ciuupi_observed_value {ciuupi}

For given observed response vector y, compute the confidence interval that utilizes the uncertain prior information (CIUUPI)

Description

If \sigma is provided then, for given observed response vector y, compute the confidence interval, with minimum coverage probability 1-\alpha, for the parameter \theta =a^{\top}\beta that utilizes the uncertain prior information that the parameter \tau=c^{\top} \beta-t (specified by the vector c and the number t) takes the value 0. If \sigma is not provided then, as long as n-p \ge 30, replace \sigma by its estimate to compute an approximation to the CIUUPI for \theta.

Usage

ciuupi_observed_value(a, c, X, alpha, bs.list, t, y, sig = NULL)

Arguments

a

The p-vector a that specifies the parameter of interest \theta =a^{\top}\beta

c

The p-vector c used in the specification of the parameter \tau=c^{\top} \beta-t. The uncertain prior information is that \tau=0

X

The n \times p design matrix X, with linearly independent columns

alpha

1 - \alpha is the desired minimum coverage probability of the confidence interval for \theta

bs.list

A list that includes the following components: natural, d, q and the vector bsvec (b(h),...,b((q-1)h), s(0),s(h)...,s((q-1)h)), where h=d/q, that specifies the CIUUPI for all possible values of the random error variance and the observed response vector

t

The number t used to specify the parameter \tau=c^{\top} \beta-t. The uncertain prior information is that \tau = 0

y

The n-vector of observed responses

sig

Standard deviation of the random error. If a value is not specified then, provided that n-p \ge 30, sig is estimated from the data.

Details

Suppose that

y = X \beta + \varepsilon

where y is a random n-vector of responses, X is a known n \times p matrix with linearly independent columns, \beta is an unknown parameter p-vector and \varepsilon has components that are iid normally distributed with zero mean and known variance. Suppose that \theta=a^{\top} \beta is the parameter of interest, where a is a specified vector. Let \tau=c^{\top} \beta -t, where c is a specified vector, t is a specified number and a and c are linearly independent vectors. Also suppose that we have uncertain prior information that \tau = 0. For given observed response vector y and a design matrix X, ciuupi_observed_value computes the confidence interval, with minimum coverage probability 1-alpha, for \theta that utilizes the uncertain prior information that \tau = 0.

The example below is described in Discussion 5.8 on p.3426 of Kabaila and Giri (2009). This example is obtained by extracting a 2 \times 2 factorial data set from the 2^3 factorial data set described in Table 7.5 of Box et al. (1963).

Value

If \sigma is provided then a data frame of the lower and upper endpoints of the confidence interval, with minimum coverage probability 1-\alpha, for the parameter \theta that utilizes the uncertain prior information that \tau = 0. If \sigma is not provided then, as long as n-p \ge 30, a data frame of the lower and upper endpoints of an approximation to this confidence interval.

References

Box, G.E.P., Connor, L.R., Cousins, W.R., Davies, O.L., Hinsworth, F.R., Sillitto, G.P. (1963) The Design and Analysis of Industrial Experiments, 2nd edition, reprinted. Oliver and Boyd, London.

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

Examples

a <- c(0, 2, 0, -2)
c <- c(0, 0, 0, 1)
x1 <- c(-1, 1, -1, 1)
x2 <- c(-1, -1, 1, 1)
X <- cbind(rep(1, 4), x1, x2, x1*x2)
alpha <- 0.05
t <- 0
y <- c(87.2, 88.4, 86.7, 89.2)
sig <- 0.8
ciuupi_observed_value(a, c, X, alpha, bs.list.example, t, y, sig=sig)