wald: Confidence Interval - Wald
In jeksterslabds/jeksterslabRboot: Jekster's Lab Linear Bootstrap Utilities

Description Usage Arguments Details Value Author(s) References Examples

Calculates symmetric Wald confidence intervals.

wald(
  thetahat,
  sehat,
  null = 0,
  alpha = c(0.001, 0.01, 0.05),
  dist = "z",
  df,
  eval = FALSE,
  theta = 0
)

`thetahat`	Numeric. Parameter estimate ≤ft( \hat{θ} \right).
`sehat`	Numeric. Estimated standard error of `thetahat` ≤ft( \widehat{\mathrm{se}} ≤ft( \hat{θ} \right) \right).
`null`	Numeric. Hypothesized value of `theta` ≤ft( θ_{0} \right). Set to zero by default.
`alpha`	Numeric vector. Significance level ≤ft( α \right) . By default, `alpha` is set to conventional significance levels `alpha = c(0.001, 0.01, 0.05)`.
`dist`	Character string. `dist = "z"` for the standard normal distribution. `dist = "t"` for the t distribution.
`df`	Numeric. Degrees of freedom (df) if `dist = "t"`. Ignored if `dist = "z"`.
`eval`	Logical. Evaluate confidence intervals using `zero_hit()`, `theta_hit()`, `len()`, and `shape()`.
`theta`	Numeric. Population parameter ≤ft( θ \right) .

As the sample size approaches infinity ≤ft( n \to ∞ \right), the distribution of \hat{θ} approaches the normal distribution with mean equal to θ and variance equal to \widehat{\mathrm{Var}} ≤ft( \hat{θ} \right)

\hat{θ} \mathrel{\dot\sim} \mathcal{N} ≤ft( θ, \widehat{\mathrm{Var}} ≤ft( \hat{θ} \right) \right) .

As such, \hat{θ} can be expressed in terms of z-scores from a standard normal distribution

\frac{ \hat{θ} - θ } { \widehat{\mathrm{se}} ≤ft( \hat{θ} \right) } \mathrel{\dot\sim} \mathcal{N} ≤ft( 0, 1 \right)

where \widehat{\mathrm{se}} ≤ft( \hat{θ} \right) = √{ \widehat{\mathrm{Var}} ≤ft( \hat{θ} \right) } .

To form a confidence interval around \hat{θ}, the z-score associated with a particular alpha level can be plugged-in the equation below.

\hat{θ} \pm z_{\frac{α}{2}} \times \widehat{\mathrm{se}} ≤ft( \hat{θ} \right)

Note that this is valid only when n \to ∞. In finite samples, this is only an approximation. Gosset derived a better approximation in the context of \hat{θ} = \bar{x}

\frac{ \hat{θ} - θ } { \widehat{\mathrm{se}} ≤ft( \hat{θ} \right) } \mathrel{\dot\sim} t ≤ft( ν \right)

where t is the Student's t distribution and ν, the degrees of freedom n - 1, is the Student's t distribution parameter. As such, the symmetric Wald confidence interval is given by

\hat{θ} \pm t_{ ≤ft( \frac{ α } { 2 } , ν \right) } \times \widehat{\mathrm{se}} ≤ft( \hat{θ} \right) .

Note that in large sample sizes, t converges to z.

Returns a vector with the following elements:

statistic: Square root of Wald test statistic.
p: p-value.
se: Estimated d=standard error of thetahat ≤ft( \widehat{\mathrm{se}} ≤ft( \hat{θ} \right) \right).
ci_: Estimated confidence limits corresponding to alpha.

If eval = TRUE, also returns

zero_hit_: Logical. Tests if confidence interval contains zero.
theta_hit_: Logical. Tests if confidence interval contains theta.
length_: Length of confidence interval.
shape_: Shape of confidence interval.

Ivan Jacob Agaloos Pesigan

Wikipedia: Confidence interval

#############################################################
# Generate sample data from a normal distribution
# with mu = 0 and sigma^2 = 1.
#############################################################
set.seed(42)
n <- 10000
x <- rnorm(n = n)

#############################################################
# Estimate the population mean mu using the sample mean xbar.
#############################################################
# Parameter estimate xbar
thetahat <- mean(x)
thetahat
# Closed form solution for the standard error of the mean
sehat <- sd(x) / sqrt(n)
sehat

#############################################################
# Generate Wald confidence intervals
# for alpha = c(0.001, 0.01, 0.05).
#############################################################
wald(
  thetahat = thetahat,
  sehat = sehat
)