cistandard2 {ciuupi2}

Compute the usual confidence interval

Description

Compute the usual 1 - alpha confidence interval

Usage

cistandard2(X, a, y, alpha)

Arguments

X

A known n by p matrix

a

A p-vector used to specify the parameter of interest

y

The n-vector of observed responses

alpha

1 - alpha is the coverage probability of the confidence interval

Details

Suppose that

Y = X \beta + \epsilon

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, where a is a specified p-vector. Then cistandard2 computes the usual 1 - alpha confidence interval for \theta, for given n-vector of observed responses y.

In the examples, we continue with the same 2 x 2 factorial example described in the documentation for find_rho, for the vector of observed responses y = (-1.3, 0.8, 2.6, 5.8, 0.3, 1.3, 4.3, 5.0, -0.4, 1.0, 5.2, 6.2).

The design matrix X and the vector a (denoted in R by a.vec) are entered into R using the commands in the following example.

Value

The usual 1 - alpha confidence interval.

References

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

See Also

find_rho

Examples

col1 <- rep(1,4)
col2 <- c(-1, 1, -1, 1)
col3 <- c(-1, -1, 1, 1)
col4 <- c(1, -1, -1, 1)
X.single.rep <- cbind(col1, col2, col3, col4)
X <- rbind(X.single.rep, X.single.rep, X.single.rep)
a.vec <- c(0, 2, 0, -2)
y <- c(-1.3, 0.8, 2.6, 5.8, 0.3, 1.3, 4.3, 5.0, -0.4, 1.0, 5.2, 6.2)

# Calculate the usual 95% confidence interval
res <- cistandard2(X, a=a.vec, y, alpha = 0.05)
res

# The usual 1 - alpha confidence interval for theta is (-0.08185, 3.08185)