The Wald interval is obtained by inverting the acceptance region of the Wald
large-sample normal test.
The Wilson interval, which is the default, was introduced by Wilson (1927) and is
the inversion of the CLT approximation to the family of equal tail tests of p = p0.
The Wilson interval is recommended by Agresti and Coull (1998) as well as by
Brown et al (2001).
The Agresti-Coull interval was proposed by Agresti and Coull (1998) and is a slight
modification of the Wilson interval. The Agresti-Coull intervals are never shorter
than the Wilson intervals; cf. Brown et al (2001).
The Jeffreys interval is an implementation of the equal-tailed Jeffreys prior interval
as given in Brown et al (2001).
The modified Wilson interval is a modification of the Wilson interval for x close to 0
or n as proposed by Brown et al (2001).
The modified Jeffreys interval is a modification of the Jeffreys interval for
x == 0 | x == 1
and x == n-1 | x == n
as proposed by
Brown et al (2001).
The Clopper-Pearson interval is based on quantiles of corresponding beta
distributions. This is sometimes also called exact interval.
The arcsine interval is based on the variance stabilizing distribution for the binomial
distribution.
The logit interval is obtained by inverting the Wald type interval for the log odds.
The Witting interval (cf. Beispiel 2.106 in Witting (1985)) uses randomization to
obtain uniformly optimal lower and upper confidence bounds (cf. Satz 2.105 in
Witting (1985)) for binomial proportions.
For more details we refer to Brown et al (2001) as well as Witting (1985).