# cpciuupi2: Compute the coverage probability of the Kabaila & Giri (2009)... In ciuupi2: Kabaila and Giri (2009) Confidence Interval

## Description

Evaluate the coverage probability of the Kabaila & Giri (2009) confidence interval that utilizes uncertain prior information (CIUUPI), with minimum coverage 1 - `alpha`, at `gam`.

## Usage

 `1` ```cpciuupi2(gam, bsvec, alpha, m, rho, natural = 1) ```

## Arguments

 `gam` A value of gamma or vector of gamma values at which the coverage probability 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 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 and `t` is a specified number. `rho` is the known correlation between the least squares estimators of θ and τ. It is determined by the n by p design matrix X and the p-vectors a and c using `find_rho`.

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 coverage probability 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.

`find_rho`, `bsciuupi2`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22``` ```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 coverage probability function gam <- seq(0, 10, by = 0.1) cp <- cpciuupi2(gam, bsvec, alpha, m, rho = -0.7071068) plot(gam, cp, type = "l", lwd = 2, ylab = "", las = 1, xaxs = "i", main = "Coverage Probability", col = "blue", xlab = expression(paste("|", gamma, "|")), ylim = c(0.9490, 0.9510)) abline(h = 1-alpha, lty = 2) ```