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Terminating Statistical Analysis

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Terminating Statistical Analysis

By Dr. Jason Merrick

- Random input leads to random output (RIRO)
- Run a simulation (once) — what does it mean?
- Was this run “typical” or not?
- Variability from run to run (of the same model)?

- Need statistical analysis of output data
- Time frame of simulations
- Terminating: Specific starting, stopping conditions
- Steady-state: Long-run (technically forever)
- Here: Terminating

Simulation with Arena — Intermediate Modeling and Terminating Statistical Analysis

- Suppose we are trying to estimate an output measure E[Y] = based upon a simulated sample Y1,…,Yn
- We come up with an estimate
- For instance

- How good is this estimate?
- Unbiased
- Low Variance (possibly minimum variance)
- Consistent
- Confidence Interval

Simulation with Arena — Intermediate Modeling and Terminating Statistical Analysis

- The t-statistic is given by
- If the Y1,…,Ynare normally distributed and then the t-statistic is t-distributed
- If the Y1,…,Ynare not normally distributed, but then the t-statistic is approximately t-distributed thanks to the Central Limit Theorem
- requires a reasonably large sample size n

- We require an estimate of the variance of denoted

Simulation with Arena — Intermediate Modeling and Terminating Statistical Analysis

- An approximate confidence interval for is then
- The center of the confidence interval is
- The half-width of the confidence interval is
- is the 100(/2)% percentile of a t-distribution with f degrees of freedom.

Simulation with Arena — Intermediate Modeling and Terminating Statistical Analysis

- Case 1: Y1,…,Ynare independent
- This is the case when you are making n independent replications of the simulations
- Terminating simulations
- Try and force this with steady-state simulations

- Compute your estimate and then compute the sample variance
- s2 is an unbiased estimator of the population variance, so s2/n is an unbiased estimator of with f = n-1 degrees of freedom

- This is the case when you are making n independent replications of the simulations

Simulation with Arena — Intermediate Modeling and Terminating Statistical Analysis

- Case 2: Y1,…,Ynare not independent
- This is the case when you are using data generated within a single simulation run
- sequences of observations in long-run steady-state simulations

- s2/n is a biased estimator of
- Y1,…,Ynis an auto-correlated sequence or a time-series
- Suppose that our point estimator for is , a general result from mathematical statistics is

- This is the case when you are using data generated within a single simulation run

Simulation with Arena — Intermediate Modeling and Terminating Statistical Analysis

- Case 2: Y1,…,Ynare not independent
- For n observations there are n2 covariances to estimate
- However, most simulations are covariance stationary, that is for all i, j and k
- Recall that k is the lag, so for a given lag, the covariance remains the same throughout the sequence
- If this is the case then there are n-1 lagged covariances to estimate, denoted k and

Simulation with Arena — Intermediate Modeling and Terminating Statistical Analysis

Positively correlated sequence with lag 1

Positively correlated sequence with lags 1 & 2

Positively correlated, covariance non-stationary sequence

Negatively correlated sequence with lag 1

Simulation with Arena — Intermediate Modeling and Terminating Statistical Analysis

- Case 2: Y1,…,Ynare not independent
- What is the effect of this bias term?
- For primarily positively correlated sequences B < 1, so the half-width of the confidence interval will be too small
- Overstating the precision => make conclusions you shouldn’t

- For primarily negatively correlated sequences B > 1, so the half-width of the confidence interval will be too large
- Underestimating the precision => don’t make conclusions you should

Simulation with Arena — Intermediate Modeling and Terminating Statistical Analysis

- For terminating case, make IID replications
- Simulate module: Number of Replications field
- Check both boxes for Initialization Between Reps.
- Get multiple independent Summary Reports
- Different random seeds for each replication

- How many replications?
- Trial and error (now)
- Approximate no. for acceptable precision
- Sequential sampling

- Save summary statistics (e.g. average, variance) across replications
- Statistics Module, Outputs Area, save to files

Simulation with Arena — Intermediate Modeling and Terminating Statistical Analysis

- Prefer smaller confidence intervals — precision
- Notation:
- Confidence interval:
- Half-width =

Want this to be “small,” say

< h where h is prespecified

Simulation with Arena — Intermediate Modeling and Terminating Statistical Analysis

- To improve the half-width, we can
- Increase the length of each simulation run and so increase the mi
- What does increasing the run length do?
- Increase the number of replications

Simulation with Arena — Intermediate Modeling and Terminating Statistical Analysis

- Set half-width = h, solve for
- Not really solved for n (t, s depend on n)
- Approximation:
- Replace t by z, corresponding normal critical value
- Pretend that current s will hold for larger samples
- Get

- Easier but different approximation:

s = sample standard

deviation from “initial”

number n0 of replications

n grows quadratically

as h decreases.

h0 = half width from “initial”

number n0 of replications

Simulation with Arena — Intermediate Modeling and Terminating Statistical Analysis