Page 578 - Probability and Statistical Inference

12. Large-Sample Inference 555

The Remark 12.3.2 again holds in essence in the case of the Poisson prob-
lem. Also, one can easily tackle the associated two-sample problems in the
Poisson case. We leave out the details as exercises.

In practice, a sample of size thirty or more is considered large.

Exercise 12.3.8 Customer arrivals at a train station’s ticket counter have a
Poisson distribution with mean λ per hour. The station manager wished to
check whether the rate of customer arrivals is 40 per hour or more at 10%
level. The data was collected for 120 hours and we found that 5,000 custom-
ers arrived at the ticket counter during these 120 hours. With α = .10 so that

z = 1.28, we wish to test H : λ = 40 versus H : λ > 40. We have = 5000/
0
.10
1
120 ≈ 41.6667. Now, in view of (12.3.38), we obtain
≈ 2.8868 which is larger than z . Thus,
.10
we reject H : λ = 40 in favor of H : λ > 40 at approximate 10% level. That is,
1
0
it appears that customers arrive at the ticket counter at a rate exceeding 40 per
hour. !
12.4 The Variance Stabilizing Transformations

Suppose that we have a sequence of real valued statistics {T ; n ≥ 1} such
n
that n (T − θ) N(0, σ ) as n → ∞. Then, the Mann-Wald Theorem from
½
2
n
Section 5.3.2 will let us conclude that

if g(.) is a continuous real valued function and g′(θ) is finite and nonzero.
When σ itself involves θ, one may like to determine an appropriate function
2
g(.) such that for large n, the approximate variance of the associated trans-
formed statistic g(T ) becomes free from the unknown parameter θ. Such a
n
function g(.) is called a variance stabilizing transformation.
In the case of both the Bernoulli and Poisson problems, discussed in the
previous section, the relevant statistic T happened to be the sample mean
n
and its variance was given by p(1 − p)/n or λ/n as the case may be. In what
follows, we first exhibit the variance stabilizing transformation g(.) in the
case of the Bernoulli and Poisson problems. We also include the famous arctan
transformation due to Fisher, obtained in the context of asymptotic distribu-
tion of the sample correlation coefficient in a bivariate normal population.