Estimation of Mittag-Leffler Parameters

06/07/2018 ∙ by Dexter Cahoy, et al. ∙ 0

We propose a procedure for estimating the parameters of the Mittag-Leffler (ML) and the generalized Mittag-Leffler (GML) distributions. The algorithm is less restrictive, computationally simple, and necessary to make these models usable in practice. A comparison with the fractional moment estimator indicated favorable results for the proposed method.

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1 Introduction

For the last two decades, the Mittag-Leffler function has gained popularity in many scientific areas. For instance, the Mittag-Leffler (ML) distribution originally introduced by [Pillai(1990)] has now been used to model random phenomena in finance and economics [Meerschaert and Scalas(2006), Scalas(2006)]. In addition, [Kozubowski(2001)] studied the Mittag-Leffler ML

distribution where the integral and series representations of the probability density function are

correspondingly, is the scale parameter, and

is the generalized Mittag-Leffler function. The ML density function has the Laplace transform

and is showed to be geometric stable. It can also be easily seen as a generalization of the density of an exponential distribution with parameter

. Furthermore, [Kozubowski(2001)] constructed the following structural representation of a ML

distributed random variable

as

(1.1)

where has the standard exponential distribution , and has the density function

It is also shown that the fractional moment of the random variable is:

(1.2)

The parameter estimation problem for this model was first addressed by [Kozubowski(2001)]. They proposed fractional moment estimators for the ML distribution which require appropriate constants prior to the calculation of the estimates. Note that the pre-selection of appropriate values requires information about the true or unknown parameter a priori, which is not feasible in practice. As a direct consequence, it is expected that the above estimators will perform poorly when the restrictions are violated. Thus, it is mainly these drawbacks that motivate us to propose an estimation procedure that avoids this difficulty and uses all the available data possible.

More recently, [Jose et. al(2010)]

used the generalized Mittag-Leffler (GML) distribution in astrophysics and time series modeling. They specifically constructed GML processes which are autoregressive time series models with GML as the stationary marginal distribution. Moreover, the cumulative distribution function of the generalized Mittag-Leffler distribution GML

is given by

When

, we get the gamma distribution while

yields [Pillai(1990)]’s ML distribution. If , we get the exponential density. The GML probability density function has the specific form

where is the cumulative distribution function of a strictly positive stable random variable with as the Laplace transform of the corresponding probability density function. The Laplace transform of the GML probability density function above is This distribution can be considered as the positive counterpart of [Pakes(1998)]’s generalized Linnik distribution with the probability density function having the Laplace transform . Furthermore, the mixture representation of a GML distributed random variable is

(1.3)

where is gamma distributed with scale parameter 1 and shape parameter , and its probability density function given by

The fractional moments of the GML distributed random variable for are derived by [Pillai(1990)] as

while [Lin(1998a), Lin(1998b)] obtained the expression

(1.4)

for and . Note that the moments are infinite for order .

The main goal of this paper is to propose a procedure to estimate the model parameters in the Mittag-Leffler (ML) and the generalized Mittag-Leffler (GML) distributions that uses all available information. This is necessary in order for these models to be usable in practice. The rest of the paper is organized as follows: In Sections 2 and 3, we derive the first few moments of the log-transformed Mittag-Leffler distributed random variables. In Section 4, we propose procedures to estimate the parameters of the ML and the GML distributions. In Section 5, some key points and extensions of our methodology are discussed. In Section 6, some computational test results are shown and interval estimators for the ML parameters are derived using the asymptotic normality of the point estimators.

2 Moments of the log-transformed ML random variable

We now derive the first four log-moments of the random variable . Applying the log-transformation to the mixture representation (1.1), we obtain

(2.1)

where , , and . Following [Cahoy et. al(2010)], it is straightforward to show the following first four non-central moments of the random variables and :

where is the Euler’s constant, and is the Riemann zeta function evaluated at ,

Taking the expectation of (2.1

) and using the above moments, we get the mean and variance

(2.2)

respectively. Observe that the mean does not involve the parameter which is surprising and is due to the expected value of being zero. Moreover, a similar calculation gives the third and fourth central moments as

respectively which will be used in the derivation of the interval estimates in the appendix.

3 Moments of the log-transformed GML random variable

Taking the logarithm of the mixture representation of the GML distributed random variable in (1.3) yields

where , , and is a one-sided -stable distributed random variable with the Laplace transform of the probability density function given as . From [Zolotarev(1986)], the first four log-moments of can be deduced as

But to calculate the first four moments of we also need to know the moments of . The moments of

can now be derived as follows: The characteristic function of

can be easily shown as

where Using the logarithmic expansion of the gamma function ([Abramowitz and Stegun(1965)]), we get the cumulant-generating function

where the th cumulant is given by

Please note that the mean and variance of are given by

which are commonly known as the digamma and trigamma functions, respectively. For , the th cumulant is the polygamma function of order evaluated at . The th integer-order moments can be calculated using the recursive relation

This implies that , and so forth. We can now derive estimating equations using the first two moments of . More specifically, it can easily be shown that the mean and variance of are

(3.1)

and

(3.2)

correspondingly. However, a more direct procedure is to consider the characteristic function of the log-transformed random variable . This simply suggests that

Taking the logarithmic expansion of the preceding equation yields the following cumulant-generating function of

where the th cumulant is given by

It is easy to check that

where and . Also, using the recursive relation between the cumulant and the th moment above, we can easily derive an expression for the third moment of the random variable as

The expression for the fourth moment easily follows but we omit it here.

