Exact inbreeding coefficient and effective size of finite populations under partial sib mating

An exact recurrence equation for inbreeding coefficient is derived for a partially sib-mated population of N individuals mated in N/2 pairs. From the equation, a formula for effective size (Ne) taking second order terms of 1/N into consideration is derived. When the family sizes are Poisson or equally distributed, the formula reduces to Ne = [(4 - 3 beta) N/(4 - 2 beta)] + 1 or Ne = [(4 - 3 beta) N/(2 - 2 beta)] - 8/(4 - 3 beta), approximately. For the special case of sib-mating exclusion and Poisson distribution of family size, the formula simplifies to Ne = N + 1, which differs from the previous results derived by many authors by a value of one. Stochastic simulations are run to check our results where disagreements with others are involved.