Bapat–Beg Theorem
   HOME

TheInfoList



OR:

In
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 ...
, the Bapat–Beg theorem gives the
joint probability distribution A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
of
order statistics In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Important ...
of
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...
but not necessarily identically distributed
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 ...
s in terms of 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 ...
s of the random variables.
Ravindra Bapat Ravindra B. Bapat is an Indian mathematician known for the Bapat–Beg theorem. Education He obtained B.Sc. from University of Mumbai, M.Stat. from the Indian Statistical Institute, New Delhi and Ph.D. from the University of Illinois at Chic ...
and M.I. Beg published the theorem in 1989, though they did not offer a proof. A simple proof was offered by Hande in 1994. Often, all elements of the
sample Sample or samples may refer to: * Sample (graphics), an intersection of a color channel and a pixel * Sample (material), a specimen or small quantity of something * Sample (signal), a digital discrete sample of a continuous analog signal * Sample ...
are obtained from the same population and thus have the same
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 ...
. The Bapat–Beg theorem describes the order statistics when each element of the sample is obtained from a different
statistical population In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a hyp ...
and therefore has its own
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 ...
.


Statement

Let X_1,X_2,\ldots, X_n be independent real valued random variables with
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 ...
s respectively F_1(x),F_2(x),\ldots,F_n(x). Write X_,X_,\ldots, X_ for the order statistics. Then the joint probability distribution of the n_1, n_2\ldots, n_k order statistics (with n_1 and x_1) is :\begin F_(x_1,\ldots,x_k) & = \Pr ( X_\leq x_1 \land X_\leq x_2 \land\cdots\land X_ \leq x_k) \\ & = \sum_^n \cdots\sum_^ \sum _^\frac, \end where : \begin P_(x_1,\ldots,x_k) = \operatorname \begin F_1(x_1) \cdots F_1(x_1) & F_1(x_2)-F_1(x_1) \cdots F_1(x_2)-F_1(x_1) & \cdots & 1-F_1(x_k) \cdots 1-F_1(x_k) \\ F_2(x_1) \cdots F_2(x_1) & F_2(x_2)-F_2(x_1) \cdots F_2(x_2)-F_2(x_1) & \cdots & 1-F_2(x_k) \cdots 1-F_1(x_k )\\ \vdots & \vdots & & \vdots \\ \underbrace_ & \underbrace_ & \cdots & \underbrace_ \end \end is the permanent of the given
block matrix In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix w ...
. (The figures under the braces show the number of columns.)


Independent identically distributed case

In the case when the variables X_1,X_2,\ldots, X_n are
independent and identically distributed Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...
with
cumulative probability 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 ...
F_i=F for all ''i'' the theorem reduces to : \begin F_(x_1,\ldots,x_k) = \sum_^n \cdots \sum_^ \sum_^ n! \frac \frac \prod\limits_^k \frac. \end


Remarks

* No assumption of continuity of the cumulative distribution functions is needed. * If the inequalities ''x''1 < ''x''2 < ... < ''x''''k'' are not imposed, some of the inequalities "may be redundant and the probability can be evaluated after making the necessary reduction."


Complexity

Glueck and co-authors note that the Bapat‒Beg formula is computationally intractable, because it involves an exponential number of permanents of the size of the number of random variables. However, when the random variables have only two possible distributions, the complexity can be reduced to O(m^). Thus, in the case of two populations, the complexity is polynomial in m for any fixed number of statistics k.


References

{{DEFAULTSORT:Bapat-Beg Theorem Theorems in probability theory Theorems in statistics