4 Parameter estimation

4.1 Estimation for ML distribution

One way of estimating the parameters of the ML distribution is to derive the method-of-moments estimators using formula (1.2) for the fractional moments as in [Nikias and Shao(1995)] and [Kozubowski(2001)]. That is, select two values of , and say, and compute and numerically using the following two equations:

We re-emphasize that we need to choose appropriate numbers and beforehand, which are required to be less than to be able to use the fractional moment estimators. This restriction suggests that we need to know or have information about a priori to be able to estimate the parameters and .

To overcome this difficulty, we propose estimators of and based on the mean and variance of the variable . From (2.2), the method-of-moments estimators for and are

(4.1)

respectively. Note that and are the sample mean and variance of the log-transformed data , correspondingly. Moreover, the preceding estimators are non-negative as desired. Another advantage of our estimation procedure is that it is computationally simple as we do not need to numerically find the unique solutions of a system of equations as the parameter estimates. The proposed estimators above are also shown to be asymptotically unbiased (see appendix).

We now compare the proposed procedure with the fractional moment estimators ( and ). In particular, we used bias and root-mean-square error (RMSE) as bases for the comparison. Following [Kozubowski(2001)], we assumed , and used the same constants and . The fractional moment estimator of is computed by numerically solving the equation

Then the fractional estimator of is calculated as

In the root calculation above, we used the uniroot function of the statistical software R with tolerance limit of . We also performed 10000 simulation runs for each combination of the , and sample size values. Table 1 in the appendix shows the computational test results. Clearly, the proposed estimators ( and ) outperformed the fractional moment estimators ( and ) even when the sample size is as large as 25000. When , the bias ratio of the proposed to the fractional estimator ranges from 10.77% to 48.64%. This demonstrates the larger bias the fractional estimator is producing in estimating for small samples. However, the bias difference seemingly becomes negligible as the sample size increases. A similar observation can be made for the bias incurred in estimating . The RMSE’s also generally shows that our procedure produces more homogeneous estimators that are closer to the true parameter values than the fractional moment method. These results certainly added another desirable characteristic of the proposed computationally simple approach.

4.2 Estimation for GML distribution

We now propose a similar estimation procedure for the GML distribution, and compare it with the fractional moment method. Using the mean and variance of the log-transformed GML distributed random variable from Section 3, we can calculate parameter estimates and using the following two equations:

(4.2)

and

(4.3)

In this paper, we only consider an approximation based on the first few terms of the series representation of the digamma function for performance comparison. A major advantage of using these estimating equations is that both digamma and trigamma functions are monotone in . Hence, finding the solutions is straightforward. Recall that

Thus, we approximate as

This results to the following approximation of the trigamma function :

Solving the system of two equations (4.2) and (4.3) using the preceding approximations of the digamma and trigamma functions will yield the parameter estimates for the GML distribution.

For the fractional moment technique, we assumed , and to compute and numerically using the two equations:

In the comparison, we used the function optim in R for both procedures with identical tolerance limits () and initial conditions. We also generated 10000 random samples of size each from the GML distribution, and computed the bias and the root-mean-square error (RMSE). The same conclusions from the preceding subsection can be drawn from Table 2 in the appendix. The table clearly shows that the proposed procedure outperformed again the fractional moment method even for large sample sizes.

Overall, Tables 1-2 in the appendix strongly indicate that the proposed point estimators using the log-transformed data performed better in our comparisons.

5 Concluding remarks

We have derived the first few moments of the log-transformed Mittag-Leffler distributed random variables. The log-moments led to systems of equations which are then used to estimate the parameters of the ML and GML distributions. A major advantage of our method over the other moment estimators (e.g., fractional moment estimators) is that we do not need to choose appropriate constants (e.g., ) a priori to be able to calculate the parameter estimates. The calculations involved are straightforward. Approximate interval estimates for the parameters of the ML distribution are derived. Furthermore, the testing and comparison have illustrated the superiority of our estimators.

Although some work have already been done, there are still a few things that need to be pursued. For instance, the development of estimators using Hill-type, regression, and likelihood approaches would be a worthy pursuit. The application of these methods in practice particularly in economics and finance would be of interest also.

