ESTIMATION OF VARIANCE IN HETEROSCEDASTIC DATA

Abstract

Data which exhibit none constant variance is considered. Smoothing procedures are applied to estimate these none constant variances. In these smoothing methods the problem is to establish how much to smooth. The choice of the smoother and the choice of the bandwidth are explored. Kernel and Spline smoothers are compared using simulated data as well as real data. Although the two seem to work very closely, Kernel smoother comes out to be slightly better.

KEY WORDS: Smoothing, Kernel, Spline, Heteroscedastic, Bandwidth, Variance. 1. Introduction

Let us have the observations .The mean is given by , . The deviations of each observation away
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Given the errors are independently distributed. However if any of these assumptions are violated the estimates obtained under the classical or usual assumption are not good. Therefore we hope to obtain better estimates of when the estimation of the variance is incorporated.

Therefore, we need to investigate and incorporate the information about the variance estimates of the errors which are needed for better understanding of the variability of the data. In heteroscedastic regression models the variance is not constant. Often as in the case with the mean, the heteroscedasity is believed to be in functional form which is referred to as variance function. We try to understand the structure of the variances as a function of the predictors such as time, height, age and so on. Two procedures of estimating the variance function includes the parametric and nonparametric methods. The parametric variance function estimation may be defined as a type of regression problem in which we see variance as a function of estimable quantities. Thus, the heteroscedasticity is modeled as a function of the regression and other structural parameters. This function is completely known, specified up to these unknown parameters. Estimation of these parameters is what entails parametric methods.

However, for many practical problems the degree to which components of the statistical model can be specified in a parametric form varies

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