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By Fearn T., Brown P.J., Besbeas P.

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Extra resources for A Bayesian decision theory approach to variable selection for discrimination

Example text

U consists of the single point -x. O. We then Let us distinguish several cases. t. fA, ~- t. ~ .. ~ and we have '(, ~4 (~ 2) Rt. ,)'l. "t. t.. a,.. ~ ~ i. , if n,~-tRkQ,l. Here, the minimum is attained for ~ .. mM1, \ 1'\" -( tu,Q,1-1! and we have ~ ::" li - \ RR,IlID'J. tt t Therefore, admissible values of e exist when ~o.. >-t and 'tllla,:#-o when RLw .. - t. In case 1, the radius is given by 'It .. , ... o. (l = - if In cases 2 and 3, admissible values ofe always exist whenQ, does not belong to any closed circle of radius !

If we add the Rem ark 2. 51). R em ark 3. 2). For example, there are no suche when points belonging to llie on M. + ~6~ is positive, then for sufficiently small A'IJ (and a sufficiently large n. 2) is always solvable. e in a number of special cases. It is convenient to consider two half-planes of the variable e separately. In the half-plane ~ e " 1;, the admissible values of e satisfy the geometric condition Let us determine the range of admissible values of whereas in the half-plane IU e ~ t we have I e\ I..

J. 5), we can rewrite the last formula as 9 tl + 'It 11+1 \ o t5vtt)-Bvun) r. 20). R em ark. 23) or G ~I R,y:~-/'i\ 5y ·Utld'\t)dt ... 25) Lemma 6. u.. ~ H~) . \ l-) ~ ~9) f 'J - n (~+e).. d. ) ~ + ' [ . ej)t l-~ ~e J=l J. V. e f\ lb t lV) (X,+ t) dt. KALININ Proof. 9 >u,. 26) to Here, the first sum on the right-hand side is a part of a Taylor's series of the function 'l:,T9·1oI '! 9) J. =""', • n ... / \I , :t ..... , we have ~ =a -0. 26). Theorem 6. ~. , ... 28) SPECIAL FUNCTIONS AND LIMIT PROPERTIES OF PROBABILITY DISTRIBUTIONS 29 Proof.

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