Ph.D. Qualifying Examination in Statistics 0.35 January 2008

Let \(X_i\) be a random sample of size \(n\) with pdf \(f(x;\eta )=e^{-(x-\eta )}\) (\(x>\eta \), \(\eta >0\)).

1a) (5 points) Find the MLE \(\hat \eta \) and compute its pdf. Do you recognize it?

1b) (5 points) Is \(\hat \eta \) unbiased, asymptotically unbiased, MSE consistent, simple consistent?

Let \(X_i\) be a random sample of size \(n\) with pdf \(f(x;\theta )=\frac {1}{\theta }e^{-(x-\theta )/\theta }\) (\(x>\theta \), \(\theta >0\)).

2a) (5 points) What is the MME of \(\theta \)? Is it a reasonable estimator?

2b) (5 points) What is the MLE of \(\theta \)?

(10 points) Let \(X_i\sim \operatorname {POI}(\mu )\) be a random sample of size \(n\). Find the MLE of \(e^{-\mu }\). Is it unbiased, asymptotically unbiased, MSE consistent, simple consistent? [Hint: The MGF of \(\operatorname {POI}(\mu )\) is \(e^{\mu (e^t-1)}\).]

Let \(X_i\) be a random sample of size \(n\) with pdf \(f(x;\theta )=\theta x^{\theta -1}\) (\(0<x<1\), \(\theta >0\).)

4a) (5 points) Find a complete and sufficient statistic.

4b) (5 points) Find a uniformly minimum variance unbiased estimator (UMVUE) of \(1/\theta \).

Let \(X\) and \(Y\) be independent standard normals and consider \((U,V)=(aX+bY,cX+dY)\). Assume that at least one of \(a\) and \(b\) is nonzero, and at least one of \(c\) and \(d\) is nonzero. (Otherwise either \(U\) or \(V\) would be 0 and the problem is not interesting.)

5a) (5 points) Show that if \(ad=bc\), then \(U\) and \(V\) cannot be independent random variables.

5b) (5 points) Assume \(ad\ne bc\) and find the joint pdf of \((U,V)\).

5c) (5 points) Find a necessary and sufficient condition for \(U\) and \(V\) to be independent.

Consider the pdf \(f(x;\theta )=\theta x^{-(\theta +1)}\) (\(x\geq 1\)). Let \(X_i\) be a random sample of size \(n\) with pdf \(f(x;\theta _1)\). Let \(Y_i\) be a random sample of size \(m\) with pdf \(f(x;\theta _2)\). The two sets of random variables are independent. We wish to test \[ H_0:\theta _1=\theta _2 \quad \text {against}\quad H_a:\theta _1\ne \theta _2. \]

6a) (5 points) If one considers just the \(X\)-data, find the MLE \(\hat \theta _1\).

6b) (5 points) Derive a formula for the critical region obtained using the generalized likelihood ratio method.

Let \(X_i\sim \operatorname {EXP}(\theta )\) be a random sample of size \(n\).

7a) (5 points) Find the MGF of \(\frac {2n\bar X}{\theta }\) and identify its distribution.

7b) (10 points) Derive the generalized likelihood ratio test of \(H_0:\theta =\theta _0\) against \(H_a:\theta \ne \theta _0\).

7c) (5 points) The following data are times (in hours) between failures of air conditioning equipment in a particular airplane: 74, 57, 48, 29, 502, 12, 70, 21, 29, 386, 59, 27, 153, 26, 326. Test \(H_0:\theta =125\) against \(H_a:\theta \ne 125\).

Let \(X_i\) be a random sample of size \(n\) with pdf \(f(x;\theta )=\frac {2x}{\theta ^2}\) (\(0<x<\theta \)).

8a) (5 points) Use the MME \(\tilde \theta \) to find an unbiased estimator of \(\theta \).

8b) (10 points) Use the MLE \(\hat \theta \) to find another unbiased estimator of \(\theta \).

8c) (5 points) Using the factorization criterion, find one sufficient statistic for \(\theta \). Which of the two unbiased estimators you have found has a lower variance?

8d) (5 points) Calculate \(\operatorname {Var}(\tilde \theta )\) and \(\operatorname {Var}(\hat \theta )\). Does this confirm your answer?

(5 points) Let \(\zeta _n\) and \(\eta _n\) be two sequences of random variables. Prove or give counterexamples to the following statements:

9a) If \(\zeta _n\xrightarrow {(d)}\zeta \) and \(\eta _n\xrightarrow {(d)}\eta \), then \(\zeta _n+\eta _n\xrightarrow {(d)}\zeta +\eta \).

9b) If \(\zeta _n\xrightarrow {P}\zeta \) and \(\eta _n\xrightarrow {P}\eta \), then \(\zeta _n+\eta _n\xrightarrow {P}\zeta +\eta \).

9c) If \(\zeta _n\xrightarrow {(d)}\zeta \) and \(\eta _n\xrightarrow {(d)}\eta \), then \(\zeta _n+\eta _n\xrightarrow {P}\zeta +\eta \).

(\(\xrightarrow {(d)}\) and \(\xrightarrow {P}\) denote convergence in distribution and in probability, respectively.)