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PRIFYSGOL ABERYSTWYTH/ABERYSTWYTH UNIVERSITY
ARHOLIADAU / EXAMINATIONS
SEFYDLIAD MATHEMATEG A FFISEG
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 2 EXAMINATION
MA36010: Comparative
Time allowed: 2 hours
All questions may be attempted
Marks gained from questions in Section B will be given greater consideration in assessing a first class performance.
Calculators are permitted, provided they are silent, self communications facilities, 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.
Statistical Tables will be provided
PRIFYSGOL ABERYSTWYTH/ABERYSTWYTH UNIVERSITY
ARHOLIADAU / EXAMINATIONS
SEFYDLIAD MATHEMATEG A FFISEG
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 2 EXAMINATION MAI / MAY MEHEFIN/JUNE
Comparative Statistical Inference
All questions may be attempted Marks gained from questions in Section B will be given greater consideration in assessing a first class performance. Calculators are permitted, provided they are silent, self-powered, without communications facilities, and incapable of holding text or other material 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. l be provided
PRIFYSGOL ABERYSTWYTH/ABERYSTWYTH UNIVERSITY
MEHEFIN/JUNE 2010
Marks gained from questions in Section B will be given greater consideration in
powered, without and incapable of holding text or other material 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.
Section A
- Suppose that we observe 36 arrivals in 20 minutes where N(t) , the number in t minutes is Poisson, Po(λt). a. Write down the likelihood and find the maximum likelihood estimate (mle) of λ. [5] b. A Bayesian statistician’s prior beliefs about λ are that P (λ = 2) = 0.2 and P (λ = 3) = 0.8. Find her posterior probability of each λ and say how the observations have altered her beliefs about the arrival rate. [8]
- The likelihood associated with a set of n observations has been calculated as
䙲⡰ゕ䙳
㊉ ㄘ 〢ㄘ ㄉㄙ㊉^ exp 䙨㎘^
〤 ⡰ㄉㄘ} for statistics U and W with φ > 0. a. Identify a sufficient statistic for φ. [2] b. Calculate the score function and deduce from it the expected value of one of these statistics. [5] c. Calculate the minimum variance bound (MVB) for unbiased estimation of φ. [5] d. An estimator T, which is unbiased for φ, is known to have variance φ^2 /n. What is its efficiency relative to the MVB? [2]
- In a trial of a new drug for asthmatic patients, 32 out of 36 were observed to have their symptoms successfully alleviated within 1 minute of treatment. A Bayesian statistician has a Beta prior, β(4,2), for the probability of success, p. a. Assuming a Binomial distribution for the number of patients successfully treated, write down the likelihood and the prior density function and deduce the posterior distribution. [5] b. Explain why the Beta family is said to be conjugate for this situation. [2] c. Calculate the posterior mean and standard deviation of p. [3]
- The statistic T has a sampling distribution such that 〡む ᔳ ‱䙰⡩⡲䙱⡰^ , where α is an unknown
parameter. a. What property of 〡む makes it a pivotal function? [1] b. Using Statistical Tables, evaluate the confidence coefficient of the interval 䙦0.04222 㐀 ᡆ, 0.12837 㐀 ᡆ䙧. [4] c. Derive in terms of T a 99% confidence lower bound for α. [3]
- Disgraced banker, Sir Freddie Saversall, is considering an alternative career in politics. He has evaluated his posterior distribution for p , the probability that he is elevated to the House of Lords, thus becoming Lord Saversall. If this is a Beta distribution, β(2,6), calculate a 95% credible posterior upper bound for p. [7]
