probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

, Chernoff's distribution, named after

Herman Chernoff Herman Chernoff (born July 1, 1923) is an American applied mathematician, statistician and physicist. He was formerly a professor at University of Illinois Urbana-Champaign, Stanford, and MIT, currently emeritus at Harvard University. Early lif ...

, is the

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 the

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 ...

Z =\underset\ (W(s) - s^2),

where ''W'' is a "two-sided"

Wiener process In mathematics, the Wiener process (or Brownian motion, due to its historical connection with Brownian motion, the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener. It is one o ...

(or two-sided "

Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...

") satisfying ''W''(0) = 0. If :

V(a,c) = \underset \ (W(s) - c(s-a)^2),

then ''V''(0, ''c'') has density :

f_c(t) = \frac g_c(t) g_c(-t)

where ''g''_''c'' has

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

given by :

\hat_c (s) = \frac, \ \ \ s \in \mathbf

and where Ai is the

Airy function In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(''x'') and the related function Bi(''x''), are Linear in ...

. Thus ''f''_''c'' is symmetric about 0 and the density ''ƒ''_''Z'' = ''ƒ''₁. Groeneboom (1989) shows that :

f_Z (z) \sim \frac \frac \exp \left( - \frac , z, ^3 + 2^ \tilde_1 , z,  \right)
\textz \rightarrow \infty

where

\tilde_1 \approx -2.3381

is the largest zero of the Airy function Ai and where

\operatorname' (\tilde_1 ) \approx 0.7022

. In the same paper, Groeneboom also gives an analysis of the

process A process is a series or set of activities that interact to produce a result; it may occur once-only or be recurrent or periodic. Things called a process include: Business and management * Business process, activities that produce a specific s ...

\

. The connection with the statistical problem of estimating a monotone density is discussed in Groeneboom (1985). Chernoff's distribution is now known to appear in a wide range of monotone problems including

isotonic regression In statistics and numerical analysis, isotonic regression or monotonic regression is the technique of fitting a free-form line to a sequence of observations such that the fitted line is non-decreasing function, non-decreasing (or non-increasing) ...

. The Chernoff distribution should not be confused with the Chernoff geometric distribution (called the Chernoff point in information geometry) induced by the Chernoff information.

History

Groeneboom, Lalley and Temme state that the first investigation of this distribution was probably by Chernoff in 1964, who studied the behavior of a certain

estimator In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...

of a

mode Mode ( meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * MO''D''E (magazine), a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is the setting fo ...

. In his paper, Chernoff characterized the distribution through an analytic representation through the

heat equation In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...

with suitable

boundary conditions In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...

. Initial attempts at approximating Chernoff's distribution via solving the heat equation, however, did not achieve satisfactory precision due to the nature of the boundary conditions. The computation of the distribution is addressed, for example, in Groeneboom and Wellner (2001). The connection of Chernoff's distribution with Airy functions was also found independently by Daniels and Skyrme and Temme, as cited in Groeneboom, Lalley and Temme. These two papers, along with Groeneboom (1989), were all written in 1984.

References

Continuous distributions Stochastic processes {{statistics-stub