Asymptotic Variance Formulas, Gamma Functions, and Order Statistics

Transcription

1 Statistical Models and Methods for Lifetime Data, Second Edition by Jerald F. Lawless Copyright 2003 John Wiley & Sons, Inc. APPENDIX Β Asymptotic Variance Formulas, Gamma Functions, and Order Statistics B.l ASYMPTOTIC VARIANCE FORMULAS The following results are often used in developing large-sample inference procedures. Proofs can be found, for example, in Rao (1973, Ch. 2). Here means "converges in distribution to." THEOREM Β1. Let T\ n,..., T kn be statistics such that as η -* oo MTu - 0i., 7* - 0*) -3 /V(0, Σ) where Σ = (σ,-;)***. If g(*i,...,**) is a function whose first derivatives all exist, then as η oo y/h[g(tin,, T kn ) - Ε(θι 0*)] -5- Ν (θ, ΣΣ] σ '^^) (B1) where 3g/d0 ( - means 3g(0i,..., 0*)/90/ (/ = 1,...,Jfc). Remarks 1. Often the following terminology is used: Asvar[g(7i,..., 7*»)] = T T f-^-ascov(r,, T jn ), (B2) where Asvar and Ascov denote variances and covariances in the asymptotic distributions of the indicated variables. Strictly speaking, this notation is 539

2 540 ASYMPTOTIC VARIANCE FORMULAS, Γ FUNCTIONS, ORDER STATISTICS improper since the distributions of the 7},, and g{t\,..., 7^ ) are degenerate as η -> oo. However, it is used as a convention for indicating asymptotic-based approximations that are used for finite but large n. (In (B2), Ascov(7}, Tj ) is n~ i aij, for example.) 2. The results in this section are often used when (T[,..., Tkn) = (θ\,...,6k) is a vector of maximum likelihood estimates (m.l.e.'s), based on a sample of size n. 3. The results here are stated for statistics with asymptotic normal distributions. Expressions like (B2) also hold under the weaker conditions that the variances and covariances of T\ n,..., T kn are 0(n~ r ), where r > An important special case of Theorem Β1 is given by k = 1: if *Jn(T 0) -5 N(0, a 1 ) as η oo, then if g(x) has first derivative g'(x), y/h~ [g(t n ) - 8(θ)] $ Ν [θ, g'(0)v]. (B3) This implies that, in the notation of (B2), Asvar[g(r )] = s'(0) 2 Asvar(r ). (B4) 5. The preceding results can be proved with what is sometimes referred to as the 8 method, based on Taylor series expansions. For example, the function g(t\,tkn) has expansion 8(T U,..., T^) = g(9u., 0 k ) + V <5r ; higher-order terms, where <57/ = T; n ft. The results follow from this and simple convergence results for random variables. Theorems Β1 and B2 below are sometimes referred to as delta theorems. Theorem Β1 can be generalized to the case in which there are several functions of T\,..., Tk n, a s follows. THEOREM B2. Let (T\ n,..., T k ) be statistics defined as in Theorem Bl and let gi(x\,...,x k ), i = I,..., p, be functions, all of whose first derivatives exist. Then the joint distribution of *Jn[gi(T\ n,tk ) - gi(9\,..., 0*)], i = 1,..., p, is asymptotically p-variate normal with mean 0 and covariance matrix G2G', where G has (i, j) entry G tj = dgi/d9j. Remark For two functions g\{x\,...,xk) and gi{x\,.,xk), the theorem gives, in the notation of (B2), Ascov In (Ti T k ), gi(t in,t kn )] = Τ Σ 3r l?ascov(7i, Tj ). f=t y=i W> D0 J (B5)

