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This is the Exam of Statistical Science which includes Applied Statistics, Independent, Medals Winners, Following List, Medals, Proportion, Russia, China, Australia etc. Key important points are: Applied Multivariate Analysis, Dimensional Vector, Distributed, Components, Conditional, Expression, Variance, Conditional, Correlation, Random Sample
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Wednesday 2 June, 2004 9:00 to 11:
Attempt THREE questions. There are five questions in total.
The questions carry equal weight.
1 Suppose that the p-dimensional vector X is distributed as Np(μ, V ).
(i) Show that if we partition X into components X 1 , X 2 , so that XT^ = (X 1 T , X 2 T ), then the covariance matrix of X 1 conditional on X 2 = x 2 is V 11 − V 12 V 22 − 1 V 21 , where
(ii) Hence or otherwise find an expression for the variance of (X 1 |X 2 = x 2 ) in terms of V −^1 , when X 1 , X 2 are of dimensions 1, p − 1 respectively.
(iii) If now p = 3, and XT^ = (X 1 , X 2 , X 3 ), derive an expression for the correlation of X 1 , X 2 conditional on X 3 = x 3 , in terms of (ρij ), where ρij = corr(Xi, Xj ) for 1 ≤ i < j ≤ 3.
2 (i) Let x 1 ,... , xn be a random sample from the distribution Np(μ, V ). Find an expression for the loglikelihood function `(μ, V ) in terms of the standard statistics ¯x, S, and state without proof the form of the maximum likelihood estimators ˆμ, Vˆ.
(ii) Now suppose that we have independent observations from g distinct groups, with x( 1 ν ),... , x( nνν)
being the sample from the νth group, which is assumed to be a random sample from N (μ(ν), V ), for 1 ≤ ν ≤ g. Using the results of (i) above, show that the generalized likelihood ratio test of H 0 : μ(1)^ =... = μ(g)^ = μ say
with μ, V both unknown, is of the form:
reject H 0 if log
constant,
where W, B are matrices that you should define.
(iii) Describe briefly the use of the matrices W, B in discriminant analysis.
Applied Multivariate Analysis
4 Write brief essays, which should include appropriate sketch graphs, on two of the following three S-Plus functions,
princomp()
tree() cmdscale()
The second function may be replaced by
rpart() if you prefer.
5 Suppose we have two known classes, C 1 and C 2 , and our observation x is known to have arisen from one of C 1 or C 2 , with prior probabilities π 1 , π 2 respectively. The corresponding known probability densities are those of N (μ 1 , V ), N (μ 2 , V ) respectively.
(i) Show that the Bayes rule for assigning x to C 1 or C 2 is of the form:
assign x to C 1 if aT^ x > b where you should determine a, b.
(ii) In the case π 1 = π 2 = 1/2, show that the above rule has error probabilities
P (assign x to C 1 |x is from C 2 ) = P (assign x to C 2 |x is from C 1 ) = p
say, where p = p(δ), δ > 0 and
δ^2 = (μ 1 − μ 2 )T^ V −^1 (μ 1 − μ 2 ).
Applied Multivariate Analysis
