Point Estimation

• The point estimator of $\theta$ based on the data $Y_1, Y_2, ... Y_n$ is defined by
  
  $$\hat{\theta} = \frac{1}{n} \sum_{i=1}^n Y_i$$
  
  

• The point estimator $\hat{\theta}$ is said to be unbiased for $\theta$ if
  
  $$E(\hat{\theta}) = \theta$$
  
  
  In general
  
  $$E(\hat{\theta}) = \theta + b$$
  
  
  i.e. there is a drifting, or bias.

• For continuous data, the point estimator of $\phi$ based on data $\{ Y(t), 0 \le t \le T_E\}$, where $T_E$ is the simulation run length, is defined by
  
  $$\hat{\phi} = \frac{1}{T_E} \int_0^{T_E} Y(t) dt$$
  
  
  and is called a time average of Y(t) over $[0, T_E]$.

• In general
  
  $$E(\hat{\phi}) = \phi + b$$
  
  
  if b = 0, $\hat{\phi}$ is said to be unbiased for $\phi$

• One performance measure of these estimators (point or interval) is a quantile or a percentile.
  • Quantiles describe the level of performance that can be delivered with a given probability p
  • Assume Y represents the delay in queue a customer experiences, then the 0.85 quantile (or 85% percentile) of Y is the value $\gamma$ such that
    
    $$P(Y \le \gamma) = p = 0.85$$