Automatic Random Variate Generation

Automatic Random Variate Generation

1. Introduction

   • Density of the input
     • Fast generator
       • Simple code
   • Automatic
     • Black box
       • Avoid design of special algorithms
       • Work for large classes of distributions
       • Evaluate the probability density function
   • No Density
     • Observations from stochastic inputs
   • Choose input distribution
     • Johnson translation system
     • Bezier distributions

2. Universal Methods

   • Transformed Density Rejection
     • Given density
       • Transformed
       • TDR
     • Black box algorithm
     • Random variate x is generated
       • by inversion
         • Random variate generation
           • Correlation induction is easier
         • Random numbers recycled
   • Construction Points
     • Adaptive rejection sampling
       • Ratio q
         • Converges to 1
           • When n tends to infinity
         • Random variable
   • Higher dimensional distributions
     • Transformation T
     • Transformed density is concave
     • Construct tangential hyperplanes
     • Sweep-plane technique
       • Generate random variates from the hat

3. Generating a sample

   • Statistical problem
     • Modeling probability distributions from data
   • Parametric approach
     • Estimating parameters
       • Suitable standard distribution
   • Non-parametric approach
     • Estimating unknown distributions
     • Generating variates from empirical distributions
       • Generalising a sample
         • Linear interpolation
     • Profound a priori reasons
       • Favoring a standard distribution
   • Kenel density estimation
     • Naive Resampling
       • Randomly choose numbers of the sample
         • With replacement
       • Continous random
         • Small number of different values
     • Smoothed bootstrap
       • Noise to resampled numbers
         • Expectation 0
         • Small variance
         • Density
           • W
             • Kemel -> k(x)
               • Symmetric around the origin
         • Bandwidth
           • Smoothing parameter
             • Change variable bW
         • Minimise mean
           • Integrated square error
             • Silverman optimal b
         • Choise of kernel
           • Generator
             • Speed
             • Simplicity
           • Rectangular
             • Best choice
             • Density is not continuous

4. Generalising a Vector-sample

   • Model dependences
     • Specify multidimensional distribution of that random vector
   • Sampling from multidimensional KDE
     • Transform data
       • Unit covariance matrix
         • Simple reference method
           • Smoothing parameter
     • Smaller b if we assume unknown distribution is non-normal