Linear Regression with Non-Normal Error Terms

TLDR: In this article, the authors investigate the model y,-=a + bx, +,l i6 = 1,..., n, where the manifest observations are the (xi, y) pairs, and where the error random variables, the ui's, are independent, identically distributed according to (1) with,u = 0.-00 O and 9 > 0.

Abstract: -00 O and 9 > 0. (1c) The density functions are centered about the location parameter, p; ais the scale parameter. The parameter 0 measures the degree of peakedness (kurtosis) of the distribution. The multiplicative constant, k (a-, 0), which incorporates the well-tabulated gamma function, serves as a normalizing factor to insure that the area under the density curve is one.' For the normal distribution 0 = 2; 0 = 1 gives the double exponential distribution; where 0 tends to oo, the distribution tends to the rectangular. We shall investigate the model y,-=a + bx, + ,l i6 = 1, . .. , n, where the manifest observations are the (xi, y,) pairs, and where the error random variables, the ui's, are independent, identically distributed according to (1) with ,u = 0. The parameters of the model 2 are a, b, a-, and 0. Given the observations, the likelihood of the parameters is found by multiplying together the likelihoods for the individual error random variables. The likelihood of the parameters given the entire sample is