Final Examination a. STA 532: Statistical Inference. Wednesday, 2015 Apr 29, 7:00 10:00pm. Thisisaclosed bookexam books&phonesonthefloor.

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1 Final Examination a STA 532: Statistical Inference Wednesday, 2015 Apr 29, 7:00 10:00pm Thisisaclosed bookexam books&phonesonthefloor Youmayuseacalculatorandtwo pagesofyourownnotes Do not share calculators or notes Show your work Neatness counts Boxing answers helps Simplify all expressions for full credit No unevaluated sums, integrals, maxima, or unreduced fractions Dist n & pdf/pmf tables and blank worksheet are attached Good luck! Print Name Clearly:

2 1 /20 5 /20 2 /20 6 /20 3 /20 7 /20 4 /20 8 /20 /80 /80 Total: /160

3 Problem 1: The random variables {X i } iid Po(θ) have the Poisson distribution with mean θ, for 1 i n a) (2) Find the Likelihood Function Simplify! L(θ x) = b) (3) Find the Fisher Information for a sample of size n and the Reference (Jeffreys Rule) prior density: I n (θ) = π J (θ) c) (4) Find the posterior distribution for this sample & prior in terms of the sample sum S = n i=1 X i, by giving its name and parameter values: π J (θ x) d) (4) Find the Variance and Bias for the posterior mean θ J n := E J (θ x): Var( θ J n ) = β(θ) = e) (4) Find the squared-error Risk Function for this estimator: R( θ J n,θ) = f) (3) Does θ J attain the Information Inequality lower bound for biased estimators? Yes No Why? Spring Apr 29, 8:00 10:00pm

4 Problem 2: Let {X j } iid Ex(θ) for 1 j n a) What is the likelihood function? (Simplify!) f(x n θ) b) What is a one-dimensional sufficient statistic S(x n )? S(x n ) = c) Using a Gamma prior distribution θ Ga(α,λ), find the posterior distribution, by name & parameter values: π(θ x n ) d) Find the predictive pdf for one additional observation, using the same Ga(α, λ) prior: f(x n+1 x n ) = Spring Apr 29, 8:00 10:00pm

5 Problem 3: In each part of this problem the following statistics are from a random sample of n iid random variables {X i } from some specified probability distribution, but the specific distribution may vary: 1 1 Xi =A n logxi =C maxx n i =E 1 (Xi X n n ) 2 1 =B log(1 Xi )=D minx n i =F Ineachcase, findastatistict thatisafunctionofoneormoreofn,a,b,c,d,e,f for which the Rejection Region of the indicated LRT or GLRT is of the form R = {x : T(x) t} Express T explicitly as a function of n,a,b,c,d,e,f Note T isn t unique; full credit for any function that works Simplify! a) {X i } iid Po(θ); H 0 : θ = 3 vs H 1 : θ = 2 T = b) {X i } iid Be(α,β); H 0 : α = 1,β = 2 vs H 1 : α = 2, β = 1 T = c) {X i } iid Ga(α,4); H 0 : α = 1 vs H 1 : α = 4 T = d) {X i } iid Ga(α,λ); H 0 : α = 1 vs H 1 : α = 4 (λ unknown) T = e) H 0 : {X i } iid Ex(2) vs H 1 : {X i } iid Ga(2,1) T = Spring Apr 29, 8:00 10:00pm

6 Problem 4: Random variables {X i } iid No(µ 1,σ 2 ) and {Y i } iid No(µ 2,σ 2 ) are normally distributed for 1 i n = 12 with the same (unknown) variance, but the X i sandy i smight not beindependent of eachother Their differences d i = (X i Y i ) are also normally distributed and iid Summary statistics for these n = 12 pairs include: Xi = 18 (Xi X n ) 2 =110 Yi = 24 di = 6 (Yi Ȳn) 2 = 88 (di d n ) 2 =154 a) Find a 90% Confidence Interval for µ 1 b) A90%ConfidenceIntervalforµ 2 basedonthesedatais[05337,34663] Explain precisely what event it is that has probability 90% c) Find a 90% Confidence Interval for (µ 1 µ 2 ), under the assumption that the X i s and Y i s are independent, so all the (X i µ 1 ) and (Y i µ 2 ) are iid No(0,σ 2 ) Spring Apr 29, 8:00 10:00pm

