Weighted Least Squares With Multiplicative Conditional Variance

Linear model and generalized linear model are two kinds of widely used statistical models.Lots of phenomena can be depicted by linear model or generalized linear model in many fields.During the research of the linear model,Parameter estimation is always the most important and difficult problem.In this dissertation,we mainly study the problems of the Least Squares Estimation with some constraints in the classical Gauss-Markov model.In this paper,we start with the classic linear regression model,we discuss the case where the equal variances assumption is not satisfied in classical linear model,then we introduced generalized least squares regression.Because the generalized linear regression model does not make strict assumptions about the data,so it can be applied more widely to case where the variance structure is unknown.We mainly discuss the weighted least squares regression,Since it has one crucial disadvantage,its theoretical nature is based on the fact that the weights or conditional variance are already known.However,this would not be the case in real applications.One existing solution to solve this problem is the iterated reweighted least squares method.Based on this idea,We take the most classical Gauss-Markov model as the research object and propose the iterative kernel weighted least square method under the multiplicative conditional variance model,where the kernel smoothing method involves estimation of the conditional variance.The most important step of this algorithm is that we decompose the p-dimensional kernel problem into one-dimensional univariate kernel problems in the estimation of conditional variances.The “curse of dimensionality” is avoided and the estimation performance is improved.Simulation studies were done to confirm the efficiency of our iterated kernel weighted least squares algorithm comparing with the ordinary least squares.Moreover,the algorithm is verified to be still reliable in situations where the multiplicative assumption is violated.Finally,in order to dispel the concerns about the distribution of input variables,the algorithm continues to be modified,we can obtain the effectiveness of the modificated algorithm through comparison with the ordinary least squares.