An Alternative Method for Estimating the Parameters of Log-Cauchy
Distribution
MOHAMMAD A. AMLEH1*, AHMAD AL-NATOOR2, BAHA’ ABUGHAZALEH2,
1Department of Mathematics, Faculty of Science, Zarqa University, Zarqa, JORDAN
2Department of Mathematics, Faculty of Science, Isra University, Amman, JORDAN
*Corresponding author
Abstract: - In this paper, we discuss the estimation problem of the location and scale parameters of the log-Cauchy
distribution, a member of the super heavy-tailed distributions family. We consider several methods of estimation,
including a new percentile method, maximum likelihood estimation and robust estimators. A Monte Carlo
simulation experiment is conducted to compare the proposed estimation methods. Further, we illustrate the
estimation methods via a real life example.
Key-Words: - Heavy-tailed distributions; Log-Cauchy distribution; Percentile estimators, Maximum likelihood
estimators; Robust estimators
Received: October 27, 2022. Revised: December 22, 2022. Accepted: January 19, 2023. Published: February 23, 2023.
1 Introduction
Heavy-tailed distributions play an essential role in
the study of rare events, as there can be cases such
that the probability of extremely large observations
cannot be ignored. Therefore, phenomena describing
the occurrence of extreme values with a high relative
probability can be modeled by these types of
distributions. In many different fields such as
computer science, networks, communications,
economics, and finance, it is common to find
examples of heavy tail datasets. For more details on
the heavy-tailed distributions, one may refer to [1] –
[5].
The log-Cauchy distribution is one of the heavy-
tailed distributions. It is considered a special case of
the generalized beta distribution of type II. In fact, it
is a special case of the log-student distribution. It can
be transformed from Cauchy distribution as follows.
If is a Cauchy distribution, then has a log-
Cauchy distribution. Because the expected value and
variance of the Cauchy distribution are not defined,
the Cauchy distribution is sometimes referred to as
the primary example of a pathological distribution. In
fact, the moment generating function of the Cauchy
distribution has not been defined, see for example,
[6] and [7]. Therefore, it is a matter of priority that
these exotic properties are achieved in the log-
Cauchy distribution.
In the literature concerning the problem of estimating
parameters, several estimation techniques have been
proposed for the Cauchy distribution, see for
example, [8] – [11]. However, it is noteworthy that
no attention was paid to estimating the parameters of
the log-Cauchy distribution. In this paper, we
consider the estimation problem of the parameters of
the log-Cauchy distribution. We present a robust
alternative method for estimating the required
parameters. Two additional methods are presented
for comparison purposes. A simulation study is
conducted to compare the effectiveness of the
estimation techniques and a real dataset is considered
for illustrative purposes.
2 log-Cauchy Distribution
The log-Cauchy distribution is sometimes seen as an
example of a "super heavy-tailed" distributions,
because its tail is considered heavier than the Pareto-
type heavy tail. Its tail is decaying logarithmically,
see [1]. The probability density function (pdf) of the
log-Cauchy distribution is defined as:
where is the location parameter and
represents the scale parameter. The cumulative
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DOI: 10.37394/23206.2023.22.18
Mohammad A. Amleh, Ahmad Al-Natoor, Baha’ Abughazaleh
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distribution function (cdf) of the log-Cauchy
distribution is given by
The survival and hazard rate functions of the log-
Cauchy distribution are expressed as:
and
respectively.
3 Estimating the Parameters of the log-
Cauchy Distribution
In this section, we discuss the problem of estimating
the location and scale parameters of the log-Cauchy
distribution. Three methods of estimation are
presented.
3.1 Percentiles Estimators
Percentiles play an essential role in statistical
inference. They are used recently in estimating
parameters, see [12]. The technique is similar to the
moment method estimation, but instead of equating
population moments to the sample moments, it is
based on equating population percentiles to the
sample percentiles and then solving the obtained
equations simultaneously.
In this context, we propose new percentile estimators
based on the popular quartiles as follows. Assume
that is a random sample of size n from
log-Cauchy distribution. The quantile function of the
log-Cauchy distribution can be written as:
Therefore, if is a log-Cauchy distributed, based on
Eq. (3), the median is given by:
The lower and upper quartiles of are given as:
and
respectively. The new approach is based on equating
the population quartiles in Eqs. (4) to (6) to the
sample quartiles as follows:
where is the sample median, and represent
the sample lower and upper quartiles, respectively.
Solving the system of equations in (7) gives:
and
Eqs. (8) and (9) are the percentiles estimators
required to estimate the parameters ,
respectively.
