1. Basic simulation methodology
Simulating multivariate distributions
Simulating random sequences
Importance sampling
Antithetic sampling
Quasi random numbers
Computational statistics, lecture 2

2. Methods for simulating multivariate distributions
Transforming pseudo random numbers (PRNs) having a multivariate distribution that is easy to simulate
Using factorization of the multivariate density into univariate density functions
Using envelope-rejection techniques
Computational statistics, lecture 2

3. Illustrations of independent and dependent normal distributions
Computational statistics, lecture 2

4. Computational statistics, lecture 2
Theory of transforming a normal distribution to another normal distribution
Let X be a random vector having a zero mean m-dimensional normal distribution with covariance matrix C
Let Y = B X where B is an arbitrary k x m matrix
Then Y has a k-dimensional zero mean normal distribution with covariance matrix B C BT
Computational statistics, lecture 2

5. Computational statistics, lecture 2
Generating a bivariate normal distribution with a given covariance matrix: method 1
Let Y be a random vector having a bivariate normal distribution with covariance matrix C
Then, C can be decomposed into a product C = B BT
Furthermore, the random vector B X, where X has a standard bivariate normal distribution, has a bivariate normal distribution with covariance matrix C
Example:
Computational statistics, lecture 2

6. Computational statistics, lecture 2
Generating a bivariate normal distribution with a given covariance matrix: method 2
Let Y be a zero mean bivariate normal distribution with density
Decompose the probability density into
Note that the conditional distribution is normal with
Example:
Computational statistics, lecture 2

7. Random number generation: method 3 - the envelope-rejection method
Generate x from a probability density g(x) such that cg(x)  f(x) where c is a constant
Draw u from a uniform distribution on (0,1)
Accept x if u < f(x)/cg(x)
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Justification:
Let X denote a random number
from the probability density g.
Then
How can we generate normally distributed random numbers?
Computational statistics, lecture 2

8. Simulation of random sequences
Example 1: Random walk
Example 2: Autoregressive process
Note: A burn-in period is needed
Computational statistics, lecture 2

9. Computational statistics, lecture 2
Simulating rare events by shifting the probability mass to the event region
Assume that we would like to estimate pt = P(X > t) where X is a random variable with density f(x)
Let f* be an alternate probability density
Then
and we can estimate pt by computing
where Xi has density f*
Computational statistics, lecture 2

10. Simulating rare events by scaling or translation
Assume that we would like to estimate p = P(X > t) where X has probability density f(x)
Scaling:
Translation:
Computational statistics, lecture 2

11. Simulating rare events by scaling: a simple example
Assume that we would like to estimate p = P(X > 4) where X has a standard normal distribution.
Let f* be the probability density of 10X
Then
and we can estimate pt by computing
where Xi is normal with mean zero and standard deviation 10
Computational statistics, lecture 2

12. Computational statistics, lecture 2
Antithetic sampling
Use the same sequence of underlying random variates to generate a second sample in such a way that the estimate of the quantity of interest from the second sample will be negatively correlated with the estimate from the original sample.
Computational statistics, lecture 2

13. Antithetic sampling – a simple example
Use Monte-Carlo simulation to estimate the integral
How can we apply the principle of antithetic sampling?
Computational statistics, lecture 2

14. Quasi random numbers (minimal discrepancy sequences)
Quasi-random numbers give up serial independence of subsequently generated values in order to obtain as uniform as possible coverage of the domain
This avoids clusters and voids in the pattern of a finite set of selected points
Computational statistics, lecture 2

15. Pseudo and quasi random numbers
Pseudo random numbers
Quasi random numbers
Computational statistics, lecture 2