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# ASYMPTOTIC APPROXIMATIONS

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ASYMPTOTIC APPROXIMATIONS book

# ASYMPTOTIC APPROXIMATIONS

DOI link for ASYMPTOTIC APPROXIMATIONS

ASYMPTOTIC APPROXIMATIONS book

## ABSTRACT

Despite the many simple examples we have dealt with, closed form computation of risks

in terms of known functions or simple integrals is the exception rather than the rule. Even

if the risk is computable for a specific P by numerical integration in one dimension, the qualitative behavior of the risk as a function of parameter and sample size is hard to ascer-

tain. Worse, computation even at a single point may involve high-dimensional integrals. In

particular, consider a sampleX1, . . . , Xn from a distribution F , our setting for this section and most of this chapter. If we want to estimate µ(F ) ≡ EFX1 and use X¯ we can write,

MSEF (X¯) = σ2(F )

n . (5.1.1)

This is a highly informative formula, telling us exactly how the MSE behaves as a

function of n, and calculable for any F and all n by a single one-dimensional integration. However, consider med(X1, . . . , Xn) as an estimate of the population median ν(F ). If n is odd, ν(F ) = F−1

) , and F has density f we can write

MSEF (med(X1, . . . , Xn)) =

( x− F−1 ( 12))2 gn(x)dx (5.1.2)

where, from Problem (B.2.13), if n = 2k + 1,

gn(x) = n

( 2k k

) F k(x)(1 − F (x))kf(x). (5.1.3)

Evaluation here requires only evaluation of F and a one-dimensional integration, but a different one for each n (Problem 5.1.1). Worse, the qualitative behavior of the risk as a function of n and simple parameters of F is not discernible easily from (5.1.2) and (5.1.3). To go one step further, consider evaluation of the power function of the one-sided t test of Chapter 4. If X1, . . . , Xn are i.i.d. N (µ, σ2) we have seen in Section 4.9.2 that√ nX¯/s has a noncentral t distribution with parameter µ/σ and n− 1 degrees of freedom.

This distribution may be evaluated by a two-dimensional integral using classical functions