bisection function

Bisection algorithm for bootstrap calibration of prediction intervals

This helper function returns a bootstrap calibrated coefficient for the calculation of prediction intervals (and limits).

bisection(
  y_star_hat,
  pred_se,
  y_star,
  alternative,
  quant_min,
  quant_max,
  n_bisec,
  tol,
  alpha,
  traceplot = TRUE
)

Arguments

• y_star_hat: a list of length $B$ that contains the expected future observations. Each entry in this list has to be a numeric vector of length $M$.
• pred_se: a list of length $B$ that contains the standard errors of the prediction. Each entry in this list has to be a numeric vector of length $M$.
• y_star: a list of length $B$ that contains the future observations. Each entry in this list has to be a numeric vector of length $M$.
• alternative: either "both", "upper" or "lower". alternative specifies if a prediction interval or an upper or a lower prediction limit should be computed
• quant_min: lower start value for bisection
• quant_max: upper start value for bisection
• n_bisec: maximal number of bisection steps
• tol: tolerance for the coverage probability in the bisection
• alpha: defines the level of confidence ($1-\alpha$)
• traceplot: if TRUE: Plot for visualization of the bisection process

Returns

This function returns $q^{calib}$ in the equation above.

Details

This function is an implementation of the bisection algorithm of Menssen and Schaarschmidt 2022. It returns a calibrated coefficient $q^{calib}$ for the calculation of pointwise and simultaneous prediction intervals

lower prediction limits

or upper prediction limits

that cover all of $m=1, ... , M$ future observations.

In this notation, $\hat{y}^*_m$ are the expected future observations for each of the $m$ future clusters, $q^{calib}$ is the calibrated coefficient and $\hat{se}(Y_m - y^*_m)$

are the standard errors of the prediction.

References

Menssen and Schaarschmidt (2022): Prediction intervals for all of M future observations based on linear random effects models. Statistica Neerlandica.

tools:::Rd_expr_doi("10.1111/stan.12260")