Stat 5101 Lecture Notes - School of Statistics

158 Stat 5101 (Geyer) Course NotesthenZ ∼ Multi l (n, q)where the parameter vector q has componentsq j = p i1 + ···+p imjis formed by collapsing the categories in the same way as in forming Z from Y.No wonder Lindgren felt the urge to sloppiness here. The correct statementis a really obnoxious mess of notation. But the idea is simple and obvious. If wecollapse some categories, that gives a different (coarser) partition of the samplespace and a multinomial distribution with fewer categories.Example 5.4.1.Consider the multinomial random vector Y associated with i. i. d. sampling of acategorical random variable taking values in the set (5.36). Let Z be the multinomialrandom vector associated with the categorical random variable obtainedby collapsing the categories on the ends, that is, we collapse the categories“strongly agree” and “agree” and we collapse the categories “strongly disagree”and “disagree.” ThuswhereandY ∼ Multi 5 (n, p)Z ∼ Multi 3 (n, q)Z 1 = Y 1 + Y 2Z 2 = Y 3Z 3 = Y 4 + Y 5q 1 = p 1 + p 2q 2 = p 3q 3 = p 4 + p 5The notation is simpler than in the theorem, but still messy, obscuring thesimple idea of collapsing categories. Maybe Lindgren has the right idea. Slopis good here. The marginals of a multinomial are sort of, but not precisely,multinomial. Or should that be the sort-of-but-not-precisely marginals of amultinomial are multinomial?Recall that we started this section with the observation that one-dimensionalmarginal distributions of a multinomial are binomial (with no “sort of”). Buttwo-dimensional multinomial distributions must also be somehow related to thebinomial distribution. The k = 2 multinomial coefficients are binomial coefficients,that is, ( ) n= n! ( ) ( )n ny 1 ,y 2 y 1 !y 2 ! = =y 1 y 2