pc: Confidence Interval - Percentile
In jeksterslabds/jeksterslabRboot: Jekster's Lab Linear Bootstrap Utilities

Description Usage Arguments Details Value Author(s) References See Also

Calculates percentile confidence intervals.

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

`thetahatstar`	Numeric vector. The bootstrap sampling distribution ≤ft( \boldsymbol{\hat{θ}^{}} \right), that is, the sampling distribution of `thetahat` estimated for each `b` bootstrap sample. \hat{θ}_{≤ft( 1 \right)}, \hat{θ}_{≤ft( 2 \right)}, \hat{θ}_{≤ft( 3 \right)}, \hat{θ}_{≤ft( b \right)}, …, \hat{θ}_{≤ft( B \right)}* .
`thetahat`	Numeric. Parameter estimate ≤ft( \hat{θ} \right) from the original sample data.
`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)`.
`wald`	Logical. If `TRUE`, calculates the square root of the Wald test statistic and p-value. The estimated bootstrap standard error is used. The arguments `null`, `dist`, and `df` are used to calculate the square root of the Wald test statistic and p-value and are NOT used in constructing the bootstrap confidence interval. If `FALSE`, returns `statistic = NA` and `p = NA` If `FALSE`, the arguments `null`, `dist`, and `df` are ignored.
`null`	Numeric. Hypothesized value of `theta` ≤ft( θ_{0} \right). Set to zero by default.
`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) .

The estimated bootstrap standard error is given by

\widehat{\mathrm{se}}_{\mathrm{B}} ≤ft( \hat{θ} \right) = √{ \frac{1}{B - 1} ∑_{b = 1}^{B} ≤ft[ \hat{θ}^{*} ≤ft( b \right) - \hat{θ}^{*} ≤ft( \cdot \right) \right]^2 }

where

\hat{θ}^{*} ≤ft( \cdot \right) = \frac{1}{B} ∑_{b = 1}^{B} \hat{θ}^{*} ≤ft( b \right) .

Note that \widehat{\mathrm{se}}_{\mathrm{B}} ≤ft( \hat{θ} \right) is the standard deviation of \boldsymbol{\hat{θ}^{*}} and \hat{θ}^{*} ≤ft( \cdot \right) is the mean of \boldsymbol{\hat{θ}^{*}} .

The percentile confidence interval is given by

≤ft[ \hat{θ}_{\mathrm{lo}}, \hat{θ}_{\mathrm{up}} \right] = ≤ft[ \hat{θ}^{*}_{z_{≤ft( \frac{α}{2} \right)}}, \hat{θ}^{*}_{z_{≤ft( 1 - \frac{α}{2} \right)}} \right] .

For more details and examples see the following vignettes:

Notes: Introduction to Nonparametric Bootstrapping

Notes: Introduction to Parametric Bootstrapping

Returns a vector with the following elements:

statistic: Square root of Wald test statistic. NA if wald = FALSE.
p: p-value. NA if wald = FALSE.
se: Estimated bootstrap standard error ≤ft( \widehat{\mathrm{se}}_{\mathrm{B}} ≤ft( \hat{θ} \right) \right).
ci_: Estimated percentile confidence limits corresponding to alpha from the bootstrap sampling distribution thetahatstar ≤ft( \boldsymbol{\hat{θ}^{*}} \right).