sel2ciuupi2 {ciuupi2} | R Documentation |
Compute the second definition of the scaled expected length of the Kabaila & Giri (2009) CIUUPI
Description
Evaluate the second 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
sel2ciuupi2(gam, bsvec, alpha, m, rho, natural = 1)
Arguments
gam |
A value of gamma or vector of gamma values at which the second 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 |
alpha |
The minimum coverage probability is 1 - |
m |
Degrees of freedom |
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
|
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 second definition of the scaled expected length of the Kabaila and
Giri(2009) CIUUPI is the expected value of the ratio of the length of this
confidence interval divided by the length of the usual confidence interval,
with coverage probability 1 - alpha
, computed from the same data.
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 second 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
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 second definition of the
# scaled expected length (takes about 30 mins to run):
bsvec <- bsciuupi2(alpha, m, rho = -0.7071068, obj = 2)
# The result bsvec is (to 7 decimal places) the following:
bsvec <- c(-0.0344224, -0.2195927, -0.3451243, -0.3235045, -0.1060439,
1.9753281, 2.0688684, 2.3803642, 2.6434660, 2.6288564, 2.4129931)
# Graph the squared scaled expected length function
gam <- seq(0, 10, by = 0.1)
sel <- sel2ciuupi2(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)