5 Lt X1 Xn be a sample from beta0 1 distribution e Compute

5. Lt X1, ...., Xn be a sample from beta(0, 1) distribution. (e) Compute the empirical power corresponding to this sample. Compare it to the nominal power.

Solution

The distribution of each Xi is Beta distribution with parameters theta, 1

f(xi)= x^alpha-1(1-x)^beta-1

f(cxi) = c^alpha(c)^beta-1x^alpha-1(1-x)^beta-1

i.e. f(cx) is directily proportional to f(x) and constant of proportionality is c^{alpha+beta -1}

Nominal power

if cx =y

f(y) = same as above.