Parameter Estimation of Pareto Type II Distribution with NLMIXED in SAS

In several previous posts, I’ve shown how to estimate severity models under the various distributional assumptions, including Lognormal, Gamma, and Inverse Gaussian. However, I am not satisfied with the fact that the supporting domain of aforementioned distributions doesn’t include the value at ZERO.

Today, I had spent some time on looking into another interesting distribution, namely Pareto Type II distribution, and the possibility of estimating the regression model. The Pareto Type II distribution, which is also called Lomax distribution, is a special case of the Pareto distribution such that its supporting domain starts at ZERO (>= 0) with a long tail to the right, making it a good candidate for severity or loss distributions. This distribution can be described by 2 parameters, a scale parameter “Lambda” and a shape parameter “Alpha” such that prob(y) = Alpha / Lambda * (1 + y / Lambda) ^ (-(1 + Alpha)) with the mean E(y) = Lambda / (Alpha – 1) for Alpha > 1 and Var(y) = Lambda ^ 2 * Alpha / [(Alpha – 1) ^ 2 * (Alpha – 2)] for Alpha > 2.

With the re-parameterization, Alpha and Lambda can be further expressed in terms of E(y) = mu and Var(y) = sigma2 such that Alpha = 2 * sigma2 / (sigma2 – mu ^ 2) and Lambda = mu * ((sigma2 + mu ^ 2) / (sigma2 – mu ^ 2)). Below is an example showing how to estimate the mean and the variance by using the likelihood function of Lomax distribution with SAS / NLMIXED procedure.

data test;
  do i = 1 to 100;
    y = exp(rannor(1));
    output;
  end;
run;

proc nlmixed data = test tech = trureg;
  parms _c_ = 0 ln_sigma2 = 1;
  mu = exp(_c_);
  sigma2 = exp(ln_sigma2);
  alpha = 2 * sigma2 / (sigma2 - mu ** 2);
  lambda = mu * ((sigma2 + mu ** 2) / (sigma2 - mu ** 2));
  lh = alpha / lambda * ( 1 + y/ lambda) ** (-(alpha + 1));
  ll = log(lh);
  model y ~ general(ll);
  predict mu out = pred (rename = (pred = mu));
run;  

proc means data = pred;
  var mu y;
run;

With the above setting, it is very doable to estimate a regression model with the Lomax distributional assumption. However, in order to make it useful in production, I still need to find out an effective way to ensure the estimation convergence after including co-variates in the model.

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