3 GAMMA FUNCTIONS 541 B.2 GAMMA FUNCTIONS We summarize a few results about the gamma function and other functions and probability distributions related to it. More details on these topics can be found in the books by Abramowitz and Stegun (1965), and Johnson et al. (1994, 1995). The gamma function is denned as / OO Γ(ζ) = / u z - x e-"du ζ > 0. (B6) Jo We note the well-known results (see Abramowitz and Stegun 1965, Ch. 6) Γ(ζ + 1) = ζγ(ζ) ζ > 0 (B7) Γ (\λ = π 1 / 2 = (Β8) log Γ(ζ) = (ζ - 1) log ζ - ζ + \ log(2*) + ± - ^ +. (Β9) It follows from (Β6) and (B7) that for ζ a positive integer, Γ(ζ+1) = ζ! The digamma function is defined as The poly gamma functions are rflogr(z) Γ(ζ) ψ (ζ) = = -p r z > 0 - ( B1 ) dz Γ(ζ) ^"Vr(z) (B) ^ (z) = j-r «=1-2,... αζ" The case η = 1 is called the trigamma function. Two useful results are ψ (l) = -γ = , 7r 2 ^'(1) = ^ (BID The incomplete gamma function is defined in this book as 1 C x l(k, x ) = u k - x e' u du k>0 x> 0. (B12) Γ (Λ) JO This is the distribution function for the one-parameter gamma distribution (1.3.17) denoted Ga(k), and takes on values between 0 and 1. I(k, x) is related to the distribution function for the χ 2 distribution with υ degrees of freedom (denoted χ 2 υ)) as follows:

4 542 ASYMPTOTIC VARIANCE FORMULAS, Γ FUNCTIONS, ORDER STATISTICS Fv(x) = Pr(X 2 v) <X) X > 0 I χ ζυ/2-\ β-ζ dz Jo 2"/ 2 Γ>/2) -'(1-1) Many software packages provide values of Γ(ζ), ψ(ζ), (B12), and (B13). B.3 ORDER STATISTICS A few results about order statistics are given here. An extended treatment and references can be found in the book by Arnold et al. (1992). Suppose that X has continuous probability density function (p.d.f.) f(x) and distribution function F(x) and that X\,..., X is a random sample from this distribution. The Xi, rearranged in order of magnitude, and denoted X(\) < X(2) < < X(n), are called the order statistics of the sample. The joint p.d.f. of..., X(t k ), where \ < l\ < li < < Ik < η and 1 < k < n, can be shown to be / *+i \k+\ k \ ι = Ι / i=l ( = 1 (B14) where X(^) < X( l2 ) < < X(i k ) and where, for convenience, we define ίο = 0, 4+1 = η + l,x(t 0 ) = -oo, and X(t k+l ) = +oo. Important special cases of (B14) are the following. 1. The joint p.d.f. of X(\),..., X( r ) (r < n) is T^jj (iv^) [1 - F(x (r) )T r. (B15) 2. The p.d.f. of X (i) (1 < i < n) is (f-dkn-o!^ 0 ^^'"' [ 1 " F ( ^» ) ] " " ' ( B 1 6 ) 3. The joint p.d.f. of Xu) and X(j), for i < j, is n! (i-i)iu-i-i)lo^nxm)fixw)f(xv)) i-l χ [F(x (j) ) - FCxw)]'-'- 1 [1 - F( Xij) )] n - j. (B17)

5 ORDER STATISTICS 543 These expressions and (B14) can be obtained directly. For example, (B17) can be found from the probability that of the η values X\,..., X n, i 1 are less than X(i), one is in (*(, ), *(, ) + Δ), j i 1 are between *(, ) + Δ and X(j), one is in (x(j), X(j) + Δ), and η j are above X(j) + A. Moments of order statistics are useful in some applications, though it is usually not possible to get simple analytical expressions. Two exceptions are for the uniform and exponential distributions, for which the following results are easily established. 1. For the uniform distribution C/(0, 1), with p.d.f. f(x) = 1 (0 < χ < 1), ( X (,. ) ) = _ i _ Var(X (,. ) ) = ( w + ^ ( n + 2 ). (B18) 2. For the standard exponential distribution, with p.d.f. /(x) = e~ x (x > 0), i i Ε(Χ (ί) ) = Σ{η-ί + \)-\ Var(X (0 ) = ^(' i=\ e=i ) _ 2 - ( B 1 9 > Several types of asymptotic results can be established for order statistics. Some involve the extreme order statistics and X( ) (e.g., Arnold et al. 1992). We mention only the case of X(,), where ι = np and 0 < ρ < 1 as η oo. The pth quantile of the distribution of X is x p = F~' (p), and it can be shown that if f(x) = F'(x) is continuous at x p and f(x p ) φ 0, then *Jn(Xu) x p ) is asymptotically normal with mean 0 and variance!7(w (B20) There is also a multivariate generalization of this.