Page 562 - Probability and Statistical Inference

12

Large-Sample Inference

12.1 Introduction

In the previous chapters we gave different approaches of statistical infer-
ence. Those methods were meant to deliver predominantly exact answers,
whatever be the sample size n, large or small. Now, we summarize approxi-
mate confidence interval and test procedures which are meant to work when
the sample size n is large. We emphasize that these methods allow us to con-
struct confidence intervals with approximate confidence coefficient 1 − α or
construct tests with approximate level α.
Section 12.2 gives some useful large-sample properties of a maximum
likelihood estimator (MLE). In Section 12.3, we introduce large-sample con-
fidence interval and test procedures for (i) the mean µ of a population having
an unknown distribution, (ii) the success probability p in the Bernoulli distri-
bution, and (iii) the mean λ of a Poisson distribution. The variance stabilizing
transformation is introduced in Section 12.4 and we first exhibit the two
customary transformations and used respectively in the case
of a Bernoulli(p) and Poisson(λ) population. Section 12.4.3 includes Fisher’s
tanh (ρ) transformation in the context of the correlation coefficient ρ in a
-1
bivariate normal population.

12.2 The Maximum Likelihood Estimation
In this section, we provide a brief introductory discussion of some of the
useful large sample properties of the MLE. Consider random variables X , ...,
1
X which are iid with a common pmf or pdf f(x; θ) where x ∈ χ ⊆ ℜ and θ ∈
n
Θ ⊆ ℜ. Having observed the data X = X, recall that the likelihood function is
given by

We denote it as a function of θ alone because the observed data X = (x , ..., x )
n
1
is held fixed.
We make some standing assumptions. These requirements are, in spirit,
similar to those used in the derivation of Cramér-Rao inequality.
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