jack_hat: Jackknife Estimates
In jeksterslabds/jeksterslabRboot: Jekster's Lab Linear Bootstrap Utilities

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

Calculates jackknife estimates.

jack_hat(
  thetahatstarjack,
  thetahat,
  alpha = c(0.001, 0.01, 0.05),
  eval = FALSE,
  theta = 0
)

`thetahatstarjack`	Numeric vector. Jackknife sampling distribution, that is, the sampling distribution of `thetahat` estimated for each `i` jackknife sample. \hat{ θ }_{ ≤ft( 1 \right) }, \hat{ θ }_{ ≤ft( 2 \right) }, \hat{ θ }_{ ≤ft( 3 \right) }, …, \hat{ θ }_{ ≤ft( n \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)`.
`eval`	Logical. Evaluate confidence intervals using `zero_hit()`, `theta_hit()`, `len()`, and `shape()`.
`theta`	Numeric. Population parameter ≤ft( θ \right) .

The jackknife estimate of bias is given by

\widehat{ \mathrm{ bias } }_{ \mathrm{ jack } } ≤ft( θ \right) = ≤ft( n - 1 \right) ≤ft( \hat{ θ }_{ ≤ft( \cdot \right) } - \hat{ θ } \right)

where

\hat{ θ }_{ ≤ft( \cdot \right) } = \frac{ 1 } { n } ∑_{ i = 1 }^{ n } \hat{ θ }_{ ≤ft( i \right) } .

The jackknife estimate of standard error is given by

\widehat{ \mathrm{ se } }_{ \mathrm{ jack } } ≤ft( \hat{ θ } \right) = √{ \frac{ n - 1 } { n } ∑_{ i = 1 }^{ n } ≤ft( \hat{ θ }_{ ≤ft( i \right) } - \hat{ θ }_{ ≤ft( \cdot \right) } \right)^2 } .

The bias-corrected jackknife estimate is given by

\hat{ θ }_{ \mathrm{ jack } } = \hat{ θ } - \hat{ \mathrm{ bias } }_{ \mathrm{ jack } } ≤ft( θ \right) = n \hat{ θ } - ≤ft( n - 1 \right) \hat{ θ }_{ ≤ft( \cdot \right) } .

Pseudo-values can be computed using

\tilde{ θ }_{ i } = n \hat{ θ } - ≤ft( n - 1 \right) \hat{ θ }_{ ≤ft( i \right) } .

The standard error can be estimated using the pseudo-values

\widehat{ \mathrm{ se } }_{ \mathrm{ jack } } ≤ft( \tilde{ θ } \right) = √{ ∑_{ i = 1 }^{ n } \frac{ ≤ft( \tilde{ θ }_{ i } - \tilde{ θ } \right)^2 } { ≤ft( n - 1 \right) n } }

where

\tilde{ θ } = \frac{ 1 } { n } ∑_{ i = 1 }^{ n } \tilde{ θ }_{ i } .

An interval can be generated using

\tilde{ θ } \pm t_{ \frac{ α } { 2 } } \times \widehat{ \mathrm{ se } }_{ \mathrm{ jack } } ≤ft( \tilde{ θ } \right)

with degrees of freedom ν = n - 1 .

Returns a list with the following elements:

hat: Jackknife estimates.
ps: Pseudo-values.
ci: Confidence intervals using pseudo-values.

The first list element hat contains the following:

mean: Mean of thetahatstarjack ≤ft( \hat{θ}_{≤ft( \cdot \right) } \right).
bias: Jackknife estimate of bias ≤ft( \widehat{\mathrm{bias}}_{\mathrm{jack}} ≤ft( θ \right) \right).
se: Jackknife estimate of standard error ≤ft( \widehat{\mathrm{se}}_{\mathrm{jack}} ≤ft( \hat{θ} \right) \right).
thetahatjack: Bias-corrected jackknife estimate ≤ft( \hat{θ}_{\mathrm{jack}} \right).

Ivan Jacob Agaloos Pesigan

Wikipedia: Jackknife resampling

Other jackknife functions: jack()

n <- 100
x <- rnorm(n = n)
thetahat <- mean(x)
xstar <- jack(
  data = x
)
thetahatstarjack <- sapply(
  X = xstar,
  FUN = mean
)
str(xstar, list.len = 6)
hist(
  thetahatstarjack,
  main = expression(
    paste(
      "Histogram of ",
      hat(theta),
      "*"
    )
  ),
  xlab = expression(
    paste(
      hat(theta),
      "*"
    )
  )
)
jack_hat(
  thetahatstarjack = thetahatstarjack,
  thetahat = thetahat
)