el2.test.wts {emplik2}R Documentation

Computes maximium-likelihood probability jumps for a single mean-type hypothesis, based on two independent uncensored samples

Description

This function computes the maximum-likelihood probability jumps for a single mean-type hypothesis, based on two samples that are independent, uncensored, and weighted. The target function for the maximization is the constrained log(empirical likelihood) which can be expressed as,

sum_{dx_i=1} wx_i log{μ_i} + sum_{dy_j=1} wy_j log{nu_j} - eta ( 1 - sum_{dx_i=1} μ_i ) - delta ( 1 -sum_{dy_j=1} nu_j ) - λ sum_{dx_i=1} sum_{dy_j=1} ( g(x_i,y_j)- mean ) μ_i nu_j

where the variables are defined as follows:

x is a vector of data for the first sample

y is a vector of data for the second sample

wx is a vector of estimated weights for the first sample

wy is a vector of estimated weights for the second sample

μ is a vector of estimated probability jumps for the first sample

nu is a vector of estimated probability jumps for the second sample

Usage

el2.test.wts(u,v,wu,wv,mu0,nu0,indicmat,mean)

Arguments

u a vector of uncensored data for the first sample
v a vector of uncensored data for the second sample
wu a vector of estimated weights for u
wv a vector of estimated weights for v
mu0 a vector of estimated probability jumps for u
nu0 a vector of estimated probability jumps for v
indicmat a matrix [g(u_i,v_j)-mean] where g(u, v) is a user-chosen function
mean a hypothesized value of E(g(u,v)), where E indicates ``expected value.''

Details

This function is called by el2.cen.EMs. It is listed here because the user may find it useful elsewhere.

The value of mean should be chosen between the maximum and minimum values of (u_i,v_j); otherwise there may be no distributions for u and v that will satisfy the the mean-type hypothesis. If mean is inside this interval, but the convergence is still not satisfactory, then the value of mean should be moved closer to the NPMLE for E(g(u,v)). (The NPMLE itself should always be a feasible value for mean.) The calculations for this function are derived in Owen (2001).

Value

el2.test.wts returns a list of values as follows:

u the vector of uncensored data for the first sample
wu the vector of weights for u
jumpu the vector of probability jumps for u that maximize the weighted empirical likelihood
v the vector of uncensored data for the second sample
wv the vector of weights for v
jumpv the vector of probability jumps for v that maximize the weighted empirical likelihood
lam the value of the Lagrangian multipler found by the calculations

Author(s)

William H. Barton <bbarton@lexmark.com>

References

Owen, A.B. (2001). Empirical Likelihood. Chapman and Hall/CRC, Boca Raton, pp.223-227.

Examples

 
u<-c(10, 209, 273, 279, 324, 391, 566, 785)
v<-c(21, 38, 39, 51, 77, 185, 240, 289, 524)
wu<-c(2.442931, 1.122365, 1.113239, 1.113239, 1.104113, 1.104113, 1.000000, 1.000000)
wv<-c( 1, 1, 1, 1, 1, 1, 1, 1, 1)
mu0<-c(0.3774461, 0.1042739, 0.09649724, 0.09649724, 0.08872055, 0.08872055, 0.0739222, 0.0739222)
nu0<-c(0.1013718, 0.1013718, 0.1013718, 0.1013718, 0.1013718, 0.1013718, 0.1095413, 0.1287447,
 0.1534831)
mean<-0.5

#let fun=function(x,y){x>=y}
indicmat<-matrix(nrow=8,ncol=9,c(
-0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5, 
-0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,
-0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,
-0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,
-0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,
-0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,
-0.5, -0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,
-0.5, -0.5, -0.5, -0.5,  0.5,  0.5,  0.5,  0.5,
-0.5, -0.5, -0.5, -0.5, -0.5, -0.5,  0.5,  0.5))
el2.test.wts(u,v,wu,wv,mu0,nu0,indicmat,mean)

# jumpu
# [1] 0.3774461, 0.1042739, 0.09649724, 0.09649724, 0.08872055, 0.08872055, 0.0739222, 0.0739222

# jumpv
# [1] 0.1013718, 0.1013718, 0.1013718, 0.1013718, 0.1013718, 0.1013718, 0.1095413, 0.1287447,
# [9] 0.1534831

# lam
# [1] 7.055471

[Package emplik2 version 1.00 Index]