# exactCoverageProb: Evaluate exact coverage probability In popKorn: For interval estimation of mean of selected populations

## Description

This function will evaluate the exact coverage probability, as given in equation (4) on page 7 of the paper. See Details section.

## Usage

 `1` ```exactCoverageProb(c.vec, theta.diff, lambda, c.val, sigma.2 = 1, n = 1) ```

## Arguments

 `c.vec` This is a vector of length 2. It consists of the lower and upper limits in the integral. Checking is carried out to ensure that is of length two, and that 0 <= c.vec[1] <= c.vec[2]. This parameter is ignored if `lambda` is not missing. `theta.diff` A vector of length p-1, where p is the number of populations of treatments. Coordinate [i] in theta.diff corresponds to θ_i - θ_{i+1}. See `genDelMat`. `lambda` In case the user wishes to use the shrinkage version, this parameter should be specified. It must be between 0 and 1. `c.val` In case lambda is specified, this must not be missing. This will be combined with lambda to create a c.vec. This very function will then call itself. `sigma.2` The known variance of the error terms. `n` The number of replications per population.

## Details

This function evaluates the coverage probability for an interval defined by (X_{(1)} - c_2, X_{(1)} + c_1). Note that, as specified in the reference paper, we must have that 0 ≤ c_1 ≤ c_2. This function will call integrate2. Please note the ordering of the elements in the `c.vec` argument: the first element corresponds to the upper limit of the interval, and to the negative of the lower limit of the integral.

## Value

The function returns a scalar value that is the value of the exact coverage coverage probability defined in equation (4) of page 7.

 ```1 2 3``` ```del1 <- c(2, 4) exactCoverageProb(c(1.1,1.3), del1) exactCoverageProb(theta.diff=c(2,3,4), lambda=0.9, c.val=2) ```