qrminfundtheorem {bqror}

Minimizes the negative of log-likelihood in the OR1 model

Description

This function minimizes the negative of log-likelihood in the OR1 model with respect to the cut-points \delta using the fundamental theorem of calculus.

Usage

qrminfundtheorem(deltaIn, y, x, beta, cri0, cri1, stepsize, maxiter, h, dh, sw, p)

Arguments

deltaIn

initialization of cut-points.

y

observed ordinal outcomes, column vector of size (n x 1).

x

covariate matrix of size (n x k) including a column of ones with or without column names.

beta

\beta, a column vector of size (k x 1).

cri0

initial criterion, cri0 = 1.

cri1

criterion lies between (0.001 to 0.0001).

stepsize

learning rate lies between (0.1, 1).

maxiter

maximum number of iteration.

h

change in each value of \delta, holding other \delta constant for first derivatives.

dh

change in each value of \delta, holding other \delta constant for second derivaties.

sw

iteration to switch from BHHH to inv(-H) algorithm.

p

quantile level or skewness parameter, p in (0,1).

Details

First derivative from first principle

dy/dx=[f(x+h)-f(x-h)]/2h

Second derivative from first principle

f'(x-h)=(f(x)-f(x-h))/h

f''(x)= [{(f(x+h)-f(x))/h} - (f(x)-f(x-h))/h]/h

= [(f(x+h)+f(x-h)-2 f(x))]/h^2

cross partial derivatives

f(x) = [f(x+dh,y)-f(x-dh,y)]/2dh

f(x,y)=[{(f(x+dh,y+dh) - f(x+dh,y-dh))/2dh} - {(f(x-dh,y+dh) - f(x-dh,y-dh))/2dh}]/2dh

= 0.25* [{(f(x+dh,y+dh)-f(x+dh,y-dh))} -{(f(x-dh,y+dh)-f(x-dh,y-dh))}]/dh2

Value

Returns a list with components

deltamin:

cutpoint vector that minimizes the log-likelihood function.

negsum:

negative sum of log-likelihood.

logl:

log-likelihood values.

G:

gradient vector, (n x k) matrix with i-th row as the score for the i-th unit.

H:

Hessian matrix.

See Also

differential calculus, functional maximization, mldivide

Examples

set.seed(101)
deltaIn <- c(-0.002570995,  1.044481071)
data("data25j4")
y <- data25j4$y
xMat <- data25j4$x
p <- 0.25
beta <- c(0.3990094, 0.8168991, 2.8034963)
cri0     <- 1
cri1     <- 0.001
stepsize <- 1
maxiter  <- 10
h        <- 0.002
dh       <- 0.0002
sw       <- 20
output <- qrminfundtheorem(deltaIn, y, xMat, beta, cri0, cri1, stepsize, maxiter, h, dh, sw, p)

# deltamin
#   0.8266967 0.3635708
# negsum
#   645.4911
# logl
#    -0.7136999
#    -1.5340787
#    -1.1072447
#    -1.4423124
#    -1.3944677
#    -0.7941271
#    -1.6544072
#    -0.3246632
#    -1.8582422
#    -0.9220822
#    -2.1117739 .. soon
# G
#    0.803892784  0.00000000
#   -0.420190546  0.72908381
#   -0.421776117  0.72908341
#   -0.421776117 -0.60184063
#   -0.421776117 -0.60184063
#    0.151489598  0.86175120
#    0.296995920  0.96329114
#   -0.421776117  0.72908341
#   -0.340103190 -0.48530164
#    0.000000000  0.00000000
#   -0.421776117 -0.60184063.. soon
# H
#   -338.21243  -41.10775
#   -41.10775 -106.32758