emplik2 (version 1.32)

el2.test.wts: 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, $$\sum_{dx_i=1} wx_i \log{\mu_i} + \sum_{dy_j=1} wy_j \log{\nu_j} - \eta ( 1 - \sum_{dx_i=1} \mu_i ) - \delta ( 1 -\sum_{dy_j=1} \nu_j ) - \lambda \sum_{dx_i=1} \sum_{dy_j=1} ( g(x_i,y_j)- mean ) \mu_i \nu_j $$

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

\(\mu\) 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,lamOld=0)

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

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.''

lamOld

The previous solution of lambda, used as the starting point to search for new solution of lambda.

Author

William H. Barton <[e-mail]> and modified by Mai Zhou.

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).

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