Final Examination. STA 215: Statistical Inference. Saturday, 2001 May 5, 9:00am 12:00 noon

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Lionel Farmer
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1 Final Examination Saturday, 2001 May 5, 9:00am 12:00 noon This is an open-book examination, but you may not share materials. A normal distribution table, a PMF/PDF handout, and a blank worksheet are attached to the exam. If you don t understand something in one of the questions feel free to ask me, but please do not talk to each other. You may use a calculator but not a PDA or laptop computer. You must show your work to get partial credit. Unsupported answers are not acceptable, even if they are correct. It is to your advantage to write your solutions as clearly as possible, since I cannot give credit for solutions I do not understand. Good luck. Print Name: 1. /20 2. /20 3. /20 4. /20 5. /20 6. /20 X. /00 Total: /120

2 Problem 1: Let X j Be(θ, 1) be independent beta-distributed random variables with density function f(x θ) = θ x θ 1, 0 < x < 1, for some θ Θ = (0, ) and answer the following questions: a. (5) Find the mean µ X = E[X θ] and variance σ 2 X = Var[X θ]: µ X = σ 2 X = b. (5) Change variables to find the probability density function for Y ln X: f Y (y θ) = c. (5) Give the mean and variance for Y ln X: µ Y = σ 2 Y = d. (5) Find the Fisher Information I X for a single observation of X: I X = Spring May 5, 2001

3 Problem 2: Again let X j Be(θ, 1) be independent beta-distributed random variables with density function f(x θ) = θ x θ 1, 0 < x < 1, (1) and let x = (x 1,..., x n ) X = (0, 1) n be a random sample of size n: a. (5) Find the likelihood function for θ upon observing x X. L(θ x) = b. (15) If this is an Exponential Family, find the canonical parameter η(θ), the canonical sufficient statistic T ( x) = T n ( x), and the normalizing function B(θ) in the Exponential Family representation η(θ) Tn( x) n B(θ) f( x θ) = h n ( x)e for a random sample of size n (you need not give h n ( x) = h(x j )). If it is not an exponential family, explain why. T n ( x) = η(θ) = B(θ) = Spring May 5, 2001

4 Problem 3: With the same Beta Be(θ, 1) model as above, and with x = (x 1,..., x n ) X = (0, 1) n a random sample of size n, a. (10) Show that the Gamma θ Ga(α, β) distribution with density π(θ) β α θ α 1 e β θ /Γ(α), θ > 0 is conjugate for x and find α, β (which may depend on α, β, n, and x) such that π(θ x) Ga(α, β ): α = β = b. (10) Find the Jeffreys prior density function π J (θ) and give the Jeffreys posterior distribution (upon observing x X ), either by giving its pdf or (better) by giving the distribution s name and parameters: π J (θ) π J (θ x) Spring May 5, 2001

5 Problem 4: With the same Beta Be(θ, 1) model as above, and again with x = (x 1,..., x n ) X = (0, 1) n a random sample of size n, a. (5) Find the MLE ˆθ: ˆθ = b. (5) Find a Method of Moments (MOM) estimator θ (Hint: Solve 1(a) for θ as a function of µ X, then substitute x for µ X ): θ = c. (5) Find the posterior mean θ for a Bayesian analysis with Gamma prior distribution θ Ga(α, β): θ = d. (5) The Raô-Blackwell Theorem says that one of these three estimators could be improve to reduce its risk. Which one, and how does Raô-Blackwell suggest it may be improved? Be specific (use 2b!), but no calculations are needed. Spring May 5, 2001

6 Problem 5: With the same Beta Be(θ, 1) model as above, and with n = 10 observations x = (x 1,..., x 10 ) satisfying S 1 n j=1 x j = S 2 n j=1 x j 2 = S 3 n j=1 ln(x j ) = S 4 n j=1 1/x j = S 5 min 1 j n x j = S 6 max 1 j n x j = a. (10) Find an expression for the Jeffreys posterior probability of the hypothesis H 0 : [θ 1]. Give the answer as your choice of either S-plus or Mathematica commands or as an explicit onedimensional integral expression: π J (H 0 x) = b. (10) Perform a significance test of the hypothesis H 0 : [θ 1] against the alternative H 1 : [θ > 1]. Give the p-value as your choice of either S-plus or Mathematica commands or an explicit one-dimensional integral expression (Hint: what is the distribution of your sufficient statistic T n from 2b?) P = Spring May 5, 2001

7 Problem 6: Ron thinks that the recorded failure times t j of felt dryboard marking pens have an Ex(1) distribution, with mean one hour, while Tom thinks they have a Ga(2, 2) distribution, also with mean one hour. Consider the following statistics, each based on an independent sample t = {t j } of size n: S 1 = n j=1 t j S 2 = n j=1 t j 2 S 3 = n j=1 ln(t j ) S 4 = n j=1 1/t j S 5 = min 1 j n t j S 6 = max 1 j n t j (20) How should Ron and Tom resolve their dispute? Suggest and defend a statistic based on one or more of the statistics S k = S k ( t n ) above and a procedure for deciding whether t j Ex(1) or t j Ga(2, 2). Describe briefly what computation would be required. Please present both a (10) Frequentist and a (10) Bayesian solution. Spring May 5, 2001

8 Problem X: Extra Credit: a) (+5) (Problem 6, revisited). Use a normal approximation if necessary to complete the analysis (either one) you began in Problem 6; decide whether Ron or Tom is correct, and defend your choice, based on the following data for n = 100 observations: S 1 n j=1 t j = S 2 n j=1 t j 2 = S 3 n j=1 ln(t j ) = S 4 n j=1 1/t j = S 5 min 1 j n t j = S 6 max 1 j n t j = b) (+5) (Problem 6, revisited). Set f 0 (t) = e t and f 1 (t) = 4t e 2t for t > 0, the density functions Ron and Tom believe govern {t j } in Problem 6. One can compute K(f 0, f 1 ) 0 ln f 0(t) f f 1 (t) 0(t) dt = 2 ln 2 ψ(1) K(f 1, f 0 ) 0 ln f 1(t) f f 0 (t) 1(t) dt = ln 2 + ψ(2) Approximately how large a sample would be required to resolve Ron and Tom s debate (at least, to be 99% sure), and why? c) (+0) What are Ron and Tom s last names? Spring May 5, 2001

9 Extra worksheet, if needed: Spring May 5, 2001

10 ... Name: Φ(x) = x 1 2π e t2 /2 dt: x Table 5.1Area Φ(x) under the Standard Normal Curve to the left of x. x

Midterm Examination. Mth 136 = Sta 114. Wednesday, 2000 March 8, 2:20 3:35 pm

Midterm Examination. Mth 136 = Sta 114. Wednesday, 2000 March 8, 2:20 3:35 pm Midterm Examination Mth 136 = Sta 114 Wednesday, 2000 March 8, 2:20 3:35 pm This is a closed-book examination so please do not refer to your notes, the text, or to any other books. You may use a two-sided