Spline Analysis For Two Regression Models

Compared with parametric methods, nonparametric techniques have the flexibility of constructing models based on observations, which can reduce modeling biases and help us to choose appropriate parametric models. As one of the commonly used nonparametric techniques, the spline smoothing method has the advantage of simple implementation and fast computation. In this paper, we will use the spline smoothing method to study three problems proposed in the nonparametric time series regressive model and the linear regression model with an interval-censored continuous covariate.In Chapter2, for nonparametric time series regression, we propose to apply the poly-nomial spline smoothing to estimate the conditional variance and construct its simulta-neous confidence bands. To be specific, we obtain the spline estimation of the conditional mean function in the first step. Then, the conditional variance function is estimated by applying polynomial spline smoothers to the residuals. Under a weak a-mixing condition, we obtain the uniform convergence rate. Furthermore, we construct simultaneous confi-dence bands for the conditional variance, using piecewise constant and piecewise linear splines. Simultaneous confidence bands can be used to test a parametric pattern for the variance curve. For instance, we can test whether the conditional heteroscedasticity exists or the conditional variance has the quadratic form. In the simulation, not only have we considered the case that both the conditional mean and variance functions are smooth, but also we have used the examples in which the conditional mean or the conditional variance function is rough. In the application,, we apply the proposed method to the S&P500Index daily data. Through the numerical results provided in the simulation and the application, we conclude that our method performs well enough and it is faster than the commonly used local polynomial smoothing method.In Chapter3, for time series nonparametric regression models with discontinuities, we propose to use polynomial splines to detect jumps in the conditional mean function. The number, locations, as well as magnitudes of the jumps are all assumed unknown. First, we obtain the spline estimator of the conditional mean function. Then, based on the maximal difference of the spline estimators between neighboring knots, test statistics for the existence of jumps are given and their limiting distributions are derived under the null hypothesis that the conditional mean function is continuous. Moreover, we use the knots to locate the detected jumps and apply a multiple ordered regression spline procedure to refit the data and estimate the jump magnitudes. In the simulation section, we analyze the asymptotic power of the proposed method, the frequencies of wrong detection and the frequencies of converge for true jumps over500replications.In Chapter4, we focus on the estimation of a linear regression model with an interval-censored continuous covariate. The linear regression model with an interval-censored covariate is proposed by Gomez, Espinal and Lagakos [9] in2003, which is motivated by an acquired immunodeficiency syndrome clinical trial. The censored covariate which is studied by Gomez et. al.[9] is a discrete random variable. To estimate the regression coefficients, Gomez et. al.[9] developed a likelihood approach, together with a two-step conditional algorithm. However, the algorithm of Gomez et. al.[9] is complex and time-consuming. Worse still, their method is inapplicable when the interval-censored covariate is continuous. In this chapter, we studied the tough problem that how to estimate the regression coefficients when the censored covariate is a continuous random variable. A novel and fast method is proposed to estimate the linear regression coefficients. Based on the logspline model of Stone [54], Kooperberg and Stone [55], we estimate the density and distribution functions of the interval-censored covariate in the first step. In the next step, we impute the interval-censored covariate with a conditional expectation. Then, we apply the ordinary least squared method to the linear regression model with the imputed covariate and obtain the estimated regression coefficients. In the simulation, we compare our imputation method with the midpoint imputation and the semiparametric Bayesian method. Through intensive simulation studies, we found our imputation method can give more accurate estimates and smaller MSEs when the width of the censoring interval is variable, and our imputation method is more than100times faster than the semiparametric Bayesian method.In Chapter5, we summarize the study of the whole paper by analyzing the merits and shortcomings of the proposed method. Moreover, we propose some possible directions for further study.