Description Usage Arguments Details Value References See Also Examples
Compute the Kabaila and Giri (2009) confidence interval that utilizes
uncertain prior information (CIUUPI), with minimum coverage 1 - alpha
,
for a given vector y
of observed responses.
1 |
alpha |
1 - |
X |
The n by p design matrix |
a |
A vector used to specify the parameter of interest |
c |
A vector used to specify the parameter about which we have uncertain prior information |
bsvec |
The vector (b(d/6),b(2d/6),...,b(5d/6),s(0),s(d/6),...,s(5d/6))
computed using |
t |
A number used to specify the uncertain prior information, which has the form τ = t |
y |
The n-vector of observed responses |
natural |
Equal to 1 (default) if the b and s functions are evaluated by
natural cubic spline interpolation or 0 if evaluated by clamped cubic spline
interpolation. This parameter must take the same value as that used in
|
Suppose that
y = X β + ε
where y is a random
n-vector of responses, X is a known n by p matrix
with linearly independent columns, β is an unknown parameter
p-vector and ε 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 θ = a
'
β. The uncertain prior information is that τ = c
'
β takes the value t
, where a
and c
are
specified linearly independent vectors nonzero p-vectors and t
is a specified number. Given the vector bsvec
, computed using
bsciuupi2
, the design matrix X
, the vectors a
and c
and the number t, ciuupi2
computes the confidence
interval for θ that utilizes the uncertain prior information that
τ = t
for given n-vector of observed responses
y
.
In the examples, we continue with the same 2 x 2 factorial example described
in the documentation for find_rho
, for the vector of observed
responses y = (-1.3, 0.8, 2.6, 5.8, 0.3, 1.3, 4.3, 5.0, -0.4, 1.0,
5.2, 6.2).
The Kabaila & Giri (2009) confidence interval, with minimum coverage 1
- alpha
, that utilizes the uncertain prior information.
Kabaila, P. and Giri, K. (2009) Confidence intervals in regression utilizing prior information. Journal of Statistical Planning and Inference, 139, 3419 - 3429.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | # Specify the design matrix X and vectors a and c
# (denoted in R by a.vec and c.vec, respectively)
col1 <- rep(1,4)
col2 <- c(-1, 1, -1, 1)
col3 <- c(-1, -1, 1, 1)
col4 <- c(1, -1, -1, 1)
X.single.rep <- cbind(col1, col2, col3, col4)
X <- rbind(X.single.rep, X.single.rep, X.single.rep)
a.vec <- c(0, 2, 0, -2)
c.vec <- c(0, 0, 0, 1)
# Compute the vector (b(d/6),...,b(5d/6),s(0),...,s(5d/6)) that specifies the
# Kabaila & Giri (2009) CIUUPI, with minimum coverage 1 - alpha, for the
# first definition of the scaled expected length (default)
# for given alpha, m and rho (takes about 30 mins to run):
bsvec <- bsciuupi2(alpha = 0.05, m = 8, 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)
# Specify t and y
t <- 0
y <- c(-1.3, 0.8, 2.6, 5.8, 0.3, 1.3, 4.3, 5.0, -0.4, 1.0, 5.2, 6.2)
# Find the Kabaila and Giri (2009) CIUUPI, with minimum coverage 1 - alpha,
# for the first definition of the scaled expected length
res <- ciuupi2(alpha=0.05, X, a=a.vec, c=c.vec, bsvec, t, y, natural = 1)
res
# The Kabaila and Giri (2009) CIUUPI, with minimum coverage 1 - alpha,
# is (0.14040, 2.85704).
# The usual 1 - alpha confidence interval for theta is (-0.08185, 3.08185).
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