6 Appendix

6.1 Empirical results

Bias RMSE 0.018 0.010 0.004 0.001 0.000 0.085 0.059 0.041 0.018 0.002 0.167 0.142 0.120 0.088 0.051 0.182 0.155 0.131 0.095 0.053 0.135 0.061 0.029 0.007 0.000 0.519 0.296 0.188 0.078 0.010 4.704 1.602 0.674 0.366 0.203 292.556 65.100 5.712 0.901 0.209 0.021 0.010 0.005 0.001 0.000 0.097 0.066 0.047 0.021 0.002 0.132 0.106 0.085 0.053 0.019 0.156 0.128 0.105 0.070 0.030 0.773 0.407 0.208 0.041 0.001 3.522 2.243 1.479 0.619 0.086 6.915 2.562 1.731 1.079 0.395 334.817 22.053 4.053 4.043 0.589 0.022 0.011 0.006 0.001 0.000 0.104 0.072 0.051 0.023 0.003 0.099 0.076 0.058 0.031 0.006 0.131 0.107 0.089 0.059 0.024 0.108 0.051 0.023 0.005 0.000 0.525 0.346 0.231 0.101 0.014 0.303 0.194 0.132 0.069 0.015 2.033 0.593 0.295 0.141 0.061 0.020 0.010 0.005 0.001 0.000 0.105 0.075 0.053 0.024 0.003 0.067 0.049 0.036 0.016 0.002 0.106 0.088 0.074 0.050 0.018 6.957 3.547 1.808 0.343 0.001 41.680 28.209 19.214 8.405 1.189 13.208 8.406 5.572 2.226 0.283 45.167 29.343 19.732 9.572 3.128 0.018 0.009 0.004 0.001 0.000 0.104 0.076 0.054 0.024 0.003 0.037 0.027 0.018 0.007 0.000 0.078 0.066 0.056 0.038 0.013 0.005 0.002 0.001 0.000 0.000 0.033 0.022 0.016 0.007 0.001 0.006 0.004 0.002 0.001 0.000 0.032 0.022 0.015 0.007 0.002

Table 1: Comparison of point estimators for the ML distribution using different values of and for sample sizes .

Bias RMSE 0.019 0.009 0.005 0.001 0.000 0.083 0.059 0.042 0.019 0.003 0.109 0.081 0.061 0.031 0.006 0.143 0.113 0.089 0.055 0.021 4.955 2.382 1.220 0.243 0.004 13.013 7.942 5.221 2.0252 0.319 34.751 23.6442 16.45 7.723 1.346 44.717 30.581 21.875 11.413 3.509 0.019 0.011 0.005 0.001 0.000 0.088 0.064 0.046 0.021 0.003 0.089 0.065 0.044 0.019 0.002 0.132 0.105 0.084 0.052 0.016 2.353 1.275 0.634 0.107 -0.003 6.536 4.373 3.062 1.394 0.268 12.363 8.548 5.850 2.459 0.235 16.378 11.895 8.900 4.849 1.462 0.017 0.010 0.005 0.001 0.000 0.086 0.064 0.047 0.021 0.003 0.069 0.048 0.033 0.012 0.000 0.117 0.096 0.078 0.048 0.013 0.884 0.492 0.252 0.057 0.001 2.729 1.976 1.423 0.646 0.094 3.931 2.723 1.882 0.732 0.033 5.491 4.246 3.329 1.955 0.563 0.013 0.007 0.003 0.001 0.000 0.079 0.058 0.043 0.020 0.002 0.047 0.031 0.021 0.007 0.000 0.098 0.081 0.068 0.041 0.010 0.177 0.091 0.049 0.012 0.000 0.817 0.576 0.422 0.192 0.028 0.730 0.497 0.337 0.127 0.002 1.262 0.974 0.801 0.502 0.132 0.023 0.012 0.006 0.001 0.000 0.106 0.075 0.053 0.024 0.003 0.031 0.019 0.012 0.003 0.000 0.085 0.067 0.054 0.031 0.006 0.106 0.007 0.002 0.000 0.000 0.177 0.123 0.085 0.038 0.005 0.022 0.011 0.005 0.001 0.000 0.188 0.125 0.087 0.039 0.006

Table 2: Comparison of point estimators for the GML distribution using different values of and for sample sizes .

6.2 Interval estimation for the distribution

We first study the limiting distribution of our estimators and for the distribution. If we let

then the following weak convergence holds [Ferguson(1996)], i.e.,

as , where and are defined in Section 2. Using a standard result on asymptotic theory, the weak convergence above implies that

where is a mapping from and is continuous in a neighborhood of . We now apply this result to the consistent estimator of . Letting

The gradient then becomes

This implies that

where

(1)

where the last line is obtained by plugging in for . Similarly,

where the final simplification is attained by substituting ,

and

Therefore, we have shown that our method-of-moments estimators are normally distributed (asymptotically unbiased) as the sample size

goes large. Consequently, we can now approximate the confidence interval for and as

and

respectively, where is the

th quantile of the standard normal distribution, and

.

7 Acknowledgment

The author is grateful to the Associate Editor and the reviewer for their insightful comments and suggestions that led to the improvement of the article. This research was supported under the Louisiana Board of Regents Research Competitiveness Subprogram grant LEQSF(2011-14)-RD-A-15.

8 References

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