Confidence Interval - Wald

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
)

Arguments

thetahat	Numeric. Parameter estimate \(\left( \hat{\theta} \right)\).
sehat	Numeric. Estimated standard error of thetahat \(\left( \widehat{\mathrm{se}} \left( \hat{\theta} \right) \right)\).
null	Numeric. Hypothesized value of theta \(\left( \theta_{0} \right)\). Set to zero by default.
alpha	Numeric vector. Significance level \(\left( \alpha \right)\) . 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 \( \left( \theta \right) \).

Value

Returns a vector with the following elements:

statistic

Square root of Wald test statistic.

p

p-value.

se

Estimated d=standard error of thetahat \(\left( \widehat{\mathrm{se}} \left( \hat{\theta} \right) \right)\).

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.

Details

As the sample size approaches infinity \(\left( n \to \infty \right)\), the distribution of \(\hat{\theta}\) approaches the normal distribution with mean equal to \(\theta\) and variance equal to \(\widehat{\mathrm{Var}} \left( \hat{\theta} \right)\) $$ \hat{\theta} \mathrel{\dot\sim} \mathcal{N} \left( \theta, \widehat{\mathrm{Var}} \left( \hat{\theta} \right) \right) . $$ As such, \( \hat{\theta} \) can be expressed in terms of \(z\)-scores from a standard normal distribution $$ \frac{ \hat{\theta} - \theta } { \widehat{\mathrm{se}} \left( \hat{\theta} \right) } \mathrel{\dot\sim} \mathcal{N} \left( 0, 1 \right) $$ where \( \widehat{\mathrm{se}} \left( \hat{\theta} \right) = \sqrt{ \widehat{\mathrm{Var}} \left( \hat{\theta} \right) } \).

To form a confidence interval around \(\hat{\theta}\), the \(z\)-score associated with a particular alpha level can be plugged-in the equation below. $$ \hat{\theta} \pm z_{\frac{\alpha}{2}} \times \widehat{\mathrm{se}} \left( \hat{\theta} \right) $$

Note that this is valid only when \(n \to \infty\). In finite samples, this is only an approximation. Gosset derived a better approximation in the context of \(\hat{\theta} = \bar{x}\) $$ \frac{ \hat{\theta} - \theta } { \widehat{\mathrm{se}} \left( \hat{\theta} \right) } \mathrel{\dot\sim} t \left( \nu \right) $$ where \(t\) is the Student's \(t\) distribution and \(\nu\), the degrees of freedom \(n - 1\), is the Student's \(t\) distribution parameter. As such, the symmetric Wald confidence interval is given by $$ \hat{\theta} \pm t_{ \left( \frac{ \alpha } { 2 } , \nu \right) } \times \widehat{\mathrm{se}} \left( \hat{\theta} \right) . $$ Note that in large sample sizes, \(t\) converges to \(z\).

References

Wikipedia: Confidence interval

Examples

#############################################################
# 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
#> [1] -0.01130945
# Closed form solution for the standard error of the mean
sehat <- sd(x) / sqrt(n)
sehat
#> [1] 0.01006135

#############################################################
# Generate Wald confidence intervals
# for alpha = c(0.001, 0.01, 0.05).
#############################################################
wald(
  thetahat = thetahat,
  sehat = sehat
)
#>    statistic            p           se      ci_0.05       ci_0.5       ci_2.5 
#> -1.124049082  1.739007796  0.010061348 -0.044416585 -0.037225765 -0.031029329 
#>      ci_97.5      ci_99.5     ci_99.95 
#>  0.008410431  0.014606866  0.021797686