Variance estimation for nucleotide substitution models

The current variance estimators for most evolutionary models were derived when a nucleotide substitution number estimator was approximated with a simple first order Taylor expansion. In this study, we derive three variance estimators for the F81, F84, HKY85 and TN93 nucleotide substitution models, respectively. They are obtained using the second order Taylor expansion of the substitution number estimator, the first order Taylor expansion of a squared deviation and the second order Taylor expansion of a squared deviation, respectively. These variance estimators are compared with the existing variance estimator in terms of a simulation study. It shows that the variance estimator, which is derived using the second order Taylor expansion of a squared deviation, is more accurate than the other three estimators. In addition, we also compare these estimators with an estimator derived by the bootstrap method. The simulation shows that the performance of this bootstrap estimator is similar to the estimator derived by the second order Taylor expansion of a squared deviation. Since the latter one has an explicit form, it is more efficient than the bootstrap estimator.