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Unformatted text preview: 0530001 UNIVERSITY OF YORK BA, BSc and MMath Examinations 2007 MATHEMATICS Bayesian Statistics Time Allowed: 1 1 2 hours. Answer three questions. Non-alphanumeric calculators may be used. Candidates are provided with (i) copies of Statistics Tables by H R Neave (George Allen and Unwin 1978/Routledge 1993) (ii) Appendices A and B from Bayesian Statistics: An Introduction by P M Lee (Arnold 1997) Page 1 (of 3) Turn over 0530001 1 (of 4). (a) (13 marks) Show that the mean and variance of the Rayleigh distribution which has density p ( x ) = 2 αx exp(- αx 2 ) (0 < x < ∞ ; α > 0) are given by E x = 1 2 radicalbigg π α and V x = 1 α parenleftBig 1- π 4 parenrightBig . (b) (12 marks) By considering Taylor series expansions show that if g is a reasonably smooth function and z = g ( x ) then to a reasonable approxi- mation E z = g ( E x ) and V z = V x [ g ′ ( E x )] 2 . (c) (5 marks) Show that if x has a Rayleigh distribution as above and g ( x ) = log x then to a good approximation the variance of g ( x ) does not depend on the parameter α . 2 (of 4). (i) (5 marks) Explain what is meant by saying that a statistic t = t ( x ) is sufficient for a parameter θ given data x . (ii) (5 marks) State Neyman’s Factorization Theorem. (iii) (12 marks) Prove Neyman’s Factorization Theorem. (iv) (8 marks) Suppose that x = ( x 1 , x 2 , . . ., x n ) consists of independent observations x i each of which has a Poisson distribution P( λ ) of mean λ . Find a statistic t which is sufficient for λ given x . 3 (of 4). (i) (8 marks) Define Fisher’s measure I ( θ | x ) of the information provided by an experiment and give without proof an alternative expression for it. (ii) (10 marks) Show how Jeffreys defined a prior distribution p ( θ ) which does not depend on an arbitrary choice of the scale θ in which the unknowns are measured in terms of Fisher’s measure of information. (iii) (6 marks) Find a transformation ψ = ψ ( θ ) which is so defined in terms of I ( θ | x ) that the Jeffreys prior p ( ψ ) is uniform in ψ . (iv) (6 marks) Find the Jeffreys prior for the parameter α of the Maxwell distribution p ( x | α ) = radicalbig (2 /π ) α 3 / 2 x 2 exp(- 1 2 αx 2 ) and find a transformation of this parameter in which the corresponding prior is uniform....
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