• Linear Regression
  • This we understand—best linear fit
• Logistic Regression
  • Convert binary classification into a linear regression problem via the logistic function
• Singular-Value Decomposition
  • Originally developed for data which is linearly separable in higher dimensions
    • Increase the number of dimensions to \(n\) where one can find a hyperplane of dimension \(n-1\) that separates the data as needed
  • Has been extended to non-linearly separable sets via a hinge function
    • Hinge loss is a continuous linear function, but not differentiable

• Known dependent variable

• Many independent variables (features)

• Classifying or describing in-sample

• Model can be coerced into linear format

  • LM \(\rightarrow\) GLM \(\rightarrow\) GLMM \(\rightarrow\) GAM

• Modeling outcomes with no clear or clean dependent/independent relationships

  • Hurricane-caused flood damage in Florida gulf coast

  • Property damage due to electrical power plant turbine explosion

  • Corporate liability due to faulty products

• Other kinds of modeling are necessary

Here is a summary of 25 synthetic years of commercial liability losses suffered by XYZ Manufacturing last year (assume brought to common rate and exposure level…)

##        Min.     1st Qu.      Median        Mean     3rd Qu.        Max. 
##   "194,994"   "531,144"   "907,272"   "948,864" "1,262,869" "2,386,361"

• If the XYZ wants to put aside money to cover their expected loss 99.5% of the time, how much do they need to reserve?

• Not enough other information to relate to independent features like policyholder income or state of domicile

• Need new methods to estimate probabilities

• The following slides are a bit of a simplification 8-)

• Select a probability distribution and solve for parameters which match the first \(n\) moments as needed

• Normal Distribution:

  • \(\hat{\mu} = \bar{x}\) and \(\hat{\sigma^2} = \bar{v}\) where this is the biased empirical variance
  • \(\hat{\mu}\): 948,864; \(\hat{\sigma}\): 523,150

• Gamma Distribution

  • \(f(x) = \frac{x^{\alpha - 1}}{\Gamma(\alpha)\theta^\alpha}e^{-\frac{x}{\theta}}\)
  • Shape = \(\alpha\) = 3.28969; Scale = \(\theta\) = 288,436
  • \(\hat{\mu}\): 948,864; \(\hat{\sigma}\): 523,150

Find the parameters that maximize the total likelihood of the observed data for the given distribution

• A likelihood is the value of the probability density function at the observed data point
• A data sets likelihood is the product of the likelihoods of the observations
• This gets very hard to calculate very quickly due to most values being well below zero
• Logarithms to the rescue!
  • The log of a product is the sum of the logs of the components
• Most common procedure is to minimize sum of negative log likelihoods
  • Most non-linear optimizers are calibrated to minimize and objective function

• Normal Distribution:
  • Can be shown that the MLE = MoM
  • \(\hat{\mu}\): 948,864; \(\hat{\sigma}\): 523,150
• Gamma Distribution
  • Shape = \(\alpha\) = 3.35264; Scale = \(\theta\) = 283,020
  • \(\hat{\mu}\): 948,867; \(\hat{\sigma}\): 518,217

• Gamma options have thicker tail than Normal
• In this case, MLE returned thicker tail than MoM for Gamma
  • Does tail thickness matter when trying to predict open-ended liabilities which haven’t happened yet?
• Neither family fits that well
  • A lot of parameter risk with only 25 observations
  • Try other distributional families
  • Use various goodness-of-fit measures to select best distribution
    • Information Criteria
    • Q-Q plots
    • Cramer-Von Mises
    • Anderson-Darling

True distribution was a Weibull with shape = 2 and scale \(\approx\) 1,128,379