LN_QuantReg: Bayesian estimate of the log-normal conditioned quantiles

Description

This function produces a point estimate for the log-normal distribution quantile of fixed level quant.

Usage

LN_QuantReg(
  y,
  X,
  Xtilde,
  quant,
  method = "weak_inf",
  guess_s2 = NULL,
  y_transf = TRUE,
  CI = TRUE,
  method_CI = "exact",
  alpha_CI = 0.05,
  type_CI = "two-sided",
  rel_tol_CI = 1e-05,
  nrep_CI = 1e+05
)

Value

The function returns the prior parameters and their posterior values, summary statistics of the parameters $\beta$ and $\sigma^2$, and the estimate of the specified quantile: the posterior mean and variance are provided by default. Moreover the user can control the computation of posterior intervals.

#'@source

Gardini, A., C. Trivisano, and E. Fabrizi. Bayesian inference for quantiles of the log-normal distribution. Biometrical Journal (2020).

Arguments

y: Vector of observations of the response variable.
X: Design matrix.
Xtilde: Covariate patterns of the units to estimate.
quant: Number between 0 and 1 that indicates the quantile of interest.
method: String that indicates the prior setting to adopt. Choosing "weak_inf" a weakly informative prior setting is adopted, whereas selecting "optimal" the hyperparameters are fixed trough a numerical optimization algorithm aimed at minimizing the frequentist MSE.
guess_s2: Specification of a guess for the variance if available. If not, the sample estimate is used.
y_transf: Logical. If TRUE, the y vector is assumed already log-transformed.
CI: Logical. With the default choice TRUE, the posterior credibility interval is computed.
method_CI: String that indicates if the limits should be computed through the logSMNG quantile function qlSMNG (option "exact", default), or by randomly generating ("simulation") using the function rlSMNG.
alpha_CI: Level of credibility of the posterior interval.
type_CI: String that indicates the type of interval to compute: "two-sided" (default), "UCL" (i.e. Upper Credible Limit) for upper one-sided intervals or "LCL" (i.e. Lower Credible Limit) for lower one-sided intervals.
rel_tol_CI: Level of relative tolerance required for the integrate procedure or for the infinite sum. Default set to 1e-5.
nrep_CI: Number of simulations for the C.I. in case of method="simulation" and for the posterior of the coefficients vector.

Details

The function allows to carry out Bayesian inference for the conditional quantiles of a sample that is assumed log-normally distributed. The design matrix containing the covariate patterns of the sampled units is X, whereas Xtilde contains the covariate patterns of the unit to predict.

The classical log-normal linear mixed model is assumed and the quantiles are estimated as: $$\theta_p(x)=exp(x^T\beta+\Phi^{-1}(p))$$.

A generalized inverse Gaussian prior is assumed for the variance in the log scale $\sigma^2$, whereas a flat improper prior is assumed for the vector of coefficients $\beta$.

Two alternative hyperparamters setting are implemented (choice controlled by the argument method): a weakly informative proposal and an optimal one.