Testing for Wavelet Variances

Wavelet methods such as standard thresholding techniques are commonly used to estimate a nonparametric regression function from noisy sample data, under the traditional assumption that noises have constant variance. In situations where data have nonconstant variance, these standard techniques do not work well in estimating the regression function unless the heteroscedasticity is taken into account in estimation. This raises a fundamental issue for testing for the adequacy of the assumption of homoscedasticity before one uses the standard wavelet techniques under the constant variance assumption. This paper develops an asymptotically locally optimal partial score test for testing for the suitability of a homoscedastic wavelet model against a general heteroscedastic wavelet model. As the construction of the partial score test requires a consistent estimate for the nuisance parameter, namely the constant variance estimate under the null hypothesis, we conduct a comprehensive investigation in order to choose it's best possible estimate among some competitors. The size and power performances of the partial score test are reported for testing for heteroscedasticity in a time series of ̄nite length.