When to use linear regression with heavy tails?

When to use linear regression with heavy tails?

Eawhen the error has a symmetric distribution with heavy tails. The results of our paper are exemplary rather than analytical. They describe the outcomes of an initial exploration on estimators for linear regression with heavy tails. The numerical results in the tables in Section 6 may be regarded as benchmarks.

What is the quadratic loss function for heavy tails?

Their performance for heavy tails is compared below on the basis of a quadratic loss function. The case where X= 1=Ufor a uniform variable U on (0;1) and where Y has a Cauchy distribution plays a central role, but heavier and lighter tails are also considered.

How are explanatory variables used in linear regression?

In linear regression the explanatory variables are often assumed to be equidistant on an interval. If the values are random they may be uniformly distributed over an interval or normal or have some other distribution. In the paper below the explanatory variables are random.

Are there any estimators of the regression line?

Abstract There exist several estimators of the regression line in the simple linear regression Y i= b+ aX i+ Y i i= 1;:::;n: Least Squares, Least Absolute Deviation, Right Median, Theil-Sen, Weighted Bal- ance, Least Trimmed Squares.

How is a weighted balance estimator used in linear regression?

A weight sequence is used to select a bisector which yields a good estimate of the regression line. Weighted balance estimators for linear regression may be likened to the median for univariate samples. The LAD (Least Absolute Deviation) estimator is a weighted balance estimator.

What are the problems with a heavy tailed distribution?

Outliers, as well skewed or heavy-tailed distributions, also affect the ordinary least squares regression estimator. In some ways the practical problems that arise are even more serious than those associated with the ANOVA F test. Consider two random variables, X and Y, and suppose

What does normal probability plot of residuals look like?

Here’s the corresponding normal probability plot of the residuals: This is a classic example of what a normal probability plot looks like when the residuals are normally distributed, but there is just one outlier. The relationship is approximately linear with the exception of the one data point.

How to calculate the probability of a regression?

Consider a simple linear regression model fit to a simulated dataset with 9 observations, so that we’re considering the 10th, 20th., 90th percentiles.

When is the condition of normality of the OLS residuals not?

The condition of normality of the residuals is useful when residuals are also homoskedastic. The result is then that OLS has the smallest variance between all of the estimator (linear OR non-linear). The extended OLS assumptions: $E(u|X_i = x) = 0$. $(X_i,Y_i), i=1,…,n,$ are i.i.d. Large outliers are rare. u is homoskedastic.

What are the assumptions for linear OLS regression?

In order to present the context for non-normality it may help to review the assumptions for linear OLS regression, which are: Weak exogeneity. This essentially means that the predictor variables, x, can be treated as fixed values, rather than random variables.