2. Let Y,,., Y, be independent random variables such that Y,(Yı., Yp)" and 0 = (0,.,0p)". Let = 0(Y) = (0,(Y),... , @p(Y))" be an estimatorof 0, and let g(Y) = (g(Y),... , gp(Y))" = – Y. Denote by || - || the Euclidean norm,||Y° = Y} + .. + Y.N(8), 1). Write Y =%3DSuppose that D(Y) = @g(Y)/ay, exists. Then it is known that%3DR(Ô(Y)} =+2 D(Y) + É19(Y)²=1is an unbiased estimator of the risk of 0, under squared error loss L(0, ê) = ||0 – e|P.[You are not required to show this].%3D(i) The James-Stein estimator is6.s(Y) = (1 – )Y._P-2y||Y?Show that an unbiased estimator of the risk of d Js(Y) isŘlójs(Y)) = p – (p - 2) /||YII°.Deduce that Y is inadmissible as an estimator of 0.Is ô js(Y) admissible? Justify your answer.

To show that an unbiased estimator of the risk of the James-Stein estimator iswe can use the…

Answered: 2. Let Y₁,..., Yp be independent random…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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4. Let Y₁, Y2,..., Yn be a random sample of size 5 from a normal population with mean 0 and variance 1. Also let 15 Y = 5 i=1 following: (a) W = Y. Let Y be another independent observation from the population. Determine the distribution of the 5 ΣΥΡ i=1 (b) U = (Y-Y)². (c) R: = (d) V = i=1 √5Y6 W 2 (5Ỹ²+Y²) U
Let X1, X2,.., X, be independent identically distributed random variables with each X, having a probability mas function given byP(X; = 0) = 1-p P(X; = 1) = p, where Osps1. 1 EXj, then E(Y)= j = 1 Define the random variable Y = n Select one: а. 1 b. p C. 0 d.
1. Let X1, X,.., X, random sample from distribution be a a Geometric 2....., fx (x) = p(1– p)*| F, (y,) = ? x-1 x=1,2,3,.... Fy, (y,) = ? P(Y, <1) =? а. b. С.
(a) Let X, Y be independent random variables with respective PMFS Px(x) = (-) )* > Py (Y) = (-) )", y=1, 2, 3, . . . .2 , х-0, 1, 2, 3, . . Find P(X+Y=3)
Let X1 and X2be independent exponential random variables: fX1(x1) = e−x1 and fX2(x2) = e−x2 1
Let JO, J1,..., J4 independent random variables according to the Ber (r;) law, where i = 0, 1,..., 4, respectively. We define the random variables Xi = min {JO + Ji, 1}, for i = 1, 2, 3, (a) Find the law of Xi , for each i = 1, 2, 3, 4. (b) Find the law of (X1, X2, X3, X4).
Q1. let Y₁ < Y₂ < Y3 < Y4 < Y5 are the order statistics of the random sample of size 5 from the distribution : f(x) = 3x², 0
9. If X and Y are two random variables and let g(X) be a random variable. Show that (a) E[g(X) X=x] = g(x). (b) E[g(x)Y|X=x] = g(x) E[Y|X=x]. Assume that E[g(x)] and E[Y] exist.
Z is the present-value random variable for a whole life insurance of b payable at the moment of death of (x). You are given: (i) (ii) 8 = 0.05 (iii) The net single premium for this insurance is equal to Var(Z). Calculate b. x+t = 0.01 t≥0 (A) 1.36 (B) 1.68 (C) 2.00 (D) 2.32 (E) 2.64
Let X1, X2, . . . are independent indicator variables with different probabilities of success. That is, P(Xi = 1) = pi. Define Yn = X1 + X2 + . . . + Xn. Find E(Yn), V ar(Yn) and coefficient of variation of Yn
2. Let the independent random variables X1 and X2 have Bin(0.1,2) and Bin(0.5, 3), respectively. (a) Find P(X1 = 2 and X2 = 2). (b) Find P(X1 + X2 = 1). (c) Find E(X1 + X2). (d) Find Var(X1 + X2).
Suppose that the random variables Y1,..., Y(n > 2) satisfy Y - Ba, + Ei, i= 1,2,..., T, where 1,..., En are iid N(0, o), and both B and o are unknown. (a) Assume z1,..., In are fixed known constants. Here we observe Y1 = n,., Y observed data Y = y = (y1,..., Yn). Find a two-dimensional sufficient statistic of Y = (Yı,., Yn) for (8,0?). (b) Assume now that a1,...,n are random variables with a known joint distribution m(r1,..., In), and the r,'s are independent of G's (it is traditional in the linear regression to use lower case for independent variables r;'s). In this case, the observed data (Y, x) = {(Yi, r;)}i-1,n. Find a three- dimensional sufficient statistic of (Y, x) for (B,02). Im, and the

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