From Algorithms to Z-Scores - matloff - University of California, Davis

322 CHAPTER 16. ADVANCED STATISTICAL ESTIMATION AND INFERENCE
is known to have an asymptotically multivariate normal distribution with mean 0 and nonsingular
covariance matrix Σ = (σij).
Let h be a smooth scalar function 1 of k variables, with hi denoting its i th partial derivative. Consider
the random variable
Y = h(R1, ..., Rk) (16.10)
Then √ n[Y − h(η1, ..., ηk)] converges in distribution to a normal distribution with mean 0 and
variance
provided not all of
are 0.
[ν1, ..., νk] ′ Σ[ν1, ..., νk] (16.11)
νi = hi(η1, ..., ηk), i = 1, ..., k (16.12)
Informally, the theorem says, with R, η, Σ, h() and Y defined above:
Suppose R is asymptotically multivariate normally distributed with mean η and covariance
matrix Σ/n. Y will be approximately normal with mean h(η1, ..., ηk) and
covariance matrix 1/n times (16.11).
Note carefully that the theorem is not saying, for example, that E[h(R) = h(η) for fixed, finite n,
which is not true. Nor is it saying that h(R) is normally distributed, which is definitely not true;
recall for instance that if X has a N(0,1) distribution, then X 2 has a chi-square distribution with
one degree of freedom, hardly the same as N(0,1). But the theorem says that for the purpose of
asymptotic distributions, we can operate as if these things were true.
1
The word “smooth” here refers to mathematical conditions such as existence of derivatives, which we will not
worry about here.
Similarly, the reason that we multiply by √ n is also due to theoretical considerations we will not go into here,
other than to note that it is related to the formal statement of the Central Limit Theorem in Section 5.5.2.11. If we
replace X1 + ..., +Xn in (5.60), by nX, we get
Z = √ n ·
X − m
v
(16.9)