bsspline: Evaluate the functions b and s at x

Description

Evaluate the functions b and s, as specified by (b(1),b(2),...,b(5),s(0),s(1),...,s(5)), alpha and natural, at x.

Usage

bsspline(x, bsvec, alpha, natural = 1)

Arguments

A value or vector of values at which the functions b and s are to be evaluated

bsvec

The vector (b(1),b(2),...,b(5),s(0),s(1),...,s(5))

alpha

The minimum coverage probability is 1 - alpha

natural

Equal to 1 (default) for natural cubic spline interpolation or 0 for clamped cubic spline interpolation

Value

A data frame containing x and the corresponding values of the functions b and s.

Details

The function b is an odd continuous function and the function s is an even continuous function. In addition, b(x)=0 and s(x) is equal to the \(1 - \alpha/2\) quantile of the standard normal distribution for all |x| greater than or equal to 6. The values of these functions in the interval \([-6,6]\) are specified by the vector \((b(1), b(2), \dots, b(5),s(0), s(1), \dots, s(5))\) as follows. By assumption, \(b(0)=0\) and \(b(-i)=-b(i)\) and \(s(-i)=s(i)\) for \(i=1,...,6\). The values of \(b(x)\) and \(s(x)\) for any \(x\) in the interval \([-6,6]\) are found using cube spline interpolation for the given values of \(b(i)\) and \(s(i)\) for \(i=-6,-5,...,0,1,...,5,6\).

The vector (b(1),b(2),...,b(5),s(0),s(1),...,s(5)) that specifies the confidence interval that utilizes uncertain prior information (CIUUPI) is obtained using bsciuupi.

Examples

Run this code

# NOT RUN {
alpha <- 0.05

# Find the vector (b(1),b(2),...,b(5),s(0),s(1),...,s(5)) that specifies the
# CIUUPI: (this may take a few minutes to run)
# }
# NOT RUN {
bsvec <- bsciuupi(alpha, rho = 0.4)
# }
# NOT RUN {
# The result (to 7 decimal places) is
bsvec <- c(0.129443483, 0.218926703, 0.125880945, 0.024672734, -0.001427343,
           1.792489585, 1.893870240, 2.081786492, 2.080407355,  1.986667246,
           1.958594824)

# Graph the functions b and s
x <- seq(0, 8, by = 0.1)
xseq <- seq(0, 6, by = 1)
bvec <- c(0, bsvec[1:5], 0)
quantile <- qnorm(1-(alpha)/2, 0, 1)
svec <- c(bsvec[6:11], quantile)
splineval <- bsspline(x, bsvec, alpha)

plot(x, splineval[, 2], type = "l", main = "b function",
ylab = " ", las = 1, lwd = 2, xaxs = "i", col = "blue")
points(xseq, bvec, pch = 19, col = "blue")
plot(x, splineval[, 3], type = "l", main = "s function",
ylab = " ", las = 1, lwd = 2, xaxs = "i",  col = "blue")
points(xseq, svec, pch = 19, col = "blue")

# }