The genetic distance between two loci on a chromosome is defined as the mean number of crossovers between the loci. The parameters of the crossover distribution are constrained by the parameters of the distribution of chiasmata. Ott (1996) derived the maximum likelihood estimator (MLE) of the parameters of the crossover distribution and the MLE of the mean. We demonstrate that the MLE of the mean is pointwise less than or equal to the empirical mean number of crossovers. It follows that the MLE is negatively biased. For small sample sizes the bias can be nonnegligible. We recommend reduced bias estimators.
Generalizations to many other problems involving linear constraints on parameters are made. Included in the generalizations are a variety of problems involving simplex constraints as studied recently by Liu (2000).
"On the bias in estimating genetic length and other quantities in simplex constrained models." Ann. Statist. 30 (1) 202 - 219, February 2002. https://doi.org/10.1214/aos/1015362190