Log-Cauchy distribution
None
Probability density function
![]() |
|
Cumulative distribution function
![]() |
|
Parameters |
|
---|---|
Support |
|
|
|
CDF |
|
Mean | infinite |
Median |
|
Variance | infinite |
Skewness | does not exist |
Ex. kurtosis | does not exist |
MGF | does not exist |
In probability theory, a log-Cauchy distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Cauchy distribution . If X is a random variable with a Cauchy distribution, then Y = exp( X ) has a log-Cauchy distribution; likewise, if Y has a log-Cauchy distribution, then X = log( Y ) has a Cauchy distribution. [1]
Characterization
The log-Cauchy distribution is a special case of the log-t distribution where the degrees of freedom parameter is equal to 1. [2]
Probability density function
The log-Cauchy distribution has the probability density function :
where
is a
real number
and
.
[1]
[3]
If
is known, the
scale parameter
is
.
[1]
and
correspond to the
location parameter
and
scale parameter
of the associated Cauchy distribution.
[1]
[4]
Some authors define
and
as the
location
and scale parameters, respectively, of the log-Cauchy distribution.
[4]
For
and
, corresponding to a standard Cauchy distribution, the probability density function reduces to:
[5]
Cumulative distribution function
The cumulative distribution function (
cdf
) when
and
is:
[5]
Survival function
The
survival function
when
and
is:
[5]
Hazard rate
The
hazard rate
when
and
is:
[5]
The hazard rate decreases at the beginning and at the end of the distribution, but there may be an interval over which the hazard rate increases. [5]
Properties
The log-Cauchy distribution is an example of a heavy-tailed distribution . [6] Some authors regard it as a "super-heavy tailed" distribution, because it has a heavier tail than a Pareto distribution -type heavy tail, i.e., it has a logarithmically decaying tail. [6] [7] As with the Cauchy distribution, none of the non-trivial moments of the log-Cauchy distribution are finite. [5] The mean is a moment so the log-Cauchy distribution does not have a defined mean or standard deviation . [8] [9]
The log-Cauchy distribution is infinitely divisible for some parameters but not for others. [10] Like the lognormal distribution , log-t or log-Student distribution and Weibull distribution , the log-Cauchy distribution is a special case of the generalized beta distribution of the second kind . [11] [12] The log-Cauchy is actually a special case of the log-t distribution, similar to the Cauchy distribution being a special case of the Student's t distribution with 1 degree of freedom. [13] [14]
Since the Cauchy distribution is a stable distribution , the log-Cauchy distribution is a logstable distribution. [15] Logstable distributions have poles at x=0. [14]
Estimating parameters
The
median
of the
natural logarithms
of a
sample
is a
robust estimator
of
.
[1]
The
median absolute deviation
of the natural logarithms of a sample is a robust estimator of
.
[1]
Uses
In Bayesian statistics , the log-Cauchy distribution can be used to approximate the improper Jeffreys -Haldane density, 1/k, which is sometimes suggested as the prior distribution for k where k is a positive parameter being estimated. [16] [17] The log-Cauchy distribution can be used to model certain survival processes where significant outliers or extreme results may occur. [3] [4] [18] An example of a process where a log-Cauchy distribution may be an appropriate model is the time between someone becoming infected with HIV and showing symptoms of the disease, which may be very long for some people. [4] It has also been proposed as a model for species abundance patterns. [19]
References
- 1 2 3 4 5 6 Olive, D.J. (June 23, 2008). "Applied Robust Statistics" (PDF) . Southern Illinois University. p. 86. Archived from the original (PDF) on September 28, 2011 . Retrieved 2011-10-18 .
-
↑
Olosunde, Akinlolu & Olofintuade, Sylvester (January 2022).
"Some Inferential Problems from Log Student's T-distribution and its Multivariate Extension"
.
Revista Colombiana de Estadística - Applied Statistics
.
45
(1): 209–229.
doi
:
10.15446/rce.v45n1.90672
. Retrieved
2022-04-01
.
{{ cite journal }}
: CS1 maint: multiple names: authors list ( link ) - 1 2 Lindsey, J.K. (2004). Statistical analysis of stochastic processes in time . Cambridge University Press. pp. 33 , 50, 56, 62, 145. ISBN 978-0-521-83741-5 .
- 1 2 3 4 Mode, C.J. & Sleeman, C.K. (2000). Stochastic processes in epidemiology: HIV/AIDS, other infectious diseases . World Scientific. pp. 29 –37. ISBN 978-981-02-4097-4 .
- 1 2 3 4 5 6 Marshall, A.W. & Olkin, I. (2007). Life distributions: structure of nonparametric, semiparametric, and parametric families . Springer. pp. 443 –444. ISBN 978-0-387-20333-1 .
- 1 2 Falk, M.; Hüsler, J. & Reiss, R. (2010). Laws of Small Numbers: Extremes and Rare Events . Springer. p. 80 . ISBN 978-3-0348-0008-2 .
- ↑ Alves, M.I.F.; de Haan, L. & Neves, C. (March 10, 2006). "Statistical inference for heavy and super-heavy tailed distributions" (PDF) . Archived from the original (PDF) on June 23, 2007.
- ↑ "Moment" . Mathworld . Retrieved 2011-10-19 .
-
↑
Wang, Y. "Trade, Human Capital and Technology Spillovers: An Industry Level Analysis". Carleton University: 14.
{{ cite journal }}
: Cite journal requires| journal=
( help ) - ↑ Bondesson, L. (2003). "On the Lévy Measure of the Lognormal and LogCauchy Distributions" . Methodology and Computing in Applied Probability : 243–256. Archived from the original on 2012-04-25 . Retrieved 2011-10-18 .
- ↑ Knight, J. & Satchell, S. (2001). Return distributions in finance . Butterworth-Heinemann. p. 153 . ISBN 978-0-7506-4751-9 .
- ↑ Kemp, M. (2009). Market consistency: model calibration in imperfect markets . Wiley. ISBN 978-0-470-77088-7 .
- ↑ MacDonald, J.B. (1981). "Measuring Income Inequality". In Taillie, C.; Patil, G.P.; Baldessari, B. (eds.). Statistical distributions in scientific work: proceedings of the NATO Advanced Study Institute . Springer. p. 169. ISBN 978-90-277-1334-6 .
- 1 2 Kleiber, C. & Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Science . Wiley. pp. 101 –102, 110. ISBN 978-0-471-15064-0 .
- ↑ Panton, D.B. (May 1993). "Distribution function values for logstable distributions" . Computers & Mathematics with Applications . 25 (9): 17–24. doi : 10.1016/0898-1221(93)90128-I .
- ↑ Good, I.J. (1983). Good thinking: the foundations of probability and its applications . University of Minnesota Press. p. 102. ISBN 978-0-8166-1142-3 .
- ↑ Chen, M. (2010). Frontiers of Statistical Decision Making and Bayesian Analysis . Springer. p. 12. ISBN 978-1-4419-6943-9 .
- ↑ Lindsey, J.K.; Jones, B. & Jarvis, P. (September 2001). "Some statistical issues in modelling pharmacokinetic data". Statistics in Medicine . 20 (17–18): 2775–278. doi : 10.1002/sim.742 . PMID 11523082 . S2CID 41887351 .
- ↑ Zuo-Yun, Y.; et al. (June 2005). "LogCauchy, log-sech and lognormal distributions of species abundances in forest communities". Ecological Modelling . 184 (2–4): 329–340. doi : 10.1016/j.ecolmodel.2004.10.011 .