chapter12_theory.lyx

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\begin_body

\begin_layout Chapter*
12.1 What is Simulation
\end_layout

\begin_layout Standard
Simulation is a way to use high-speed computer to substitute for analytical
 calculation.
 The law of large number tells us that if we observe a large sample of i.i.d.
 random variables with finite mean, then the average of these random variables
 should be close to their mean.
 If we can get a computer to produce such a large i.i.d.
 sample, then we can average the random variables instead of trying (and
 possibly failing) to calculate their mean analytically.
 For a specific problem, one needs to figure out what types of random variables
 one needs, how to make a computer produce them, and how many one needs
 in order to have any confidence in the numerical result.
 Each of these issues will be addressed to some extent in this chapter.
 
\end_layout

\begin_layout Section*
Proof of Concept
\end_layout

\begin_layout Standard
We begin with some simple examples of simulation to answer questions that
 we can already answer analytically just to show that simulation does what
 it advertises.
 Also, these simple examples will raise some of the issues to which one
 must attend then trying to answer more difficult questions using simulation.
\end_layout

\begin_layout Section*
Examples in which Simulation Might Help
\end_layout

\begin_layout Standard
Next, we present some examples where the basic questions are relatively
 simple to describe, by analytic solution would be tedious at best.
\end_layout

\begin_layout Section*
Summary 
\end_layout

\begin_layout Standard
Suppose that we want to know the mean of some function 
\begin_inset Formula $g$
\end_inset

 of a random variable or random vector 
\begin_inset Formula $W$
\end_inset

.
 For instance, in Example 
\begin_inset Formula $12.1.3$
\end_inset

 we can let 
\begin_inset Formula $W=(X,Y)$
\end_inset

 and 
\begin_inset Formula $g(W)=|X-Y|$
\end_inset

 .
 If a computer can supply a large number of i.i.d.
 random variables (or random vectors) with the distribution of 
\begin_inset Formula $W$
\end_inset

, one can use the average of the simulated values of 
\begin_inset Formula $g(W)$
\end_inset

 to approximate the mean of 
\begin_inset Formula $g(X)$
\end_inset

.
 One muse be careful to take the variability in 
\begin_inset Formula $g(W)$
\end_inset

 into account when deciding how much confidence to place in the approximation.
\end_layout

\begin_layout Chapter*
12.2 Why is Simulation Useful ?
\end_layout

\begin_layout Standard
Statistical simulations are used to estimate features of distributions such
 as means of functions, quantiles, and other features that we cannot compute
 in closed form.
 When using a simulation estimator, it is good to compute a measure of how
 precise the estimator is, in addition to the estimate itself.
\end_layout

\begin_layout Section*
Examples of Simulation 
\end_layout

\begin_layout Section*
Assessing Uncertainty about Simulation Results
\end_layout

\begin_layout Standard
We should always attempt to assess the uncertainty in a simulation.
 This uncertainty is most easily assessed via the simulation variance of
 the simulated quantity.
 The smaller the simulation variance is, the more certain we can be that
 our estimator 
\begin_inset Formula $Z$
\end_inset

 is close to what we are trying to estimate.
 But we need to estimate or bound the simulation before we can assess the
 amount of uncertainty.
 How we estimate the simulation variance of a result 
\begin_inset Formula $Z$
\end_inset

 depends on whether 
\begin_inset Formula $Z$
\end_inset

 is an average of simulated values.
 The square root of our estimate of the simulation variance will be called
 the simulation standard error, and it is estimate of the simulation standard
 deviation of 
\begin_inset Formula $Z$
\end_inset

.
 The simulation standard error is a popular way to summarize uncertainty
 about a simulation for two reasons.
 Firsts, it has the same units of measurement as the quantity that was estimated
 (unlike the simulation variance).
 Second, the simulation standard error is useful for saying how likely it
 is that the simulation estimator is close to the parameter being estimated.
 We shall explain this second point in more detail after we show how to
 calculate the simulation standard error in several common cases.
\end_layout

\begin_layout Section*
Summary 
\end_layout

\begin_layout Standard
If we wish to compute the expected value 
\begin_inset Formula $\theta$
\end_inset

 of some random variable 
\begin_inset Formula $Y$
\end_inset

, but cannot perform the necessary calculation in closed form, we can use
 simulation.
 In general, we would simulate a large random sample 
\begin_inset Formula $Y^{(1)},...,Y^{(v)}$
\end_inset

 from the same distribution as 
\begin_inset Formula $Y$
\end_inset

, and then compute the sample mean 
\begin_inset Formula $Z$
\end_inset

 as our estimator.
 We can also estimate a quantile 
\begin_inset Formula $\theta_{p}$
\end_inset

 of a distribution in a similar fashion.
 If 
\begin_inset Formula $Y^{(1)},...,Y^{(v)}$
\end_inset

 is a large sample from the distribution, we can compute the sample 
\begin_inset Formula $p$
\end_inset

 quantile 
\begin_inset Formula $Z$
\end_inset

.
 It is always a good idea to compute some measure of how good a simulation
 estimator is.
 One common measure is the simulation standard error of 
\begin_inset Formula $Z$
\end_inset

, an estimate of the standard deviation of the simulation distribution of
 
\begin_inset Formula $Z$
\end_inset

.
 Alternatively, one could perform enough simulations to make sure that the
 probability is high that the 
\begin_inset Formula $Z$
\end_inset

 is close to the parameter being estimated.
\end_layout

\end_body
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