# minxent.multiple: Minimum Cross Entropy Distribution under Multiple Constraints In minxent: Entropy Optimization Distributions

## Description

minxent.multiple estimates the MinxEnt distribution under given moment constraints.

## Usage

 ```1 2``` ```## S3 method for class 'multiple' minxent(q, G, eta, lambda) ```

## Arguments

 `q` a priori distribution. `G` matrix of moment vector functions. `eta` vector of moment constraints. `lambda` initial points for langrangian multipliers.

## Details

MinxEnt distribution is obtained by Kullback's minimum cross entropy principles. This principle is introduced by Kullback (1957) which minimizes Kullback-Leibler divergence subject to given constraints. If a priori distribution is taken to be uniform distribution then minimizing Kullback-Leibler divergence is equivalent to maximizing Shannon's entropy subject to same given constraints. In the other words, in this special case MaxEnt distribution introduced by Jaynes (1957) is equivalent to MinxEnt distribution. For various application see Kapur&Kesavan(1992).

## Value

minxent.multiple returns an estimate of Lagrange multipliers and minimum cross entropy distribution under multiple constraints which is specified by user.

## Warning

Since first Lagrange multiplies is a function of the others, there exists (m-1) constraints. (See. Kapur&Kesavan(1992) pp.44).

Senay Asma

## References

Jaynes, E. T. (1957). Information Theory and Statistical Mechanics. Physical Reviews, 106: 620-630. Kapur, J.N. and Kesavan, H.K.(1992), Entropy Optimization Principle with Applications, Academic Press. Kullback, S. (1959). Information Theory and Statistics. John Wiley, New York.

`minxent.single`
 ```1 2 3 4 5``` ```xi <- 1:6 eta<-c(1,4,19) #expected moment constraints q<-c(rep(1/6),6) #a priori distribution G<-matrix(c(rep(1,6),xi,xi^2),byrow=TRUE,nrow=3) #matrix of moment vector function of observed data minxent.multiple(q=q,G=G,eta=eta,c(0,0)) #estimates of lagrangian multipliers and MinxEnt distribution ```