probability theory Probability theory 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 expressing it through a set o ...

, 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 ...

, is the

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 the

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 ...

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

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

Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It i ...

(or two-sided "

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

") 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 A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

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 and the related function , are linearly independent soluti ...

. 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 se ...

\

. 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 (or non-increasing) everywhere, and lies as ...

. 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 observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the ...

of a mode. In his paper, Chernoff characterized the distribution through an analytic representation through the heat equation with suitable

boundary conditions In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...

. 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