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The Chinese Uanel‘SlEX of Hong Kong 0 we Course Examination 1nd Term 2006 2007 Course Code & Title : Time allowed Student ID. No. 1. Considering the problem of using only one real number m to represent a set of scalar samples {)9 Lil as follws: N
(a) (5%) Design a criterion J [m  {x, } {:1} such that it is minimized by m = x,, . . 1 N
in what a sense this m = —Z xr represents {x, L"; ?
(=1 (b) (5%) Seek a denisty p(x  m) and show that its maximum likelihood (ML) estimate 1 N
is also m=ﬁz x,. t=1
(c) (5%) Explain the key points of designing J [m l {x, } {:1} ?
(d) (5%) Explain the counterparts of the above key points in making the ML estimate. 2. Given a set of samples {x, Lil in R" and a subspace S originated at m 6 Rd and spanned by aie Rd,i=1,,nSd, (a) (5%) Let 9?, e S denote a projection of x, e R" on S and l, is the length of the straightline segment Xi, , prove that l, reaches its minimum 1: at it: e S such that A*
xix, J. S , where _L means ‘being orthogonal to’. (b) (5%) Given the expression of l: in term of x, E Rd and at. 6 Rd ,i=1,,n. N
(c) (10%) Determine me Rd and ai e R",i =1,,nSdsuch that 2 l: is minimized. t=l Does this minimum value exist ? Is the solution on m 6 Rd and
ai e R" ,i =1,,n S d unique, why ? if not, give one unique solution. 3. Divid a set of samples {xt },':1,x, 6 Rd into k clusters, on each cluster you can get a criterion J [m j [{x, },N=1] in the same way as the above 1(a) such that this cluster can be best represented by its mean m J. e R d of the cluster, (a) (8%) In total there are how many such a k—clusterpartitioning ? Give your criterion
J [{m j }’;=1  (x, },':1] for seeking the best partitioning. Explain it (i.e., in what a sense). Course Code : CSC5150 Page 2 of 2 (b) (12%) Extend the above (a) such that each cluster is represented in a sense of the
above 2(c), and provide an algorithm for seeking a solution. Explain whether your
algorithm gives the best partitioning , why ? (a) (5%) Prove that maximizing Ip(x) 1n q(x)dx with respect to q(x) leads to
£100 = p(x) (b) (5%) Solve mum...) jpo I x)p(x)1n[q(x y)q(y>1dxdy. (c) (5%) Discuss the generalization error versus fitting error, and explain what is the
two step procedure for statistical learning. (d) (5%) From the minimum description lenghth perspective, discuss the model selection problem. (a) (10%) Assume the prevailing interest rate in the capital market is r% . Explain and
illustrate how investors’ investment decisions should be made in consideration of
their different consumption preferences by means of the budget line and indifference curves. What is the physical interpretation of the slope the budget line? (b) (10%) In classical finance, two famous models, respectively the arbitrage pricing
theory (APT) and the capital asset pricing model (CAPM), have been used for asset
pricing. Discuss the rationale(s) for using each model in asset pricing and state the
relative merit(s)/demerit(s) of using APT as compared with CAPM for pricing stock securities from both the theoretical and implementation perspectives. — End of Paper 
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