| A MONTE CARLO EXPERIMENT TO EVALUATE THE EFFICIENCY
OF THE MANDELBROT-LÉVY CHARACTERISTIC EXPONENT (a)
AS A TEST STATISTIC FOR CAUCHY DISTRIBUTED RANDOM VARIABLES
Robert F. Mulligan, Ph.D. (SUNY Binghamton) Institut für Weltwirtschaft - Kiel Institute of World Economics
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ABSTRACT
A refinement of Mandelbrot's technique for estimating the Mandelbrot-Lévy characteristic exponent (a ) is applied to Cauchy distributed pseudorandom samples. The approximate empirical distribution of a under the null hypothesis of a Cauchy population is obtained, and tables of probability values are calculated. Once the distribution of a under the Cauchy null is established, estimated a can be used as a test statistic for the Cauchy distribution, and evaluated using the tables of probability values calculated here.
Key words: Cauchy distribution, characteristic exponent, characteristic function, Mandelbrot-Lévy distribution.
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1. INTRODUCTION
The Mandelbrot-Lévy distributions are a family of probability
distributions which generally cannot be written as probability functions,
(i.e., the probability functions cannot be expressed in analytical form,
though there are three important exceptions.) Instead of being written
as probability functions, the Mandelbrot-Lévy distributions are
identified by their characteristic functions, which are written in logarithms
as:
Three special cases of the Mandelbrot-Lévy distribution can be written as probability functions:
2. If a = 2, ß = 0, the distribution is normal or Gaussian; and
3. If a = 1/2, ß = 1, and g = 1, the distribution governs tosses of a fair coin, and is sometimes called the Lévy distribution.
The Mandelbrot-Lévy distributions exhibit the property of stability, which means the distribution is invariant under addition. Stability is demonstrated by the fact that the log characteristic function of the sum of n independent and identically distributed Mandelbrot-Lévy random variables is:
This is the same as log f(t), except that d (location) and g (scale) are multiplied by n. Stability of the distribution under addition means that a (characteristic exponent) and ß (skewness) are constant. Stability holds as long as a and ß remain constant over all random variables being summed, even if d and g vary.
In the Calcul des Probabilités (1925), Paul Lévy proved that Mandelbrot-Lévy distributions are the only possible limiting distributions for sums of independent and identically distributed random variables. Gnedenko and Kolmolgorov (1954, 164) give a proof in English. The conventional central limit theorem is a special case restricting the summation of independent random variables to normality by requiring each variable have finite variance. Lévy's more general central limit theorem applies to sums of random variables with finite or infinite variance.
2. ESTIMATING THE CHARACTERISTIC EXPONENT (a)
In his famous series of papers on the statistical distribution of incomes (in which he introduced the stable Paretian hypothesis (1960, 1961, 1962),) and the distribution of speculative prices (1963a, 1963b, 1967, 1971b), and related papers (1971a, 1972, 1973), Benoit Mandelbrot used a graphical technique to estimate the characteristic exponent a . He defined U as a vector of observations of a random variable composed of a series of u's, and F(u) as the empirical cumulative density function. Using doubly logarithmic paper, Mandelbrot graphed log(u) on the horizontal axis, and log[1 - F(u)] on the vertical axis. The slope in the linear region of the upper tail of the marginal distribution is -a . The variance is finite if a = 2 and the series is normal. Mandelbrot found estimated a was not clustered closely around two for normal samples, but for nearly Cauchy samples, a was close to its theoretical value of one. He concluded that his method was highly accurate when the true a was less than 1.7.
The characteristic exponent a is an attractive test statistic for the Cauchy distribution both because a is simple to calculate, and can be estimated most accurately for Cauchy distributed samples. Mandelbrot (1963a, 1963b, 1967) suggested all economic variates (not just regression residuals) are drawn from some stable Mandelbrot-Lévy distribution for which a ranges from 1 (Cauchy) to 2 (normal). A variety of well-known tests for normality are available, but there is no statistical test for the Cauchy distribution.
