Order Statistics Exercise 1. For the continuous component of a random variable X, in the neighbourhood of the quantile qa, a Taylor Series of the order statistic Xanlin is obtained through its inverse cdf, X =F-1(U). The random variable U ~ U(0, 1) is uniformly distributed between zero and one. In Section 4.6 of [3), a Taylor series of Xanlin = F-(Uanin) is expanded about its expected value E[U janlin] = [anl/(n+1). Only the first three derivatives, (F-1) (0), 1 = 1, 2,3, are required to exist and be continuous. I have used the Lagrange Remainder Theorem and a random variable C := C(U fanlin) in a neighbourhood of fan]/(n + 1) such that F -' (U jan]in ) = F-1(or )+ (Ufanlin - jan ) (F-1) (1) ( [an] (Ufanlin - [an] ) ( F-1) (2) ( [an] n+ 1 mon +6 (Ujanlin - [an] n+ 1 ( F - 1) (3) ( C ) . (1) (a) Justify the validity of writing the Taylor polynomial (1) both in terms of writing a stochastic tion 2.5.4 in [7). Taylor polynomial and integrability of the polynomial random variable. Hint: See Sec- (b) Show that: (i) Denoting the quantile, qo = F-1(a). Verify that d F-1(x) 1 dx f (qa ) (ii) For r = 1, ...,n, and l E N, show that for a collection of n i.id. uniform u(0, 1) random variables E[(Ur:)'] = II ti -1 i=1 nti Hint: The beta function, for a > 0, b > 0 has an integral representation B(a, b) = fora-1(1 -x)6-1dx. The first four moments are needed for part (c). (iii) Show that via the Mean Value Theorem that