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Statistical Information Risk. Gary G. Venter Guy Carpenter Instrat. Statistical Information Risk. Estimation risk Uncertainty related to estimating parameters of distributions from data Projection risk Uncertainty arising from the possibility of future changes in the distributions

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statistical information risk

Statistical Information Risk

Gary G. Venter

Guy Carpenter Instrat

statistical information risk1
Statistical Information Risk
  • Estimation risk
    • Uncertainty related to estimating parameters of distributions from data
  • Projection risk
    • Uncertainty arising from the possibility of future changes in the distributions
  • Both quantifiable to some extent
topics
Topics
  • Impact of information risk
  • Estimation risk 1 – asymptotic theory
  • Estimation risk 2 – Bayesian updating
  • Estimation risk 3 – both together
  • Projection risk
impact of information risk
Impact of Information Risk
  • Collective risk theory
    • L = Si=1N Xi
    • E(L) = E(N)E(X)
    • Var(L) = E(N)Var(X) + E(X)2Var(N)
    • CV(L)2 = Var(X)/[E(N)E(X)2] + Var(N)/E(N)2
    • = [CV(X)2 + VM(N)] / E(N)
    • = [ 49 + 1 ] / E(N)
    • Goes to zero as frequency gets large
  • Add projection risk
    • K = JL, E(J) = 1, so E(K) = E(L)
    • CV(K)2 = [1+CV(J)2]CV(L)2 + CV(J)2
    • Lower limit is CV(J)2
estimation risk 1 asymptotic theory
Estimation Risk 1 – Asymptotic Theory
  • At its maximum, all partial derivatives of log-likelihood function with respect to parameters are zero
  • 2nd partials are negative
  • Matrix of negative of expected value of 2nd partials called “information matrix”
  • Usually evaluated by plugging maximizing values into formulas for 2nd partials
  • Matrix inverse of this is covariance matrix of parameters
  • Distribution of parameters is asymptotically multivariate normal with this covariance
estimation risk 1 asymptotic theory1
Estimation Risk 1 – Asymptotic Theory
  • See Loss Models p. 63 for discussion
  • Simulation proceeds by simulating parameters from normal, then simu-lating losses from the parameters
  • See Loss Models p. 613 for how to simulate multivariate normals
estimation risk 2 bayesian updating
Estimation Risk 2 – Bayesian Updating
  • Bayes Theorem:
    • f(x|y)f(y) = f(x,y) = f(y|x)f(x)
    • f(x|y) = f(y|x)f(x)/f(y)
    • f(x|y)  f(y|x)f(x)
  • Diffuse priors
    • f(x)  1
    • f(x)  1/x
estimation risk 2 bayesian updating frequency example
Estimation Risk 2 – Bayesian UpdatingFrequency Example
  • Poisson distribution with diffuse prior f(l) 1/l
    • f(k| l) = e- llk/k!
    • f(l|k)  f(k| l)/l e- llk-1
    • Gamma distribution in k, 1
    • That’s posterior; predictive distribution of next observation is mixture of Poisson by that gamma, which is negative binomial
    • After n years of observation, predictive distribution is negative binomial with same mean as sample and variance = mean*(1+1/n)
    • Parameter distribution for l is gamma with mean of sample and variance = mean/n
    • Can simulate from negative binomial or from gamma then Poisson
estimation risk 3 bayes information
Estimation Risk 3 – Bayes + Information
  • Tests developed by Rodney Kreps
  • Likelihood function is proportional to f(data|parameters)
  • Assume prior for parameters is proportional to 1
  • Then likelihood function is proportional to f(parameters|data)
testing assumptions of parameter distribution
Testing Assumptions of Parameter Distribution
  • Use information matrix to get covariance matrix of parameters
  • This is quadratic term of expansion of likelihood function around max
  • Compare bivariate normal and bivariate lognormal parameter distributions to scaled likelihood function
quantifying projection risk
Quantifying Projection Risk
  • Regression 100q% confidence intervals
    • t(q/2;N-2) is the upper 100(q/2)% point from a Student “T” distribution with N-2 degrees of freedom
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