The first step in performing the Monte Carlo study was generating a Cauchy variable. Pseudo-random numbers were generated by the multiplicative congruential method described by L'Ecuyer (1990) and Hall (1992a, 1992b), with modulus 231 - 1 and multiplier 41358, initially seeded with 9378214.
Next, the random variable was sorted so the marginal distribution could be examined. To estimate a , it was necessary to use only the upper tail of the empirical distribution. This meant that once the sample was sorted from least to greatest, it was necessary to decide how many of the largest observations to use to estimate a . Different percentages of the largest observations were tried before 20% was selected.
The 20% figure was selected by performing 100 iterations of the experiment with different percentages of the largest observations, ranging from 5% to 40%. In these preliminary trials, a non-linear, second-order term was added to the regression to estimate a , to see if the upper tail was linear. These preliminary test regressions had the form
log[1 - F(u)] = a + b log(u) + c [log(u)]2 + e;
t-statistics for c greater than 1.96 were counted as evidence that the tail was too big - i.e., so large it included some of the non-linear part of the distribution. Table 1. gives the results of this procedure.
The characteristic exponent a was estimated as the slope coefficient b in a regression of the form log[1 - F(u)] = a + b log(u) + e. The procedure was performed in a do-loop 1000 times to provide a sample of 1000 a 's. This sample was then used to calculate the table of approximate percentage points of a under the null of a Cauchy distributed population, with sample sizes ranging from 50 to 10,000.
3. MONTE CARLO RESULTS
Tables 2 through 9 present the approximate percentage points of the distribution of the test statistic a under the null hypothesis of a Cauchy distributed population, calculated from 1000 samples of 50, 100, 250, 500, 1000, 2500, 5000, and 10,000 observations. It is interesting that the mean and median values of a are both always below the theoretical value of one, though both the mean and the median appear to be converging to one.
Table 10 compares the means, medians, standard deviations, and ten percent critical values of a for different sample sizes. The test statistic appears to converge asymptotically to its theoretical value a = 1. The test statistic is less dispersed than the Jarque-Bera (1980) efficient normality test statistic, even for samples as small as fifty.
The tables of percentage points can be used by researchers to test if a variable is Cauchy distributed. The raw data should be sorted, and the 20% largest observations used in a regression to estimate the Mandelbrot-Lévy characteristic exponent a , following the procedure described above. Then the table for the same size sample gives the approximate probability level for the test statistic a under the null hypothesis. The null hypothesis is that the sample used to calculate a was drawn from a Cauchy population. The researcher decides to accept or reject the Cauchy null based on the probability level of the estimated a .
The author thanks Robert L. Basmann, Charles W. Bischoff, Clint Cummins,
Edward C. Kokkelenberg, Gene Martin, and Wentie L. Wang. I am responsible
for errors.
upper tail significant sample standard coefficient
area (%) t statistics on c deviation of variation
(avg. in 10 trials)
.05 6.7 3.71 55.42
.10 4.9 2.02 45.00
.15 5.7 2.00 35.14
.20 4.0 2.26 56.52
.25 5.4 2.27 42.05
.30 4.4 1.84 41.77
.35 3.5 1.90 54.29
.40 4.6 1.65 35.80
Note: A local minimum sample mean number of significant second-order coefficients occurred at .20 which was selected as the size of the upper tail used to calculate a .
as a function of sample size
sample size mean a median a
s.d. 10% crit values (90%) (two-tailed) range
1. 50 .926 .864 .357 .464 1.521 1.057
2. 100 .927 .882 .257 .556 1.354 .890
3. 250 .947 .931 .166 .691 1.215 .524
4. 500 .959 .955 .118 .765 1.153 .388
5. 1000 .968 .965 .0887 .825 1.108 .283
6. 2500 .976 .975 .0583 .876 1.066 .190
7. 5000 .981 .981 .0421 .906 1.048 .142
8. 10000 .986 .985 .0296 .935 1.034 .099
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