Abstract
This thesis aims to develop the varying local correlation function of two variables $X$ and $Y$ given another variable $T$ in a neighborhood of $t_0$, which can capture the relationships between $X$ and $Y$ among defferent $t_0$'s. In the literature, Bjerve and Doksum (1993) introduced the correlation curve of two variables $X$ and $Y$ given $X$ in a neighborhood of $x_0$, which applies to two variables with a nonliner relationship. First, we reveiw the relationships between the Pearson correlation and the simple linear regression, and extend it to the relationships between the weighted Pearson correlation and the weighted simple linear regression. We replace the weights of the weighted Pearson correlation by the kernel function to form an estiamtor of the local correlation between $X$ and $Y$ given $X$ in a neighborhood of $x_0$. Then we show that the local correlation enjoys some properties similar to the weighted Pearson correlation, and it has a connection with the correlation curve when estimating by local linear regression. Similarly, we replace the bivariate kernel function of the local dependence function (Jones (1996)) by the univariate kernel function to serve as the theoretic definition of the local correlation. For the varying local correlation function, it is based on the bivariate local dependence function (Jones (1996)) but with kernel weights assigned in a neighborhood of $T=t_0$. We discuss the properties of the local correlation function and the varing local correlation function and derive some asymptotic properties. Finaliy, we verify the theoretical results through simulations and a real-data example.