7 Problem 4 (cont d): Still for 1 i 12 the X i s, Y i s, and d i = (X i Y i ) have Xi = 18 (Xi X n ) 2 =110 Yi = 24 di = 6 (Yi Ȳn) 2 = 88 (di d n ) 2 =154 d) Find a 90% Confidence Interval for (µ 1 µ 2 ), if the X i s and Y i s are expected to be positively correlated 1 e) One way to test the hypothesis H 0 : µ 1 = µ 2 against H 1 : µ 1 µ 2 is to Reject if zero is not in the confidence interval you found in parts c) or d) above Which test would be more powerful? The one based on interval from: Explain: c) d) Both the same power It depends 1 For example, (X i,y i ) might be mileages for the ith car on two different tries; or responses of the ith subject to two different drug treatments; or sunburn scores on left and right arms of ith subject in a sunscreen test Note there is no need to know the correlation coefficient it should not appear in your answer Spring Apr 29, 8:00 10:00pm

8 Problem 5: Let X Bi(n,p) be the number of successes in n independent trials, all with success probability p, and let φ := log ( p 1 p) be the log odds a) (5) Find the maximum likelihood estimator for φ: ˆφ(x) = b) (10) For large n, ˆφ(X) has approximately a normal distribution Use the delta method to find the approximate mean and variance µ x = σ 2 x = c)(5) One way to quantify the difference between the success probabilities p and q for independent Binomial trials with X Bi(n,p) and Y Bi(m,q) is through the log odds ratio ε := log { p 1 p } 1 q, q which will be zero if p = q but positive (resp, negative) if p > q (resp, p < q) The MLE of this too will be approximately normally-distributed Give the mean and variance: E[ˆε] Var[ˆε] Spring Apr 29, 8:00 10:00pm

9 Problem 6: Sandy, a lazy student, is assigned the task of rolling a(possibly loaded) die 60 times, and reporting the outcomes Sandy reports finding: 1 s 2 s 3 s 4 s 5 s 6 s a) We wish to test H 0 = [ these are 60 rolls of a fair die ] Give and evaluate a test statistic T(x) that will typically be small if H 0 is true and large under the alternative H 1 = [ die is unfair ]: T(x) = b) Give the approximate probability distribution of your test statistic T(x), if H 0 is true, by specifying its name and parameter value(s): T(x) c) Give an expression (either an integral or a bit of R code) for the P-value for H 0 Would you reject H 0 at level α = 010? Why? You should be able to answer this without a table or computer P = Reject? Yes No d) Sandy s instructor is suspicious that Sandy didn t actually perform the assigned task Why? Spring Apr 29, 8:00 10:00pm

10 Problem 7: {X i } iid 1 i n Pa(θ, 1) have the generalized Pareto distribution with shape θ > 0 and unit scale, with pdf and CDF for x > 0 f(x θ) = θ(1+x) θ 1 F(x θ) = 1 (1+x) θ a) Find a one-dimensional sufficient statistic for a sample of size n: T(x) = b) Express the likelihood function in terms of T(x) L(θ) = c) Find a conjugate parametric family of prior distributions d) Give the posterior parameters, in terms of the prior parameters, n, and the statistic T e) Give the posterior mean using the improper prior π(θ) = 1 {θ>0}, for a single observation of X 1 = 1 E[θ X 1 = 1] = Spring Apr 29, 8:00 10:00pm

11 Problem 8: a T F For two points each, True or False? No explanations needed If X χ 2 8 and Y are independent, and if X +Y χ2 12, then Y χ 2 4 b T F The best test of H 0 : {X i } iid f 0 (x) vs H 1 : {X i } iid f 1 (x) will reject H 0 for large values of T(x) := i log( f 1 (x i )/f 0 (x i ) ) c T F If ˆθ(x) is an unbiased estimator of θ, then eˆθ(x) is an unbiased estimator of e θ d T F If we observe one success in two independent tries with a uniform prior for the success probability θ [0,1], then the posterior probability distribution is also uniform on [0, 1] e T F If T(x) is a sufficient statistic, then so is S(x) := e T(x) f T F If ˆθ(x) is the MLE of θ, then eˆθ(x) is the MLE of e θ g T F In regular statistical models the MLE ˆθ n is approximately normally distributed for large n, with mean θ and variance I n (θ) 1 h T F i T F For small normal samples with known variance, use the t distribution to construct confidence intervals or hypothesis tests about the mean [17,42] is a 90% Confidence Interval for a parameter θ if and only if P[17 θ 42] = 090 j T F If {X i } iid Ga(7,λ) then X i is sufficient for λ k T F An estimator δ with MSE Q and bias β has variance Q β 2 Spring Apr 29, 8:00 10:00pm

