bsciuupi2 {ciuupi2}

Compute the vector (b(d/6),...,b(5d/6),s(0),...,s(5d/6)) that specifies the Kabaila & Giri (2009) CIUUPI

Description

Compute the vector (b(d/6),...,b(5d/6),s(0),...,s(5d/6)) that specifies the Kabaila and Giri (2009) confidence interval that utilizes uncertain prior information (CIUUPI) and has minimum coverage 1 - alpha.

Usage

bsciuupi2(alpha, m, rho, obj = 1, natural = 1)

Arguments

alpha

The minimum coverage probability is 1 - alpha

m

Degrees of freedom n - p

rho

A known correlation

obj

Equal to 1 (default) for the first definition of the scaled expected length or 2 for the second definition of the scaled expected length

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 [-d, d]

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 independent and identically normally distributed with zero mean and unknown variance. The parameter of interest is \theta = a' \beta. The uncertain prior information is that \tau = c' \beta takes the value t, where a and c are specified linearly independent vectors nonzero p-vectors and t is a specified number. rho is the known correlation between the least squares estimators of \theta and \tau. It is determined by the n by p design matrix X and the p-vectors a and c using find_rho.

The confidence interval for \theta, with minimum coverage probability 1 - alpha, that utilizes the uncertain prior information that \tau = t 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 t distribution with m degrees of freedom for all |x| greater than or equal to d, where d is a sufficiently large positive number (chosen by the function bsciuupi2). The values of these functions in the interval [-d,d] are specified by the vectors (b(d/6), b(2d/6), \dots, b(5d/6)) and (s(0), s(d/6), \dots, s(5d/6)) as follows. By assumption, b(0)=0 and b(-i)=-b(i) and s(-i)=s(i) for i=d/6,...,d. The values of b(x) and s(x) for any x in the interval [-d,d] are found using cube spline interpolation for the given values of b(i) and s(i) for i=-d,-5d/6,...,0,d/6,...,5d/6,d. The choices of d for m = 1, 2 and >2 are d=20, 10 and 6, respectively.

The vector (b(d/6), b(2d/6), \dots, b(5d/6), s(0), s(d/6), \dots, s(5d/6)) is found by numerical nonlinear constrained optimization so that the confidence interval has minimum coverage probability 1 - alpha and utilizes the uncertain prior information that \tau = t through its desirable expected length properties. The optimization is performed using the slsqp function in the nloptr package.

The first definition of the scaled expected length of the Kabaila and Giri(2009) CIUUPI is the expected length of this confidence interval divided by the expected length of the usual confidence interval with coverage probability 1 - alpha. The second definition of the scaled expected length of the Kabaila and Giri(2009) CIUUPI is the expected value of the ratio of the length of this confidence interval divided by the length of the usual confidence interval, with coverage probability 1 - alpha, computed from the same data.

In the examples, we continue with the same 2 x 2 factorial example described in the documentation for find_rho.

Value

The vector (b(d/6), b(2d/6), \dots, b(5d/6), s(0), s(d/6), \dots, s(5d/6)) that specifies the Kabaila & Giri (2009) CIUUPI, with minimum coverage 1 - alpha.

References

Kabaila, P. and Giri, R. (2009). Confidence intervals in regression utilizing prior information. Journal of Statistical Planning and Inference, 139, 3419-3429.

See Also

find_rho

Examples

# 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:
# c(-0.0287487, -0.2151595, -0.3430403, -0.3125889, -0.0852146,
#    1.9795390,  2.0665414,  2.3984471,  2.6460159,  2.6170066,  2.3925494)