The Solution According Finite Mixture Distribution by GMM Problem as One of the Modes of Expression of a Probability Density Function

Probability density estimation is an important tool in Data Analysis and many other areas, where it is often used for exploratory data analysis or as a part of another estimator. However, the population who can express to distribution of a beautiful form which appears in the statistical textbook can hardly be found out. Then it is a problem why the probability density function is expressed. The estimation of a probability density function based on a sample of independent identically distributed observations is essential in a wide range of applications. The estimation method of Probability Density Function -- (1) a parametric method (2) a nonparametric method and (3)a semi-parametric method etc. -- it is. I n this paper, GMM problem is taken up as a semi- parametric method and We use a wavelet method as a powerful new technique. Compactly supported wavelets are particularly interesting because of their natural ability to represent data with intrinsically local properties.