteroscedasticity and excess kurtosis or heavy tails have better density forecasts. Generalized autoregressive conditional heteroscedasticity significantly improves the modeling of the conditional variance and kurtosis, whereas regime switching and jumps improve the modeling of the marginal density of interest rates. Our analysis shows that the sophisticated spot rate models in the existing literature are important for applications involving density forecasts of interest rates.     

The Gaussian Mixture Dynamic Conditional Correlation Model: Parameter Estimation,Value at Risk Calculation,and Portfolio Selection

A multivariate generalized autoregressive conditional heteroscedasticity model with dynamic conditional correlations is proposed, in which the individual conditional volatilities follow exponential generalized autoregressive conditional heteroscedasticity models and the standardized innovations follow a mixture of Gaussian distributions. Inference on the model parameters and prediction of future volatilities are addressed by both maximum likelihood and Bayesian estimation methods. Estimation of the Value at Risk of a given portfolio and selection of optimal portfolios under the proposed specification are addressed. The good performance of the proposed methodology is illustrated via Monte Carlo experiments and the analysis of the daily closing prices of the Dow Jones and NASDAQ indexes.     

A note on the satterthwaite confidence interval for a variance

When using a Satterthwaite chi-squared approximation, it is generally thought that the approximation is satisfactory when it is applied to a positive linear combination of mean squares. In this note, we describe how the Williams - Tukey idea for getting a confidence interval for the among groups variance in a random one-way model can be incorporated into Satterthwaite’s procedure for getting a confidence interval for a variance. This adjusted Satterthwaite procedure insures that his chi-squared approximation is always applied to positive linear combinations of mean squares. A small simulation is included which suggests that the adjustment to the Satterthwaite procedure is effective.     

Reml and best linear unbiased prediction in state space models

This article takes a hierarchical model approach to the estimation of state space models with diffuse initial conditions. An initial state is said to be diffuse when it cannot be assigned a proper prior distribution. In state space models this occurs either when fixed effects are present or when modelling nonstationarity in the state transition equation. Whereas much of the literature views diffuse states as an initialization problem, we follow the approach of Sallas and Harville (1981,1988) and incorporate diffuse initial conditions via noninformative prior distributions into hierarchical linear models. We apply existing results to derive the restricted loglike-lihood and appropriate modifications to the standard Kalman filter and smoother. Our approach results in a better understanding of De Jong's (1991) contributions. This article also shows how to adjust the standard Kalman filter, the fixed inter- val smoother and the state space model forecasting recursions, together with their mean square errors, for he presence of diffuse components. Using a hierarchical model approach it is shown that the estimates obtained are Best Linear Unbiased Predictors (BLUP).     

On L 1-consistency of kernel-type density estimator for stationary markov processes

A convergence result for kernel type density estimators, proved by Devroye and Gyrofi (1985), is extended to stationary Markov processess satisfying (G ₂-condition introduced by Rosenblatt (1970).     

Estimation of the location and scale parameters of the extreme value distmbution based on multiply type-II censored samples

In this paper, we consider the problem of estimating the location and scale parameters of an extreme value distribution based on multiply Type-II censored samples. We first describe the best linear unbiased estimators and the maximum likelihood estimators of these parameters. After observing that the best linear unbiased estimators need the construction of some tables for its coefficients and that the maximum likelihood estimators do not exist in an explicit algebraic form and hence need to be found by numerical methods, we develop approximate maximum likelihood estimators by appropriately approximating the likelihood equations. In addition to being simple explicit estimators, these estimators turn out to be nearly as efficient as the best linear unbiased estimators and the maximum likelihood estimators. Next, we derive the asymptotic variances and covariance of these estimators in terms of the first two single moments and the product moments of order statistics from the standard extreme value distribution. Finally, we present an example in order to illustrate all the methods of estimation of parameters discussed in this paper.