Monte Carlo Statistical Methods

# Monte Carlo Statistical Methods - Monte Carlo Statistical...

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Unformatted text preview: Monte Carlo Statistical Methods/October 29, 2001 1 Monte Carlo Statistical Methods Christian P. Robert Universit´ e Paris Dauphine Monte Carlo Statistical Methods/October 29, 2001 2 Based on the book Monte Carlo Statistical Methods by Christian P. Robert and George Casella Springer-Verlag 1999 Monte Carlo Statistical Methods/October 29, 2001 3 1 Introduction 1.1 Statistical Models 1.2 Likelihood Methods 1.3 Bayesian Methods 1.4 Deterministic Numerical Methods 1.5 Simulation versus numerical analysis Monte Carlo Statistical Methods/October 29, 2001 4 • Experimenters choice before fast computers: ◦ Describe an accurate model which usually precludes computation of explicit answers ◦ or Choose a standard model which would allow such computations, but may not be a close representation of a realistic model. • Such problems contributed to the development of simulation-based inference Monte Carlo Statistical Methods/October 29, 2001 5 1.1 Statistical Models Example 1 -Censored data models- Missing data models where densities are not sampled directly. Typical simple statistical model: we observe Y 1 , ··· ,Y n ∼ f ( y | θ ) . The distribution of the sample given by the product n Y i =1 f ( y i | θ ) Inference about θ based on this likelihood. Monte Carlo Statistical Methods/October 29, 2001 6 With censored random variables, actual observations: Y * i = min { Y i , u } where u is censoring point. Inference about θ based on the censored likelihood. Monte Carlo Statistical Methods/October 29, 2001 7 For instance, if X ∼ N ( θ,σ 2 ) and Y ∼ N ( μ,ρ 2 ) , the variable Z = X ∧ Y = min( X,Y ) is distributed as • 1- Φ z- θ σ ¶‚ × ρ- 1 ϕ z- μ ρ ¶ + • 1- Φ z- μ ρ ¶‚ σ- 1 ϕ z- θ σ ¶ where ϕ and Φ are the density and cdf of the normal N (0 , 1) distribution. Monte Carlo Statistical Methods/October 29, 2001 8 Similarly, if X ∼ Weibull( α,β ) , with density f ( x ) = αβx α- 1 exp(- βx α ) the censored variable Z = X ∧ ω, ω constant , has the density f ( z ) = αβz α e- βz α II z ≤ ω + Z ∞ ω αβx α e- βx α dx ¶ δ ω ( z ) , where δ a ( · ) Dirac mass at a . Monte Carlo Statistical Methods/October 29, 2001 9 Example 2 -Mixture models- Models of mixtures of distributions : X ∼ f j with probability p j , for j = 1 , 2 ,...,k , with overall density X ∼ p 1 f 1 ( x ) + ··· + p k f k ( x ) . For a sample of independent random variables ( X 1 , ··· ,X n ), sample density n Y i =1 { p 1 f 1 ( x i ) + ··· + p k f k ( x i ) } . Expanding this product involves k n elementary terms: prohibitive to compute in large samples. Monte Carlo Statistical Methods/October 29, 2001 10 1.2 Likelihood Methods Maximum Likelihood Methods ◦ For an iid sample X 1 , ..., X n from a population with density f ( x | θ 1 ,...,θ k ), the likelihood function is L ( θ | x ) = L ( θ 1 ,...,θ k | x 1 ,...,x n ) = Y n i =1 f ( x i | θ 1 ,...,θ k ) ....
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Monte Carlo Statistical Methods - Monte Carlo Statistical...

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