Fits the log-Extreme-Value Birnbaum-Saunders (log-EVBS) regression model by joint maximum likelihood estimation. All parameters \((\beta^\top, \alpha, \gamma)^\top\) are estimated simultaneously using the BFGS algorithm with an analytic score function. Standard errors are computed from the analytic observed Fisher information matrix, which is numerically stable even for ill-conditioned design matrices.
Arguments
- x
A numeric design matrix of dimension \(n \times p\). It must include an intercept column (a column of ones). Each row corresponds to one observation and each column to one covariate.
- t
A numeric vector of length \(n\) containing the strictly positive responses (e.g. wind gust speeds). Internally the model is fit to
log(t).
Value
A list of class "evbsreg" with components:
coeffNumeric vector of length \(p+2\): the full parameter vector \((\beta_0,\ldots,\beta_{p-1}, \alpha, \gamma)\).
betahatNumeric vector of length \(p\): the regression coefficients.
alphahatScalar: the estimated shape parameter \(\hat\alpha\).
gamahatScalar: the estimated tail-shape parameter \(\hat\gamma\).
stderrorsNumeric vector of length \(p\): standard errors of the regression coefficients.
stderroralphaScalar: standard error of \(\hat\alpha\).
stderrorgamaScalar: standard error of \(\hat\gamma\).
zstatsNumeric vector: Wald z-statistics for the regression coefficients.
pvaluesNumeric vector: two-sided p-values for the regression coefficients.
muhatNumeric vector of length \(n\): fitted linear predictor on the log scale.
xi1,xi2Numeric vectors of length \(n\): helper quantities \(\xi_{i1}, \xi_{i2}\) evaluated at the MLE.
observmatrixNumeric matrix of dimension \((p+2)\times(p+2)\): the analytic Hessian \(\ddot\ell(\hat\theta)\) of the log-likelihood (negative definite at the maximum).
hessianNumeric matrix: identical to
observmatrix; provided under the conventional name. The observed Fisher information is-hessian.invNumeric matrix of dimension \((p+2)\times(p+2)\): the inverse of the observed Fisher information
-hessian, i.e. the asymptotic variance-covariance matrix of \(\hat\theta\).BNumeric matrix of dimension \(n\times n\): the influence matrix \(B = \Delta^\top (-\ddot\ell)^{-1} \Delta\) for the case-weight perturbation scheme, consumed by
cnc_diagnostics.nobsInteger: the number of observations \(n\).
nparInteger: the number of parameters \(p+2\).
The returned object has class "evbsreg" and has a
print.evbsreg method that displays a coefficient table.
Details
The EVBS regression model links the location parameter of the log-EVBS distribution to a linear predictor \(\mu_i = x_i^\top \beta\). The shape parameter \(\alpha > 0\) controls dispersion and the tail-shape parameter \(\gamma\) governs the Generalized Extreme Value tail behaviour (Frechet for \(\gamma>0\), Gumbel for \(\gamma=0\), Weibull for \(\gamma<0\)).
Initial values are obtained from lm.fit applied to
the log response, together with the moment-based starting point
\(\alpha_0 = \sqrt{(4/n)\sum \sinh^2((y_i - x_i^\top \beta_0)/2)}\)
and \(\gamma_0 = 0.01\). Optimization uses optim
with method = "BFGS", the analytic score, and tolerance
reltol = 1e-12.
The observed Fisher information is assembled analytically (see the
package vignette and the paper's appendix) and inverted via a Cholesky
factorization, falling back to solve if the matrix is not
positive definite.
References
Ospina, R., Lima, J. I. C., Barros, M., and Macedo, A. M. S. (2026). Local influence diagnostics for the extreme-value Birnbaum-Saunders regression model: methodology, validation, and application to anomalous wind gusts. Submitted.
Leiva, V., Ferreira, M., Gomes, M. I., and Lillo, C. (2016). Extreme value Birnbaum-Saunders regression models applied to environmental data. Stochastic Environmental Research and Risk Assessment, 30, 1045–1058.
Cook, R. D. (1986). Assessment of local influence. Journal of the Royal Statistical Society, Series B, 48, 133–169.
See also
cnc_diagnostics for influence diagnostics,
rqrandomized for residuals, itajai for the
example dataset.