Homework #1 (written-answer problems) 1. Suppose X1,..., X are i.i.d. with common PDF x(x) = ex...

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Homework #1 (written-answer problems) 1. Suppose X1,..., X are i.i.d. with common PDF x(x) = \e\x for x 0 and \ > 0. Let X (1) ... n X (n) denote the order statistics of X1, ..., Xn. (a) Specify the joint PDF of the order statistics, fx (1),.,X(n) (x1, . . . xn). ... 9 (b) Define Y = X(1) -0, Y2 = X(2) - X(1), Yn = X(n) X(n1) (Yi's defined this way are called exponential spacings) and derive their joint PDF Y,..., Yn (Y1, . . ., Yn) for y > 0, . . Yn > 0. = - - (c) Using part (b), show that Y,..., Yn are independent with Y ~Exp ((n i + 1)A), i = 1, . . ., n. (d) Define Zi (ni+1)Y; and show that Z1, ..., Zn are i.i.d. with each Zi ~ Exp (\). (e) Using the MGF approach, show that Z; follows a Gamma (n j + 1, ) distribution. (f) Show that for Gamma (r, B) with r known, BMLE = BMM. Using this fact, find MLE (from (e)). 2. Verify that if W ~ Pareto I(a, 0), then V = log W ~ Exp(log(0), 1/a) with PDF fv (v) = (1/a) exp{(v log(0))/a}, v log(0). 3. Suppose W1,..., Wn are i.i.d. and their common CDF F is in the Frchet MDA, which means that P{W > tw|W; > t} w for w 1 and t large. Therefore, it follows from Problem 2 that the log-spacings log(W(i)) log(W(i1)), i = j +1, ..., n, are the exponential spacings of Problem 1(b). Combine the results derived in Problems 1 and 2 and show that the MLE of a, when based on data log(w()),..., log(w(n)), is the Hill estimator: 2>> n 1 W(i) log = n j +1 w (j) i=j 1 Homework #1 (written-answer problems) 1. Suppose X1,..., X are i.i.d. with common PDF x(x) = \e\x for x 0 and \ > 0. Let X (1) ... n X (n) denote the order statistics of X1, ..., Xn. (a) Specify the joint PDF of the order statistics, fx (1),.,X(n) (x1, . . . xn). ... 9 (b) Define Y = X(1) -0, Y2 = X(2) - X(1), Yn = X(n) X(n1) (Yi's defined this way are called exponential spacings) and derive their joint PDF Y,..., Yn (Y1, . . ., Yn) for y > 0, . . Yn > 0. = - - (c) Using part (b), show that Y,..., Yn are independent with Y ~Exp ((n i + 1)A), i = 1, . . ., n. (d) Define Zi (ni+1)Y; and show that Z1, ..., Zn are i.i.d. with each Zi ~ Exp (\). (e) Using the MGF approach, show that Z; follows a Gamma (n j + 1, ) distribution. (f) Show that for Gamma (r, B) with r known, BMLE = BMM. Using this fact, find MLE (from (e)). 2. Verify that if W ~ Pareto I(a, 0), then V = log W ~ Exp(log(0), 1/a) with PDF fv (v) = (1/a) exp{(v log(0))/a}, v log(0). 3. Suppose W1,..., Wn are i.i.d. and their common CDF F is in the Frchet MDA, which means that P{W > tw|W; > t} w for w 1 and t large. Therefore, it follows from Problem 2 that the log-spacings log(W(i)) log(W(i1)), i = j +1, ..., n, are the exponential spacings of Problem 1(b). Combine the results derived in Problems 1 and 2 and show that the MLE of a, when based on data log(w()),..., log(w(n)), is the Hill estimator: 2>> n 1 W(i) log = n j +1 w (j) i=j 1