Page 576 - Probability and Statistical Inference

12. Large-Sample Inference 553

12.3.3 The Poisson Mean
Suppose that X , ..., X are iid Poisson(λ) where 0 < λ < ∞ is the unknown
1
n
parameter. The minimal sufficient estimator of ? is the sample mean which
is denoted by We immediately apply the CLT to write:

We will use as an approximate pivot since its asymptotic
distribution is N(0, 1) which is free from λ.
Confidence Intervals for the Mean
Let us first pay attention to the one-sample problems. With preassigned α
∈ (0, 1), we claim that

which leads to the confidence interval

with an approximate confidence coefficient 1 − α. Recall that z is the upper
α/2
100(α/2)% point of the standard normal distribution. See, for example, the
Figure 12.3.1.
A different confidence interval for λ is given in the Exercise 12.3.14.

Next, let us briefly discuss the two-sample problems. Suppose that the
random variables X , ..., X are iid from the i population having the Poisson(λ)
th
i1
i
ini
distribution where 0 < λ < ∞ is unknown, i = 1, 2. We suppose that the X ’s
i 1j
are independent of the X ’s and denote the sample mean obtained
2j
from the i population, i = 1, 2. In this case, one invokes the following CLT:
th
If n → ∞, n → 8 such that n /n → δ for some 0 < δ < ∞, then
1 2 1 2
For large sample sizes n and n , we should use the random variable
1 2
as an approximate pivot