STAT 431/831 Practice for Midterm
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1. In a study undertaken to model the reliability of car engines by the motor industry, the time
to failure of these engines was modelled by the density function
f(t) = 4
µ2 te!2t/µ t ! 0
(a) Show that this density is a member of the exponential family of distribution. Specify the
canonical parameter ✓, the scale parameter ", and the functions a(·), b(·), c(·; ·).
(b) Using the results from (a) find E[T ] and Var[T ]. What is the canonical link?
(c) Let Ti be a random variable following the above distribution with parameter µi, i =
1 . . . , n independently. Suppose there is a p ⇥ 1 vector of explanatory variables xi =
(xi0, xi1, . . . , xi,p!1)T associated with Ti, for i = 1, . . . , n. Derive the score vector using
the inverse link 1/µ = ⌘.
(d) (STAT 831 ONLY) Let ˆ µi be the mle’s under some constrained model of dimension p
(p < n). Find the deviance statistic D. Define any notation you introduce. State the
asymptotic distribution of D.