3.2 Maximum Likelihood Estimation
If is a random sample of size n taken
from a log-Cauchy distribution, then the likelihood
function, based on this sample, is given by:
The associated log-likelihood function may be
expressed as:
The maximum likelihood estimators (MLEs) of
can be obtained by maximizing the log-
likelihood function in Eq. (11). Consequently, the
likelihood equations are given by:
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The estimation process, through Eqs. (12) and (13),
cannot be obtained in closed form. Accordingly,
equations (12) and (13) will be solved simultaneously
using a numerical technique such as the Newton-
Raphson method. The resulting MLEs of the
parameters will be denoted by
,
respectively.
3.3 Robust Estimators
The data used to estimate the parameters of heavy-
tailed distributions are generally contaminated with
extreme values, which are called outliers. Outliers
may be considered in related applications as rare
events. By contrast, observations that are not extreme
are called inliers. An estimator is said to be robust
when these outliers do not affect the estimates of the
parameters, more details on robust estimators can be
found in [13] and [14]. Now, as mentioned above, the
log-Cauchy distribution is one of the heavy-tailed
distributions. Therefore, robust estimators of the
parameters are proposed in the literature. Hence,
based on a random sample taken from
log-Cauchy distribution, Olive [15] suggested the
following robust estimators:
(14)
where represents the median
absolute deviation of about the median of
. Clearly, if the sample size n is odd, it can
be shown that:
.
4 Simulation and Real Example
Now, in this section, we perform a simulation
experiment for computing the estimates of the
parameters based on the techniques
discussed in Section 3. A real dataset is considered to
illustrate the estimation methods.
4.1. Simulation Experiment
Here, we conduct an intensive Monte Carlo
simulation experiment for evaluating the suggested
estimators. The performance of the considered
estimators is measured in terms of the bias and the
mean square error (MSE) of the estimators, which are
expressed for any estimator of a parameter as:
and
respectively. Here, the Monte Carlo simulation is
conducted based on different sample sizes and
parameter values. For this, we generate random
samples of log- Cauchy distribution by considering
the following schemes:
Scheme 1:,
Scheme 2:,
Scheme 3:,
Samples from log-Cauchy distribution were
randomly generated under these schemes with 1000
replications of the simulation process. Using these
random samples, estimation biases and MSEs of the
estimators are obtained. The results are presented in
Tables 1 to 3.
Based on these tables, we observe the following
remarks. The biases of the three estimators are
generally small, which indicates the good
performance of the estimators in this sense. By
considering the MSE as an optimal criterion, it has
been observed that PEs are highly competitive to
MLEs and outperform REs in most of the considered
cases. It can be seen that as increases, the MSEs
for all estimators decrease. Moreover, it is noticeable
that the MSEs of the three estimators are close to
each other.
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4.2. Real Example
To clarify the estimation methods discussed in this
paper, we consider an example of real data that was
used by Alzaatreh [16]. It accounts for 157 of the
national consumer price index in Brazil. The data are
shown in Table 4. Since the log-Cauchy distribution is
defined on positive real numbers, for each sample
point ; we take the exponential
value. To test the goodness- of-fit of the
transformed data to the log-Cauchy distribution,
Kolmogorov-Smirnov (K-S) test is used. The K-S
statistic of the distance between the empirical
distribution and the fitted one, based on
estimates
, is 0.12057 and the
corresponding p-value is 0.2042. Therefore, it is
appropriate to fit the transformed data using the log-
Cauchy distribution. To see the accuracy of the log-
Cauchy distribution under the estimation methods
presented in this study, the true CDF of the data is
plotted in Fig. 1, along with the estimated CDFs. The
obtained estimates of the parameters based on
the considered methods are displayed in Table 5. It
can be observed that the values of the estimates are
close to each other.
Fig. 1: The empirical cdf (dots) and the estimated cdfs
based on the three methods.
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5 Conclusion
In this study, we have considered the estimation
problem of the parameters of log-Cauchy distribution,
one of the super heavy-tailed distributions. Three
estimators are addressed including, a new alternative
estimator based on percentiles, maximum likelihood
estimator and a robust estimator.
We have compared the performance of the estimators
using a Monte Carlo simulation experiment in terms
of the biases and MSEs for different sample sizes and
parameter values. The alternative estimators PEs are
recommended as they are computationally attractive
and have good performances based on the bias and
MSE criteria.
In this context, the proposed alternative method can
be used for estimating parameter for a wide range of
statistical distributions, which may be discussed in
future studies.
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[12] Bhatti, Sajjad Haider, et al. Efficient estimation
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Contribution of Individual Authors to the Creation
of a Scientific Article (Ghostwriting Policy)
The authors completed all aspects of the study
through joint work by proposing the new method and
comparing it with the previous methods, in addition to
work out the simulation experiments and the real data
analysis using R language. The final version has been
read and approved by all authors.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received.
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Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
