bsciuupi: Compute the vector (b(1),...,b(5),s(0),...,s(5)) that specifies the CIUUPI

Description

Compute the vector (b(1),...,b(5),s(0),...,s(5)) that specifies the confidence interval that utilizes uncertain prior information (CIUUPI).

Usage

bsciuupi(alpha, natural = 1, rho = NULL, a, c, x)

Arguments

alpha

The minimum coverage probability is 1 - alpha

natural

Equal to 1 (default) if the functions b and s are found by natural cubic spline interpolation or 0 if these functions are found by clamped cubic spline interpolation in the interval [-6,6]

rho

A known correlation

A vector used to specify the parameter of interest

A vector used to specify the parameter about which we have uncertain prior information

The n by p design matrix

Value

The vector $(b(1), b(2), \dots, b(5), s(0), s(1), \dots, s(5))$ that specifies the CIUUPI.

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 the random error with components that are iid normally distributed with zero mean and known variance. The parameter of interest is $\theta = $ a' $\beta$. The uncertain prior information is that $\tau = $ c' $\beta$ - t = 0, 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 $\tau$. The user must specify either a, c and x or rho. If a, c and x are specified then rho is computed.

The confidence interval for $\theta$, with minimum coverage probability 1 - alpha, that utilizes the uncertain prior information that $\tau = $ 0 belongs to a class of confidence intervals indexed by the functions b and s. 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 $b(1), b(2), \dots, b(5)$ and $s(0), s(1), \dots, s(5)$ 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 $(b(1), b(2), \dots, b(5), s(0), s(1), \dots, s(5))$ is found by numerical constrained optimization so that the confidence interval has minimum coverage probability 1 - alpha and utilizes the uncertain prior information through its desirable expected length properties. The optimization is performed using the slsqp function in the nloptr package.

Examples

Run this code

# NOT RUN {
# Compute the vector (b(1),...,b(5),s(0),...,s(5)) that specifies the CIUUPI,
# for given alpha and rho: (may take a few minutes to run)
# }
# NOT RUN {
bsvec <- bsciuupi(0.05, 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)
bsvec

# Compute the vector (b(1),...,b(5),s(0),...,s(5)) that specifies the CIUUPI,
# for given alpha, a, c and x
x1 <- c(-1, 1, -1, 1)
x2 <- c(-1, -1, 1, 1)
x <- cbind(rep(1, 4), x1, x2, x1*x2)
a <- c(0, 2, 0, -2)
c <- c(0, 0, 0, 1)

# The following may take a few minutes to run:
# }
# NOT RUN {
bsvec2 <- bsciuupi(0.05, a = a, c = c, x = x)
# }
# NOT RUN {
# The result (to 7 decimal places) is
bsvec2 <- c(-0.03639701, -0.18051953, -0.25111411, -0.15830362, -0.04479113,
           1.71997203, 1.79147968, 2.03881195, 2.19926399, 2.11845381,
           2.00482563)
bsvec2

# }