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 the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...

of a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

whose

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...

is distributed in accordance with a Student's t-distribution. 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), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...

: :

p(x\mid \nu,\hat,\hat) = \frac \left(1+\frac\left( \frac  \right)^2\right)^

, where

\hat

is the

location parameter In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...

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 o ...

of the underlying (non-standardized) Student's t-distribution, and

\nu

is the number of

degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

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 that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...

, 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 normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...

. Although the log-normal distribution has finite moments, for any finite degrees of freedom, the

mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the '' ari ...

and

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...

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 In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...

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 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 e^ The parameter \mu i ...

, 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 d ...

, 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 environmental watershed sustainability. A practitioner of hydrology is calle ...

and in analyzing data on

cancer Cancer is a group of diseases involving abnormal cell growth with the potential to invade or spread to other parts of the body. These contrast with benign tumors, which do not spread. Possible signs and symptoms include a lump, abnormal bl ...

remission.

Multivariate log-t distribution

Analogous to the log-normal distribution,

multivariate Multivariate may refer to: In mathematics * Multivariable calculus * Multivariate function * Multivariate polynomial In computing * Multivariate cryptography * Multivariate division algorithm * Multivariate interpolation * Multivariate optical c ...

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