ci_standard {ciuupi}

For given observed response vector y, compute the standard 1 - \alpha confidence interval

Description

If \sigma is provided then compute the standard 1 - \alpha confidence interval for \theta. If \sigma is not provided then, as long as n-p \ge 30, replace \sigma by its estimate to compute an approximate 1 - \alpha confidence interval for \theta.

Usage

ci_standard(a, X, y, alpha, sig = NULL)

Arguments

a

The vector used to specify the parameter of interest \theta = a^{\top} \beta

X

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

y

The n-vector of observed responses

alpha

1 - \alpha is the coverage probability of the standard confidence interval

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 \sim N(0, \, \sigma^2 \, I), with \sigma^2 assumed known. Suppose that the parameter of interest is \theta = a^{\top} \beta. The R function ci_standard computes the standard 1 - \alpha confidence interval for \theta.

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 standard 1 - \alpha confidence interval for \theta. 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

y <- c(87.2, 88.4, 86.7, 89.2)
x1 <- c(-1, 1, -1, 1)
x2 <- c(-1, -1, 1, 1)
X <- cbind(rep(1, 4), x1, x2, x1*x2)
a <- c(0, 2, 0, -2)
ci_standard(a, X, y, 0.05, sig = 0.8)