Empirical evaluation of initial transient deletion rules for the steady-state mean estimation problem

We propose three new initial transient deletion rules (denoted H1, H2 and H3) to reduce the bias of the natural point estimator when estimating the steady-state mean of a performance variable from the output of a single (long) run of a simulation. Although the rules are designed for the estimation of a steady-state mean, our experimental results show that these rules may perform well for the estimation of variances and quantiles of a steady-state distribution. One of the proposed rules can be applied under the only assumption that the output of interest $$\{ Y(s): s\ge 0 \}$$ { Y ( s ) : s ≥ 0 } has a stationary distribution whereas the other two rules require that $$Y(s)= f(X(s))$$ Y ( s ) = f ( X ( s ) ) for an $$\mathfrak {R}^d$$ R d -valued Markov chain $$\{ X(s): s \ge 0 \}$$ { X ( s ) : s ≥ 0 } . Our proposed rules are based on the use of sample quantiles and multivariate batch means to test the null hypothesis that a current observation Y(s) comes from a stationary distribution for $$\{ X(s): s \ge 0 \}$$ { X ( s ) : s ≥ 0 } . We present experimental results to compare the performance of the new rules against three variants of the Marginal Standard Error Rule and the Glynn-Iglehart deletion rule. When the run length was sufficiently large to provide a reliable confidence interval for the estimated parameter, one of the proposed rules (H3) provided the best reductions in Mean Square Error, so that the identification of an underlying Markov chain X for which $$Y(s)= f(X(s))$$ Y ( s ) = f ( X ( s ) ) can be useful to determine an appropriate deletion point to reduce the initial transient, and one of our proposed rules (H2) can be useful to detect that a run length is too small to provide a reliable confidence interval.