Variance function

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.

In a regression model setting, the goal is to establish whether or a not a relationship exists between a response variable and a set of predictor variables. Further, if a relationship does exist, the goal is then to be able to describe this relationship as best as possible. A main assumption in linear regression is constant variance or (homoscedasticity), meaning that different response variables have the same variance in their errors, at every predictor level. This assumption works well when the response variable and the predictor variable are jointly Normal, see Normal distribution. As we will see later, the variance function in the Normal setting, is constant, however, we must find a way to quantify heteroscedasticity (non-constant variance) in the absence of joint Normality.

When it is likely that the response follows a distribution that is a member of the exponential family, a generalized linear model may be more appropriate to use, and moreover, when we wish not to force a parametric model onto our data, a non-parametric regression approach can be useful. The importance of being able to model the variance as a function of the mean lies in improved inference (in a parametric setting), and estimation of the regression function in general, for any setting.

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