Download Posterior Distribution - Comparative Statistical Inference - Exam and more Exams Statistics in PDF only on Docsity!
PRIFYSGOL ABERYSTWYTH UNIVERSITY
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 2 EXAMINATIONS, MAY 2009
MA36010 ñComparative Statistical Inference
Time allowed ñ2 hours
All questions may be attempted.
Marks gained from questions in Section B will be given greater consider- ation in assessing a Örst class performance.
Calculators are permitted, provided they are silent, self-powered, without communications facilities, and incapable of holding text or other mater- ial that could be used to give a candidate an unfair advantage. They must be made available on request for inspection by invigilators, who are authorised to remove any suspect calculators.
An information sheet is appended. Statistical tables will be provided.
Formulae You may assume without proof
(i) that the posterior distribution for derived from a Nor- mal random sample with a Normal prior is also Normal with
E[jX] =
nx ^2
a b^2
V ar[jX]
where V ar[jX] = [ n ^2
b^2
] ^1 :
(ii) the identity
P
P (xi^ ^ )^2 =^ Sxx^ +^ n(x^ ^ )^2 ;^ where^ Sxx^ = (xi x)^2.
Section A
- Let R~ Bin( 15 ; p) and suppose we observe R = 12 successes.
(a) Write down the likelihood and Önd the value of the maximum likelihood estimate (mle) of p. [5] (b) A Bayesian statisticianís prior beliefs about p are that P (p = 0:6) = 0 : 5 , P (p = 0:7) = 0: 3 and P (p = 0:8) = 0: 2. Find her posterior probability of each p. [6] (c) Describe how the data has changed her beliefs about p. [2]
- Y 1 ; Y 2 ; :::Yn is a random sample from a distribution with probability mass function py =
^2 y^ e
2
y!
; y = 0; 1 ; 2 ; :::
(a) Find the likelihood function for and show that the score function is V =
2 S
2 n, where S =
X^ n
i=
Yi. [5]
(b) Deduce E[S] and show that Fisherís Information is 4 n. [5] (c) Find the maximum likelihood estimator of and its asymptotic dis- tribution. [5] (d) If T is an unbiased estimator of and has variance 3 ^2 =n, what is its e¢ ciency relative to the minimum variance bound? [2]
- Fourteen sensors take independent measurements of the temperature in- side a nuclear reactor, T , (in 1000 C) where Ti~N (; 0 :0175). The aver- age temperature is observed to be 1750C while extensive past experience suggests that an appropriate prior for is N (1: 5 ; 0 :0025).
(a) Evaluate the posterior distribution of : [4] (b) Calculate (i) the posterior median of ;(ii) the 99.9% posterior per- centile. [4] (c) If (1600; 1750) is quoted as a 100(1 )% posterior credible interval for 1000 , what is the value of? [3]
- The statistic T has a sampling distribution whose cumulative distribution function (cdf) is F (t) = 1 exp[ t^3 ] where t > 0 and is an unknown parameter.
(a) Show that V = T 3 is a pivotal function for : [4] (b) Derive in terms of T a 95% conÖdence lower bound for . [4]
- The random sample X 1 ; X 2 ; : : : ; X 27 are distributed as Normal N (; 9) where is unknown.
(a) Show that the maximum value of the likelihood is (18) ^27 =^2 expf 181 Sxxg where Sxx =
P
(xi x)^2 : [4] (b) Calculate a 10% relative likelihood interval for : [4] (c) What level of conÖdence would a classical statistician attach to the interval in b.? [3] (d) A Bayesian Statistician, who has a uniform prior for ; interprets the interval in b. as a 100(1 )% credible posterior interval for . Find. [3] (e) Describe the di§erent interpretations the classical and Bayesian sta- tisticians in c. and d. would place on the interval. [2]
- DeÖne the likelihood ratio for testing two simple hypotheses, H 0 and H 1. Distinguish between the size ( or signiÖcance level) and the power of a test with critical region C. State the Neyman-Pearson Lemma. [5] Suppose R~Bin(n; p) and it is required to test H 0 : p = p 0 versus H 1 : p = p 1 where p 1 > p 0. Show that the most powerful test has a critical region of the form C = fR kg. [5]
