Log-Cauchy distribution
Probability density function | |||
Cumulative distribution function | |||
Parameters |
(real) (real) | ||
---|---|---|---|
Support | |||
CDF | |||
Mean | infinite | ||
Median | |||
Variance | infinite | ||
Skewness | does not exist | ||
Excess 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
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
Since the Cauchy distribution is a
Estimating parameters
The
Uses
In
References
- ^ a b c d e f 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.
- doi:10.15446/rce.v45n1.90672. Retrieved 2022-04-01.)
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- ^ ISBN 978-981-02-4097-4.
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- ^ 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.
- Mathworld. Retrieved 2011-10-19.
- ^ Wang, Y. "Trade, Human Capital and Technology Spillovers: An Industry Level Analysis". Carleton University: 14.
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(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.
- ISBN 978-0-7506-4751-9.
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- ISBN 978-90-277-1334-6.
- ^ ISBN 978-0-471-15064-0.
- .
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- .