12 Extra worksheet, if needed:

13 STA 532 Normal Distribution Table 11 Φ(x) = x 1 2π e z2 /2 dz: x Table 51Area Φ(x) under the Standard Normal Curve to the left of x x Φ(06745) = 075 Φ(16449) = 095 Φ(23263) = 099 Φ(30902) = 0999 Φ(12816) = 090 Φ(19600) = 0975 Φ(25758) = 0995 Φ(32905) = 09995

14 Critical Values for Student s t p = tp c dt p ց (1+t 2 /ν) (ν+1)/ t p ν t 60 t 70 t 80 t 85 t 90 t 95 t 975 t 99 t 995 t 999 t 9995 t

15 Critical Values for χ 2 α = α ւ cx ν/2 1 e x/2 dx χ 2 α χ 2 α ν χ 2 50 χ 2 25 χ 2 10 χ 2 05 χ χ 2 01 χ χ χ χ

16 Name Notation pdf/pmf Range Mean µ Variance σ 2 Beta Be(α,β) f(x) = Γ(α+β) Γ(α)Γ(β) xα 1 (1 x) β 1 x (0,1) α α+β αβ (α+β) 2 (α+β+1) Binomial Bi(n,p) f(x) = ( n x) p x q (n x) x 0,,n np npq (q = 1 p) Exponential Ex(λ) f(x) = λe λx x R + 1/λ 1/λ 2 Gamma Ga(α,λ) f(x) = λα Γ(α) xα 1 e λx x R + α/λ α/λ 2 Geometric Ge(p) f(x) = pq x x Z + q/p q/p 2 (q = 1 p) HyperGeo HG(n,A,B) f(x) = (A x)( B f(y) = pq y 1 y {1,} 1/p q/p 2 (y = x+1) n x) ( A+B n ) x 0,,n np np (1 P) N n N 1 Logistic Lo(µ,β) f(x) = e (x µ)/β β[1+e (x µ)/β ] 2 x R µ π 2 β 2 /3 (P = A A+B ) Log Normal LN(µ,σ 2 ( 1 ) f(x) = x /2σ 2 x R 2πσ 2e (logx µ)2 + e µ+σ2 /2 e 2µ+σ2 e σ 1 ) 2 Neg Binom NB(α,p) f(x) = ( ) x+α 1 x p α q x x Z + αq/p αq/p 2 (q = 1 p) f(y) = ( y 1 y α) p α q y α y {α,} α/p αq/p 2 (y = x+α) Normal No(µ,σ 2 ) f(x) = 1 2πσ 2 e (x µ)2 /2σ 2 x R µ σ 2 Pareto Pa(α,ǫ) f(x) = (α/ǫ)(1+x/ǫ) α 1 x R + ǫ α 1 f(y) = αǫ α /y α+1 y (ǫ, ) ǫα α 1 ǫ 2 α (α 1) 2 (α 2) ǫ 2 α (α 1) 2 (α 2) Poisson Po(λ) f(x) = λx x! e λ x Z + λ λ Snedecor F F(ν 1,ν 2 ) f(x) = Γ(ν 1 +ν 2 2 )(ν 1 /ν 2 ) ν 1 /2 x ν Γ( ν 1 2 )Γ( ν 2 2 ) [ 1+ ν 1 ν 2 x ] ν 1 +ν 2 2 x R + ν 2 ν 2 2 ( ν2 ν 2 2 Student t t(ν) f(x) = Γ(ν+1 2 ) Γ( ν 2 ) πν [1+x2 /ν] (ν+1)/2 x R 0 ν/(ν 2) Uniform Un(a,b) f(x) = 1 a+b b a x (a,b) 2 Weibull We(α,β) f(x) = αβx α 1 e βxα x R + Γ(1+α 1 ) β 1/α (b a) 2 12 Γ(1+2/α) Γ 2 (1+1/α) β 2/α ) 2 2(ν1 +ν 2 2) ν 1 (ν 2 4) (y = x+ǫ)