Advanced Econometrics Takeshi Amemiya
Multinomial Discriminant Analysis
The DA model of Section 9.2.8 can be generalized to yield a multinomial DA model defined by
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X? l(y< =j) ~ Mjij, 2;) |
(9.3.46) |
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and |
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Р(Уі =j) = Qj |
(9.3.47) |
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for / = 1, 2,. . ., n and j = 0, 1,. . |
., m. By Bayes’s rule we obtain |
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(9.3.48) |
Р(Уі=М?)= m8j(xT)qj, X g^f)Qk
where gj is the density function of 2,). Just as we obtained (9.2.48) from
(9.2.46) , we can obtain from (9.3.48)
^р^-^Дц+^ + хГАх,*), (9.3.49)
where РЛ1), fiA2), and A are similar to (9.2.49), (9.2.50), and (9.2.51) except that the subscripts 1 and 0 should be changed to j and 0, respectively.
As before, the term xf'Axf drops out if all the 2’s are identical. If we write Дко fi'x2)xT = the DA model with identical variances can be written exactly in the form of (9.3.34), except for a modification of the subscripts of /? and x.
Examples of multinomial DA models are found in articles by Powers et al.
(1978) and Uhler (1968), both of which are summarized by Amemiya (1981).
