sel1ciuupi2 {ciuupi2}

Compute the first definition of the scaled expected length of the Kabaila & Giri (2009) CIUUPI

Description

Evaluate the first definition of the scaled expected length of the Kabaila & Giri (2009) confidence interval that utilizes uncertain prior information (CIUUPI), with minimum coverage 1 - alpha, at gam.

Usage

sel1ciuupi2(gam, bsvec, alpha, m, rho, natural = 1)

Arguments

gam

A value of gamma or vector of gamma values at which the first definition of the scaled expected length function is evaluated

bsvec

The vector (b(d/6),b(2d/6),...,b(5d/6),s(0),s(d/6),...,s(5d/6)) computed using bsciuupi2

alpha

The minimum coverage probability is 1 - alpha

m

Degrees of freedom n - p

rho

A known correlation

natural

Equal to 1 (default) if the b and s functions are obtained by natural cubic spline interpolation or 0 if obtained by clamped cubic spline interpolation. This parameter must take the same value as that used in bsciuupi2

Details

Suppose that

y = X \beta + \epsilon

where y is a random n-vector of responses, X is a known n by p matrix with linearly independent columns, \beta is an unknown parameter p-vector and \epsilon is a random n-vector with components that are independent and identically normally distributed with zero mean and unknown variance. The parameter of interest is \theta = a' \beta. The uncertain prior information is that \tau = c' \beta takes the value t, where a and c are specified linearly independent vectors and t is a specified number. rho is the known correlation between the least squares estimators of \theta and \tau. It is determined by the n by p design matrix X and the p-vectors a and c using find_rho.

The Kabaila & Giri (2009) CIUUPI is specified by the vector (b(d/6),...,b(5d/6),s(0),...,s(5d/6)), alpha, m and natural

The first definition of the scaled expected length of the Kabaila and Giri(2009) CIUUPI is the expected length of this confidence interval divided by the expected length of the usual confidence interval with coverage probability 1 - alpha.

In the examples, we continue with the same 2 x 2 factorial example described in the documentation for find_rho.

Value

The value(s) of the first definition of the scaled expected length of the Kabaila & Giri (2009) CIUUPI at gam.

References

Kabaila, P. and Giri, K. (2009) Confidence intervals in regression utilizing prior information. Journal of Statistical Planning and Inference, 139, 3419 - 3429.

See Also

find_rho, bsciuupi2

Examples

alpha <- 0.05
m <- 8

# Find the vector (b(d/6),...,b(5d/6),s(0),...,s(5d/6)) that specifies the
# Kabaila & Giri (2009) CIUUPI for the first definition of the
# scaled expected length (default) (takes about 30 mins to run):

bsvec <- bsciuupi2(alpha, m, rho = -0.7071068)


# The result bsvec is (to 7 decimal places) the following:
bsvec <- c(-0.0287487, -0.2151595, -0.3430403, -0.3125889, -0.0852146,
            1.9795390,  2.0665414,  2.3984471,  2.6460159,  2.6170066,  2.3925494)


# Graph the squared scaled expected length function
gam <- seq(0, 10, by = 0.1)
sel <- sel1ciuupi2(gam, bsvec, alpha, m, rho = -0.7071068)
plot(gam, sel^2, type = "l", lwd = 2, ylab = "", las = 1, xaxs = "i",
main = "Squared Scaled Expected Length", col = "blue",
xlab = expression(paste("|", gamma, "|")))
abline(h = 1, lty = 2)