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The Johnson's ''SU''-distribution is a four-parameter family of
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 ...
s first investigated by N. L. Johnson in 1949. Johnson proposed it as a transformation of the
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 ...
: : z=\gamma+\delta \sinh^ \left(\frac\right) where z \sim \mathcal(0,1).


Generation of random variables

Let ''U'' be 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 ...
that is uniformly distributed on the unit interval  , 1 Johnson's ''SU'' random variables can be generated from ''U'' as follows: : x = \lambda \sinh\left( \frac \right) + \xi where Φ is the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
of the
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 ...
.


Johnson's ''SB''-distribution

N. L. Johnson firstly proposes the transformation : : z=\gamma+\delta \log \left(\frac\right) where z \sim \mathcal(0,1). Johnson's ''SB'' random variables can be generated from ''U'' as follows: : y=^ : x=\lambda y +\xi The ''SB''-distribution is convenient to Platykurtic distributions (
Kurtosis In probability theory and statistics, kurtosis (from , ''kyrtos'' or ''kurtos'', meaning "curved, arching") refers to the degree of “tailedness” in the probability distribution of a real-valued random variable. Similar to skewness, kurtos ...
). To simulate ''SU'', sample of code for its
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
and cumulative distribution function is availabl
here


Applications

Johnson's S_-distribution has been used successfully to model asset returns for portfolio management. This comes as a superior alternative to using the
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 ...
to model asset returns. An R package
JSUparameters
was developed in 2021 to aid in the estimation of the parameters of the best-fitting Johnson's S_-distribution for a given dataset. Johnson distributions are also sometimes used in
option pricing In finance, a price (premium) is paid or received for purchasing or selling options. The calculation of this premium will require sophisticated mathematics. Premium components This price can be split into two components: intrinsic value, and ...
, so as to accommodate an observed
volatility smile Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given ex ...
; see Johnson binomial tree. An alternative to the Johnson system of distributions is the quantile-parameterized distributions (QPDs). QPDs can provide greater shape flexibility than the Johnson system. Instead of fitting moments, QPDs are typically fit to empirical CDF data with linear least squares. Johnson's S_-distribution is also used in the modelling of the
invariant mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
of some heavy
meson In particle physics, a meson () is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, the ...
s in the field of B-physics.As an example, see:


References


Further reading

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Preprint
* {{ProbDistributions, continuous-infinite Continuous distributions