Let Y, Y2, ..., Yn be n i.i.d random variables drawn from the population distribution of...
Let Y, Y2, Yz and Y4 be independent, identically distributed random variables from a population with mean u and variance o. Let Y = -(Y, + Y2 + Y3 +Y4) denote the average of these four random variables. i. What are the expected value and variance of 7 in terms of u and o? ii. Now consider a different estimator of u: W = y + y + y +Y4 This an example of weighted average of the Y. Show...
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.
1. Let Yı,Y2,..., Yn denote a random sample from a population with mean E (-0,) and variance o2 € (0,0). Let Yn = n- Y. Recall that, by the law of large numbers, Yn is a consistent estimator of . (a) (10 points) Prove that Un="in is a consistent estimator of . (b) (5 points) Prove that Vn = Yn-n is not a consistent estimator of (c) (5 points) Suppose that, for each i, P(Y, - of ? Prove what...
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.
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a population with Rayleigh distribution (Weibull distribution with parameters 2, θ) with density function f(y|θ) = 2y θ e −y 2/θ, θ > 0, y > 0 Consider the estimators ˆθ1 = Y(1) = min{Y1, Y2, · · · , Yn}, and ˆθ2 = 1 n Xn i=1 Y 2 i . ii) (10 points) Determine if ˆθ1 and ˆθ2 are unbiased estimators, and in...
Suppose that Y1 , Y2 ,..., Yn denote a random sample of size n from a normal population with mean μ and variance 2 . Problem # 2: Suppose that Y , Y,,...,Y, denote a random sample of size n from a normal population with mean u and variance o . Then it can be shown that (n-1)S2 p_has a chi-square distribution with (n-1) degrees of freedom. o2 a. Show that S2 is an unbiased estimator of o. b....
iid Let Yı, Y2, ..., Yn N(u,), where the population mean y and population variance o are both unknown. Show that the Method of Moments (MOM) estimators of u and o? are given by n i =Y, Y n =1 72 = n-1 S2 (Y; -Y) n n i=1 Note: In this case, (Y, S?) is a sufficient statistic for (u, o?). The MOM estimators of u and o2 are therefore functions of a sufficient statistic.
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.
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).
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.