Erlang Mixtures

The density of a mixture of Erlangs with common scale parameter:

h(x |θ, α) =M∑i=1

aix i−1e−x/θ

θi (i − 1)!

where ai is the non-negative weight of the i-th Erlang distributionin the mixture, and θ is the common scale parameter.Theoretical justification (Tijms Approximation):Let

h(x |θ) =∞∑j=1

[F (jθ)− F ((j − 1)θ)]x j−1e−x/θ

θj(j − 1)!.

Then,limθ→0

H(x |θ) = F (x),

at all the continuous points.

The Standard EM algorithm

Proposed in Dempster, et al (1977), it is an iterative algorithm forfinding the maximum likelihood estimate of the parameters of anunderlying distribution from a set of incomplete data.

The EM algorithm for Erlang MixturesLet Φ = (θ, a1, a2, · · · , aM),

p(x , y |Φ) = ayxy−1e−x/θ

θy (y − 1)!

and

q(yi |xi ,Φ(k−1)) =p(xi , yi |Φ(k−1))

p(xi |Φ(k−1)).

Then

a(k)y =

1

n

n∑i=1

q(y |xi ,Φ(k−1)), y = 1, 2, · · · ,M,

and

θ(k) =

n∑i=1

xi/n

M∑y=1

ya(k)y

.

Uniform DistributionThe uniform distribution is a benchmark to test whether a fittingalgorithm is of high quality.A set of 1000 data between 1 and 2 were generated uniformly forstudy.

Figure: Histogram of uniform distribution and line for the fitteddistribution

Uniform Distribution

Figure: PP and QQ plots for uniform (1,2) and the fitted distribution

Mixture of two gamma’s

The underlying distribution is a mixture of 2 gamma distributionswith shape parameters being 2.6 and 6.3. The corresponding scaleparameters are 0.3125 and 0.8333 having weights 0.2 and 0.8respectively.

Mixture of two gamma’s, cont’d

Figure: Histogram for a mixture of two gammas and the fitted densityusing a mixture of 3 Erlangs

Lognormal distribution

Figure: Histogram for lognormal(0.03,0.2) and the fitted density using amixture of 11 Erlangs

PCS Catastrophe Loss Data

A data set of 1271 catastrophe losses in US from 1997 to 2005with all kinds of losses: flood, wind damage, etc..The data was supplied by ISO, a leading source of informationabout property losses in the U.S.Stylized Facts:

1. The maximum value of the data is 247 times of the mean.

2. There is 9.13% of the observations categorized as outliers if1.5 IQR rule is used.

3. The skewness and kurtosis for the data are 23.04 and 619.63.

4. 56% of the data is smaller than 0.1% of the maximum valuewhile 96.6% of the data is smaller than 1% of the maximumvalue.

All points above suggest that the data is heavy-tailed.

Fitting to the PCS DataA mixture of 12 Erlangs with a common scale parameter of5830867 fits the data well with each Erlang having significantweight.

Figure: Histogram of observed loss and line for the fitted distribution

p-p and q-q plots

Moments

Quantities Empirical Fitted Fitted/Empirical Percentage Difference (%)Mean 98.33 million 98.33 million 1.0000 0.00%

Standard Deviation 825.85 million 814.00 million 0.9857 -1.43%Skewness 23.03 23.37 1.0148 1.48%Kurtosis 619.28 643.31 1.0389 3.89%

Swedish mortality data

The mortality data for the Swedish cohort of Year 1911. The datais available at www.mortality.org.Reasons for choosing the data:

I Reliability due to the high life expectancy and homogeneity ofthe Swedish population.

I Cohort 1911 presents a “noticeable” accident hump at youthages.

Swedish Mortality Data

Figure: Fitted curve for qx