4. Let X1, X2, ..., Xn be a random sample from N,(0, ), > 0. Define X = (X1X2 ...Xn). Find the distribution of U'(In - 1

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4 Let X1 X2 Xn Be A Random Sample From N 0 0 Define X X1x2 Xn Find The Distribution Of U In 1 1 (55.26 KiB) Viewed 109 times

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4. Let X1, X2, ..., Xn be a random sample from N,(0, ), > 0. Define X = (X1X2 ...Xn). Find the distribution of U'(In - 11')U, where U = (UjU2 ... Un) with u; = a'X;, i = 1, 2, ..., n, a € RP, a #0, 1' = (1,1,...,1). Find the distribution of Xb, where be R" such that b'b = 1. Hence find the distribution of b'X'S-1Xb. 6. Let X1, X2, X3, ... , Xn be a random sample from Nm(u, S), 5 > 0. Define a trans- formation Y; = AX; + B, (i=1,2,...,n) A is m x m non singular matrix of constants and B is a vector of constants. Show that Hotelling's T2 statistic for testing Ho : = Ho against HA : Ht Ho based on (X1, X2, X3, ..., X,) and that based on (Y1, Y2, Y3, ..., Yn) are the same.