### The initial transient in steady-state point estimation: Contexts, a bibliography, the mse criterion, and the MSER statistic

Raghu Pasupathy, Bruce Schmeiser
2010 Proceedings of the 2010 Winter Simulation Conference
The initial transient is an unavoidable issue when estimating parameters of steady-state distributions. We discuss contexts and factors that affect how the initial transient is handled, provide a bibliography (from the system simulation literature), discuss criteria for evaluating initial-transient algorithms, arguing for focusing on the mean squared error (mse). We discuss the MSER statistic, showing that it is asymptotially proportional to the mse and therefore a good foundation for
more » ... ansient algorithms. We suggest two new algorithms (MSER-LLM and MSER-LLM2) for using the MSER statistic and compare them, based on empirical results for M/M/1 and AR(1) data processes, to the original MSER algorithm (MSER-GM). INTRODUCTION We consider a problem that arises in determining steady-state behavior of stochastic systems. Whether the analysis is via closed-form analysis, a deterministic numerical method, or Monte Carlo simulation, analysis must distinguish between transient results, which depend upon the initial state of the system, and steady-state results, which do not. Initial-transient problems arise in Monte Carlo simulation experiments designed to estimate the unknown value of a scalar steady-state performance measure q . Multiple performance measures might be, and often are, of interest; any algorithm then can be applied to each scalar performance measure. All such problems are provided time-series data from a given simulation oracle. One replication yields data Y 0 ,Y 1 , . . . ,Y N . Any two observations, say Y i and Y j , are not identically distributed because the initial observation Y 0 is not from the steady-state distribution. The output data are possibly autocorrelated because of the state-change logic of the system (e.g., traffic, factory, finance). The sample size N might be random, such as when the data are based on numbers of transactions in a fixed amount of simulated time. In multiple-replication variations of the problem, the data-generation process is repeated, usually independently distributed and always identically distributed. When using Monte Carlo simulation to estimate a steady-state performance measure q , a classic problem is what to do about the early data that reflect the transient effect of not starting in steady state. The problem is that of choosing the point estimator, Q, for q , given the time-series data from the simulation experiment. Typically the form of the point estimator matches the form of the performance measure; for example, if q is a mean, then Q is a sample average. The choice of estimator then reduces to the weight to be given to each of the data points, with early (transient) data receiving less weight and later (close to steady state) data receiving more weight. Mostly for simplicity, the weights are often