1.36 For the multinomial (n, {;}) distribution with c > 2, a possible set of score-type simultaneous confidence limits for ; are the solutions of

1.36 For the multinomial (n, {π;}) distribution with c > 2, a possible set of score-type simultaneous confidence limits for π; are the solutions of

(π; – π;)2/[π;(1 – π;)/n] = (2a/2c)², j =

1,..., c.

a. Using the Bonferroni inequality, argue that for large n these c intervals simulta-

neously contain all {π;} with probability at least 1 α.

j

b. Show that the standard deviation of j – k is [π; + πκ – (π; - πκ)²]/n. Let a = c(c - 1)/2. For large n, explain why the probability is at least 1 a that the Wald confidence intervals (π¡ - πκ) ± Ζα/2a{[j + ûk - (πj – κ)2]/n}1/2 35 simultaneously contain the a differences (π; – πκ} (Goodman 1965).