page_title: Variance function
section_title: Non-parametric regression analysis
hierarchical_section_title: Variance function / Types / Non-parametric regression analysis
caption_attribution_description: English: A scattor plot of years in the major league against salary (x$1000). The line is the trend in the mean. The plot demonstrates that the variance is not constant.
context_page_description: In statistics, the variance function is a smooth function which depicts the variance of a random quantity as a function of its mean. The variance function plays a large role in many settings of statistical modelling. It is a main ingredient in the generalized linear model framework and a tool used in non-parametric regression, semiparametric regression and functional data analysis. In parametric modeling, variance functions take on a parametric form and explicitly describe the relationship between the variance and the mean of a random quantity. In a non-parametric setting, the variance function is assumed to be a smooth function.
context_section_description: Non-parametric estimation of the variance function and its importance, has been discussed widely in the literature
  In non-parametric regression analysis, the goal is to express the expected value of your response variable(y) as a function of your predictors (X). That is we are looking to estimate a mean function,  without assuming a parametric form. There are many forms of non-parametric smoothing methods to help estimate the function . An interesting approach is to also look at a non-parametric variance function, . A non-parametric variance function allows one to look at the mean function as it relates to the variance function and notice patterns in the data.
  An example is detailed in the pictures to the right. The goal of the project was to determine (among other things) whether or not the predictor, number of years in the major leagues (baseball,) had an effect on the response, salary, a player made. An initial scatter plot of the data indicates that there is heteroscedasticity in the data as the variance is not constant at each level of the predictor. Because we can visually detect the non-constant variance, it useful now to plot , and look to see if the shape is indicative of any known distribution. One can estimate  and  using a general smoothing method. The plot of the non-parametric smoothed variance function can give the researcher an idea of the relationship between the variance and the mean. The picture to the right indicates a quadratic relationship between the mean and the variance. As we saw above, the Gamma variance function is quadratic in the mean.
caption: A scattor plot of years in the major league against salary (x$1000). The line is the trend in the mean. The plot demonstrates that the variance is not constant.
exif: {}