Log-t Distribution
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In probability theory, a log-t distribution or log-Student t distribution is a
probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
whose
logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
is distributed in accordance with a
Student's t-distribution In probability theory and statistics, Student's  distribution (or simply the  distribution) t_\nu is a continuous probability distribution that generalizes the Normal distribution#Standard normal distribution, standard normal distribu ...
. If ''X'' is a random variable with a Student's t-distribution, then ''Y'' = exp(''X'') has a log-t distribution; likewise, if ''Y'' has a log-t distribution, then ''X'' = log(''Y'') has a Student's t-distribution.


Characterization

The log-t distribution has the
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: :p(x\mid \nu,\hat,\hat) = \frac \left(1+\frac\left( \frac \right)^2\right)^ , where \hat is the
location parameter In statistics, a location parameter of a probability distribution is a scalar- or vector-valued parameter x_0, which determines the "location" or shift of the distribution. In the literature of location parameter estimation, the probability distr ...
of the underlying (non-standardized) Student's t-distribution, \hat is the
scale parameter In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family ...
of the underlying (non-standardized) Student's t-distribution, and \nu is the number of
degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
of the underlying Student's t-distribution. If \hat=0 and \hat=1 then the underlying distribution is the standardized Student's t-distribution. If \nu=1 then the distribution is a log-Cauchy distribution. As \nu approaches
infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
, the distribution approaches a
log-normal distribution In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
. Although the log-normal distribution has finite moments, for any finite degrees of freedom, the
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
and
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
and all higher moments of the log-t distribution are infinite or do not exist. The log-t distribution is a special case of the generalized beta distribution of the second kind. The log-t distribution is an example of a
compound probability distribution In probability and statistics, a compound probability distribution (also known as a mixture distribution or contagious distribution) is the probability distribution that results from assuming that a random variable is distributed according to some ...
between the lognormal distribution and inverse gamma distribution whereby the variance parameter of the lognormal distribution is a random variable distributed according to an inverse gamma distribution.


Applications

The log-t distribution has applications in finance. For example, the distribution of stock market returns often shows fatter tails than a
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
, and thus tends to fit a Student's t-distribution better than a normal distribution. While the Black-Scholes model based on the log-normal distribution is often used to price
stock options In finance, an option is a contract which conveys to its owner, the ''holder'', the right, but not the obligation, to buy or sell a specific quantity of an underlying asset or instrument at a specified strike price on or before a specified dat ...
, option pricing formulas based on the log-t distribution can be a preferable alternative if the returns have fat tails. The fact that the log-t distribution has infinite mean is a problem when using it to value options, but there are techniques to overcome that limitation, such as by truncating the probability density function at some arbitrary large value. The log-t distribution also has applications in
hydrology Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
and in analyzing data on
cancer Cancer is a group of diseases involving Cell growth#Disorders, abnormal cell growth with the potential to Invasion (cancer), invade or Metastasis, spread to other parts of the body. These contrast with benign tumors, which do not spread. Po ...
remission.


Multivariate log-t distribution

Analogous to the log-normal distribution, multivariate forms of the log-t distribution exist. In this case, the location parameter is replaced by a vector ''μ'', the scale parameter is replaced by a matrix Σ.


References

{{Probability distributions Continuous distributions Probability distributions with non-finite variance