Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.3.∗Confidence Intervals for Parameters of Discrete Distributions 251

So the solution to the first equation isθ=0.434. In the same way, because
P(bin(30, 0 .7)≤18) = 0.1593 andP(bin(30, 0 .8)≤18) = 0.0094, the values 0. 7
and 0.8 bracket the solution to the second equation. The R segment for the solution
is:

binomci(18,30,.7,.8,.05); $solution 0.75047
Thus the confidence interval is (0. 434 , 0 .750), with a confidence of at least 90%.
For comparison, the asymptotic 90% confidence interval of expression (4.2.7) is
(0. 453 , 0 .747); see Exercise 4.3.2.

Example 4.3.3(Confidence Interval for the Mean of a Poisson Distribution).Let

X 1 ,X 2 ,...,Xnbe a random sample on a random variableXthat has a Poisson

distribution with meanθ.LetX=n−^1

∑n
i=1Xibe our point estimator ofθ.As
with the Bernoulli confidence interval in the last example, we can work withnX,
which, in this case, has a Poisson distribution with meannθ.ThecdfofXis

FX(x;θ)=

∑nx

j=0

e−nθ

(nθ)j

j!

=

1

Γ(nx+1)

∫∞

nθ

xnxe−xdx, (4.3.5)

where the integral equation is obtained in Exercise 4.3.7. From expression (4.3.5),

we immediately have

d

dθ

FX(x;θ)=

−n

Γ(nx+1)

(nθ)nxe−nθ< 0.

Therefore,FX(x;θ) is a strictly decreasing function ofθfor every fixedx.Fora
given sample, letxbe the realization of the statisticX. Hence, as discussed above,
forα 1 ,α 2 >0 such thatα 1 +α 2 < 1 /2, the confidence interval is given by (θ,θ),
where ∑
nx− 1
j=0 e

−nθ(nθ)j

j! =1−α^2 and

∑nx

j=0e

−nθ(nθ)j
j! =α^1. (4.3.6)
The confidence coefficient of the interval (θ,θ)isatleast1−α=1−(α 1 +α 2 ). As
with the Bernoulli proportion, these equations can be solved iteratively.

Numerical Illustration. Supposen= 25 and the realized value ofXisx=5;
hence,nx= 125 events have occurred. We selectα 1 =α 2 =0.05. Then, by (4.3.7),
our confidence interval solves the equations
∑ 124
j=0e

−nθ(nθ)j

j! =0.95 and

∑ 125

j=0e

−nθ(nθ)j

j! =0.^05. (4.3.7)

Our R function^2 poissonci.ruses the bisection algorithm to solve these equations.

Since ppois(124, 25 ∗4) = 0.9932 and ppois(124, 25 ∗ 4 .4) = 0.9145, for the first

equation, 4.0 and 4.4 bracket the solution. Here is the call topoissonci.ralong

with the solution (the lower bound of the confidence interval):

(^2) Download this function at the site given in the Preface.