Browsing by Subject "goodness-of-fit test"
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Item A Novel Approach to the Analysis of Nonlinear Time Series with Applications to Financial Data(2012-07-16) Lee, Jun BumThe spectral analysis method is an important tool in time series analysis and the spectral density plays a crucial role on the spectral analysis. However, one of limitations of the spectral density is that the spectral density reflects only the covariance structure among several dependence measures in the time series data. To overcome this restriction, we define two spectral densities, the quantile spectral density and the association spectral density. The quantile spectral density can model the pairwise dependence structure and provide identification of nonlinear time series and the association spectral density allows detecting periodicities on different parts of the domain of the time series. We propose the estimators for the quantile spectral density and the association spectral density and derive their sampling properties including asymptotic normality. Furthermore, we use the quantile spectral density to develop a goodness-of-fit tests for time series and explain how this test can be used for comparing the sequential dependence structure of two time series. The asymptotic sampling properties of the test statistic are derived under the null and alternative hypothesis, and a bootstrap procedure is suggested to obtain finite sample approximation. The method is illustrated with simulations and some real data examples. Besides the exploration of the new spectral densities, we consider general quadratic forms of alpha-mixing time series and derive asymptotic normality of these forms under the relatively weak assumptions.Item New Approaches in Testing Common Assumptions for Regressions with Missing Data(2014-07-30) Chown, Justin AndrewWe consider both nonparametric regression and heteroskedastic nonparametric regression models with multivariate covariates and with responses missing at random. The regression function is estimated using a local polynomial smoother, and, when necessary, the scale function is estimated using a combination of local polynomial smoothers. It is shown, for both regression models, that suitable residual-based empirical distribution functions using only the complete cases, i.e. residuals that can actually be constructed from the data, are efficient in the sense of H?jek and Le Cam. In our proofs we derive, more generally, the efficient influence function for estimating an arbitrary linear functional of the error distribution; this covers the distribution function as a special case. Our estimators are shown to admit functional central limit theorems. We do this by applying the transfer principle for complete case statistics, which makes it possible to adapt known results for fully observed data to the case of missing data. Then, we use these residual-based empirical distribution functions to test for normal errors using a martingale transform approach. Small simulation studies are conducted to investigate the performance of these tests. Our results, for the homoskedastic model, show the proposed approach to be comparable to one based on imputation, and, for the heteroskedastic model, the results are sensitive to the estimate of the scale function. Finally, we construct a test for heteroskedasticity using residuals from a nonparametric regression. The approach uses a weighted empirical process and only the completely observed data, and is shown to perform well in certain scenarios. All of the tests considered here are asymptotically distribution free, which means inference based on them does not depend on unknown parameters.