plot_cp {ciuupi}

Plot the graph of the coverage probability of the CIUUPI

Description

The input bs.list determines the functions b and s that specify the confidence interval that utilizes the uncertain prior information (CIUUPI), for all possible values of \sigma and observed response vector. The coverage probability of the CIUUPI is an even function of the unknown parameter \gamma = \tau \big/ \big(\sigma \, v_{\tau}^{1/2} \big). The R function plot_cp plots the graph of the coverage probability of the CIUUPI, as a function of |\gamma|. To provide a stringent assessment of this coverage probability, we use a fine equally-spaced grid seq(0, (d+4), by = 0.01) of values of \gamma and Gauss Legendre quadrature using 10 nodes in the relevant integrals. By contrast, for the computation of the CIUUPI, implemented in bs_ciuupi, we require that the coverage probability of this confidence interval is greater than or equal to 1-\alpha for the equally-spaced grid seq(0, (d+2), by = 0.05) of values of \gamma and we use Gauss Legendre quadrature with 5 nodes in the relevant integrals.

Usage

plot_cp(bs.list)

Arguments

bs.list

A list that includes the following components.

alpha: the desired minimum coverage is 1 - \alpha.

rho: the known correlation between \widehat{\theta} and \widehat{\tau}. This correlation is computed from the p-vectors a and c and the n \times p design matrix X using the formula \rho=a^{\top}(X^{\top}X)^{-1}c /(v_{\theta} \, v_{\tau})^{1/2}, where v_{\theta} =a^{\top}(X^{\top}X)^{-1} a and v_{\tau} =c^{\top}(X^{\top}X)^{-1} c.

natural: 1 when the functions b and s are specified by natural cubic spline interpolation or 0 if these functions are specified by clamped cubic spline interpolation

d: the functions b and s are specified by cubic splines on the interval [-d, d]

n.ints: number of equal-length intervals in [0, d], where the endpoints of these intervals specify the knots, belonging to [0, d], of the cubic spline interpolations that specify the functions b and s. In the description of bsvec, n.ints is also called q.

bsvec: the (2q-1)-vector

\big(b(h),...,b((q-1)h), s(0),s(h)...,s((q-1)h) \big),

where q=ceiling(d/0.75) and h=d/q.

Value

A plot of the graph of the coverage probability of the CIUUPI as a function of |\gamma|, where \gamma denotes the unknown parameter \tau \big/ \big(\sigma \, v_{\tau}^{1/2} \big).

Examples

plot_cp(bs.list.example)