New Jersey Institute of Technology Intro Theory of Statistics Worksheet

please finish the questions without any programming.

3)LetX, Y be two continuous random variables with joint density function

8 xy / q 4if0 < y < x< q

f(x,y½q)=

• otherwise .

We want to estimate q relative to Mean Square Error (MSE).From lecture we already know that X is a sufficient statistic forq and has density function

fX (x½q)=4 x3 / q4 if0 < x < q

0 otherwise

Consider the estimatord (x,y) =3 x y. Without computing any MSE find an estimator that will have smaller MSE than the estimatord (x,y)=3 x y and explain why that should be true.

4)We have one observation,Y,having pdf, for -1≤q≤1, equal to

2 y q + 1if-1/2

f (y½q)=

0otherwise.

NOTE:For f (y½q) to be a density function the only possible values ofq must be in the closed interval -1≤ q≤1.

Consider the estimator W(y) defined as

W( y)= 1ify³0

-1ify <0

a.IsW(y)an unbiased estimator ofq?

b..Show that W(y) is an MLEofq.