poisson.blaker.acc {BlakerCI}

Blaker's Poisson acceptability function, optionally unimodalized.

Description

Calculates values of the acceptability function for the Poisson distribution (see Blaker (2000)) in a sequence of points (for, e.g., plotting purposes). The acceptability function may optionally be “unimodalized”, i.e. replaced with the smallest greater or equal unimodal function.

Usage

poisson.blaker.acc(x, p, type = c("orig", "unimod"),
      acc.tol = 1e-10, ...)

Arguments

x

number of events.

p

vector (length 1 allowed) of hypothesized Poisson parameters. In case of more than one point, an increasing sequence required.

type

for type = "orig", original acceptability function calculated.
For type = "unimod", smallest unimodal function greater or equal to the acceptability function calculated instead.

acc.tol

numerical tolerance (relevant only for type = "unimod").

...

additional arguments to be passed to poisson.blaker.acc.single.p; in fact, just maxiter (see BlakerCI-internal).

Details

Single values of the “unimodalized” acceptability function (for type = "unimod") are computed by an iterative numerical algorithm implemented in internal function
poisson.blaker.acc.single.p. The function cited is called just once in each of the intervals where the acceptability function is continuous (namely in the leftmost one of those points of p that fall into the interval when dealing with points below x, and the rightmost one when above x). The rest is done by function cummax. This is considerably faster than calling poisson.blaker.acc.single.p for every point of p. Note that applying cummax directly to a vector of unmodified acceptability values is even faster and provides a unimodal output; it may, nevertheless, lack accuracy.

Value

Vector of acceptability values (with or without unimodalization) in points of p.

Note

Inspired by M.P. Fay (2010), mentioning “unavoidable inconsistencies” between tests with non-unimodal acceptability functions and confidence intervals derived from them. When the acceptability functions are unimodalized and the test modified accordingly (i.e. p-values slightly increased in some cases), a perfectly matching test-CI pair is obtained.

Author(s)

Jan Klaschka [e-mail]

References

Blaker, H. (2000) Confidence curves and improved exact confidence intervals for discrete distributions. Canadian Journal of Statistics 28: 783-798.
(Corrigenda: Canadian Journal of Statistics 29: 681.)

Fay, M.P. (2010). Two-sided Exact Tests and Matching Confidence Intervals for Discrete Data. R Journal 2(1): 53-58.

Examples

p <- seq(0,10,length=1001)
acc <- poisson.blaker.acc(3,p)
acc1 <- poisson.blaker.acc(3,p,type="unimod")
plot(p,acc,type="l")
lines(p,acc1,col="red")
legend(x=7,y=.8,c("orig","unimod"),col=c("black","red"),lwd=1)

## The two lines -- the unimodalized and original acceptabilities -- 
## look almost the same but some small differences are slightly 
## visible.

## They can be seen better this way:
plot(p,acc1-acc,type="l")

## Focussing on one of them:
p <- seq(5.05,5.6,length=1001)
acc <- poisson.blaker.acc(3,p)
acc1 <- poisson.blaker.acc(3,p,type="unimod")
plot(p,acc,type="l",ylim=c(.391,.396))
lines(p,acc1,col="red")
legend(x=5.4,y=.395,c("orig","unimod"),col=c("black","red"),lwd=1)