Description
Let and let x 1,X2 , X n be a random sample from a probability distribution with probability density function
- (2—x Y-1 zero otherwise.
Recall: Fx(x)
Let Y 1 < Y2 < < Y denote the corresponding order statistics.
- [Proving that Y 1 = min Xi —Y 0 is super easy, barely an inconvenience.
Let Un Yl = n min X i. Find the limiting distribution of U
“Hint”: O Find the c.d.f. of Y l , FY min X i(x).
Use FY (x) tofindthe of U n, FU
- lim FU (u). If the limit exists, and if F 00 (u) is a c.d.f.
n 00 of a probability distribution, then that is the limiting distribution of U
- Proving that Y = max Xi 2 is super easy, barely an inconvenience.
Find so that V n nß(2—Yn) = nß(2— max X i) converges in
distribution. Find the limiting distribution of V
“Hint”: 0 Use Fx(x) to find the c.d.f. of Y n, FYmax X i(x).
Use FY (x) tofindthe c.d.f. of v n, Fv
- lim Fv (v). IF the limit exists and IF F T (v) is a c.d.f.
of a probability distribution, then that is the limiting distribution of V
n a lim 1+— e . Only “interesting” case is interesting.
1
n +00 n
- Let — > 0 and let X 1 , X 2 , X n be a random sample from a probability distribution with probability density function
4 11
zero elsewhere.
Recall: The maximum likelihood estimator of is
n 3
E x 1
i=l 1
W = X 3 has a Gamma( u = 4, 0 ) distribution.
- g) Show that is asymptotically normally distributed ( as n 00).
Find the parameters.
“Hint”: O By CLT,
That is, for large n,
) is approximately N gw), [g kiw)]2 29.
n
- Let and let x I , X n be a random sample from a probability distribution with probability density function
x2 ‘ zero otherwise.
- (i) Obtain the maximum likelihood estimator of X, X. (ii) Suppose n =4, x 1 = 5, x2 = 10, x3 =3, x 4 – — 20.
Find the maximum likelihood estimate of X.
- Is X a consistent estimator of X? Justify your answer.
( NOT enough to say “because it is the maximum likelihood estimator” )
- Is X an unbiased estimator of X? If is not an unbiased estimator of X, construct an unbiased estimator of based on X. (Assume n 2. )
