Qualifying Exam MA 582 problemsApril 2015

Problem 1.

Consider the Exp() family, where  > 0 (and note that E(X) =  here).

  1. Show that no two different parameter values could give exactly the same pmf (this is “regularity condition R0”).
  2. Is the parameter space here open (this is “regularity condition R1”)? Why or why not?
  3. Is the support of the pdf here independent of the paramter (this is “regularity condition R2”)? Why or why not?
  4. Find the maximum likelihood estimator (MLE) of, call it Yn.
  5. Show whether or not Yn is unbiased for .
  6. Show whether or not Yn is a consistent estimator for .
  7. Show whether or not Yn is asymptotically normal, and if it is, identify its asymptotic normal variance.
  8. Find I(), Fisher’s Information for Is MLE() efficient?Why or why not?
  9. Find the MLE of 2. Show whether or not it is biased.
  10. Find a function g so that n1/2( g(Yn) – g()) is asymptotically standard normal.

Problem 2.

We say the rv X has the W distribution with parameter  0 (written X ~ W() ) if X has pdf f(x,)= 3x2/, for 0 < x < , and f(x) = 0, elsewhere.

Consider the parameterized W family {W() :  > 0 }.

  1. For each of the “regularity conditions” R0, R1, and R2, determine if the W family here satisfies that condition or not. (Refer to problem 1 here for a reminder of what those regularity conditions are.)
  2. Let Yn be the maximum of the random sample of size n. Show that Yn is a consistent estimator of 
  3. Find the pdf of Yn. (Hint: Find its cdf first.)
  4. Show that Yn is NOT an unbiased estimator of 
  5. Show that n(- Yn ) converges in distribution, and find its asymptotic distribution explicitly.
  6. Find an unbiased estimator of call it Tn. Show that Tn is a consistent estimator of 
  7. Show that n(- Tn ) converges in distribution, and find its asymptotic distribution explicitly. (Hint: Use parts (e) and (f) here.)