Non parametric confidence intervals for the extinction probability of a Galton–Watson branching process

ABSTRACT

It is demonstrated that the confidence intervals (CIs) for the probability of eventual extinction and other parameters of a Galton–Watson branching process based upon the maximum likelihood estimators can often have substantially lower coverage when compared to the desired nominal confidence coefficient, especially in small, more realistic sample sizes. The same conclusion holds for the traditional bootstrap CIs. We propose several adjustments to these CIs, which greatly improves coverage in most cases. We also make a correction in an asymptotic variance formula given in Stigler (1971). The focus here is on implementation of the CIs which have good coverage, in a wide variety of cases. We also consider expected CI lengths. Some recommendations are made.

KEYWORDS:

• Adjusted proportions
• bootstrap
• confidence intervals
• extinction probability
• Galton–Watson branching process
• maximum likelihood estimation.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62