Pairs of random variables (Xi, Yi) were observed. They were assumed to follow a linear regression with E(Yi|Xi) = θ1+θ2Xi but with t-distributed noise, rather than the usual normally distributed noise. More specifically, the assumed model was that conditionally, given Xi, Yi is t-distributed with mean θ1 + θ2Xi, standard deviation θ3, and degrees of freedom θ4. Also, the pairs (X1, Y1),...,(Xn, Yn) are mutually independent. The model could also be expressed as
with mean 0 and standard deviation θ3 and degrees of freedom θ4. The model was fit by maximum likelihood. The R code and output are
(a) What is the MLE of the slope of Yi on Xi?
(b) What is the standard error of the MLE of the degrees-of-freedom parameter?
(c) Find a 95 % confidence interval for the standard deviation of the noise.
(d) Did optim converge? Why or why not?