Problem 2: Let Y, Y2,. Y2o form a random sample of size n Determine the joint...
Let Y,, Y2, .., Yn denote a random sample of size n from a population whose density is given by Find the method of moments estimator for α.
5. Let Yi,Y2, , Yn be a random sample of size n from the pdf (a) Show that θ = y is an unbiased estimator for θ (b) Show that θ = 1Y is a minimum-variance estimator for θ.
QUESTION 7 Let Y, Y2, ....Yn denote a random sample of size n from a population whose density is given by (a) Find an estimator for θ by the maximum likelihood method. (b) Find the maximum likelihood estimator for E( Y4).
QUESTION 7 Let Y,, Y2,..., Yn denote a random sample of size n from a population whose density is given by (a) Find an estimator for 0 by the maximum likelihood method. (b) Find the maximum likelihood estimator for E(Y4).
QUESTION 5 Let Y , Y2, , Yn denote a random sample of size n from a population whose density is given by (a) Find the method of moments estimator for β given that α is known. Find the mean and variance of p (b) (c) show that β is a consistent estimator for β.
Let Y1<Y2<...<Yn be the
order statistics of a random sample of size n from the distribution
having p.d.f f(x) = e-y , 0<y<, zero elsewhere. Answer the following
questions.
(a) decide whether Z1 = Y2
and Z2=Y4-Y2 are
stochastically independent or not. (hint. first find the joint
p.d.f. of Y2 and Y4)
(b) show that
Z1 = nY1, Z2=
(n-1)(Y2-Y1),
Z3=(n-2)(Y3-Y2), ....,
Zn=Yn-Yn-1
are stocahstically
independent and that each Zi has the exponential
distribution.(hint use change of variable technique)
Let X, X,, ..., X, denote a random sample of size n from a population with pdf (10) = b exp(@m()).0<x<1 where (<O<0. Derive that the likelihood ratio test of H.:0=1 versus H, :0 #1 in terms of T(x) = ŽI (3)
Exercise 11. Let Xi,y, be random variables with joint PDF fXiXi. Let X2,Y2 be random variables with joint PDF fx2,Y2. Let T: R2R2 and let S: R2 -R2 so that ST(x, y) (z, y) and TS(a, y) (x, y) for every (x, y) E R2. Let J(x, y) denote the determinant of the Jacobian of S at (x,y. Using the change of variables formula from multivariable calculus, show that
QUESTION 3 Let Y1, Y2, ..., Yn denote a random sample of size n from a population whose density is given by (Parcto distribution). Consider the estimator β-Yu)-min(n, Y, where β is unknown (a) Derive the bias of the estimator β. (b) Derive the mean square error of B. , Yn).
QUESTION8 Let Y,,Y2, ..., Yn denote a random sample of size n from a population whose density is given by (a) Find the maximum likelihood estimator of θ given α is known. (b) Is the maximum likelihood estimator unbiased? (c) is a consistent estimator of θ? (d) Compute the Cramer-Rao lower bound for V(). Interpret the result. (e) Find the maximum likelihood estimator of α given θ is known.