Let Y 1 < Y 2 < < Y n be the order statistics

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Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from the exponential distribution with pdf f(x) = e−x, 0 < x < ∞, zero elsewhere.

(a) Show that Z1 = nY1, Z2 = (n−1)(Y2 −Y1), Z3 = (n−2)(Y3 −Y2), . . . , Zn = Yn−Yn−1 are independent and that each Zi has the exponential distribution.

(b) Demonstrate that all linear functions of Y1, Y2, . . . , Yn, such as Σn1 aiYi, can be expressed as linear functions of independent random variables.

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Introduction To Mathematical Statistics

ISBN: 9780321794710

7th Edition

Authors: Robert V., Joseph W. McKean, Allen T. Craig

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