Part 2 of a series

by Daniel Hanson, with contributions by Steve Su (author of the GLDEX package)

### Recap of Part 1

In our previous article, we introduced the four-parameter Generalized Lambda Distribution (GLD) and looked at fitting a 20-year set of returns from the Wilshire 5000 Index, comparing the results of two methods, namely the Method of Moments, and the Method of Maximum Likelihood.

Errata: One very important omission in Part 1, however, was not putting

require(GLDEX)

prior to the examples shown. Many thanks to a reader who pointed this out in the comments section last time.

Let’s also recall the code we used for obtaining returns from the Wilshire 5000 index, and the first four moments of the data (details are in Part 1):

require(quantmod) # quantmod package must be installed

getSymbols("VTSMX", from = "1994-09-01")

VTSMX.Close <- VTSMX[,4] # Closing prices

VTSMX.vector <- as.vector(VTSMX.Close)

# Calculate log returns

Wilsh5000 <- diff(log(VTSMX.vector), lag = 1)

Wilsh5000 <- 100 * Wilsh5000[-1] # Remove the NA in the first position,

# and put in percent format

# Moments of Wilshire 5000 market returns:

fun.moments.r(Wilsh5000,normalise="Y") # normalise="Y" -- subtracts the 3

# from the normal dist value.

# Results:

# mean variance skewness kurtosis

# 0.02824676 1.50214916 -0.30413445 7.82107430

Finally, in Part 1, we looked at two methods for fitting a GLD to this data, namely the Method of Moments (MM), and the Method of Maximum Likelihood (ML). We found that MM gave us a near perfect match in mean, variance, skewness, and kurtosis, but goodness of fit measures showed that we could not conclude that the market data was drawn from the fitted distribution. On the other hand, ML gave us a much better fit, but it came at the price of skewness being way off compared to that of the data, and kurtosis not being determined by the fitting algorithm (NA).

**Method of L-Moments (LM)**Steve Su, in his contributions to this article series, suggested the option of a “third way”, namely the Method of L-Moments. Also, as mentioned in the paper L-moments and TL-moments of the generalized lambda distribution, (William H. Asquith, 2006),

“The method of L-moments is an alternative technique, which is suitable and popular for heavy-tailed distributions. The method of L-moments is particularly useful for distributions, such as the generalized lambda distribution (GLD), that are only expressible in inverse or quantile function form.”

Additional details on the method and algorithm for computing it can be found in this paper, noted above.

As we will see in the example that follows, the result is essentially a compromise between our first two results, but the goodness of fit is still far preferable to that of the Method of Moments.

We follow the same approach as above, but using the GLDEX function fun.RMFMKL.lm(.) to calculate the fitted distribution: