Fit the model at several quantile levels, with a monotonicity/invariance check

Fits fit_copula() at each level in taus and runs two systematic diagnostics the paper relies on to avoid a known failure mode (pitfall 1: a single-start fit at tau = 0.90 can land on a spurious stationary point whose fitted quantile sits above every observation):

Usage

fit_copula_multi_tau(
  y,
  X,
  Z,
  p = 0,
  q = 0,
  family = "kuma",
  taus = c(0.5, 0.75, 0.9, 0.95, 0.99),
  tol_ll = 1,
  verbose = TRUE
)

Arguments

y

Numeric response vector in (0, 1).

X

Location-submodel design matrix.

Z

Shape-submodel design matrix.

p, q

Non-negative integers, the ARMA orders.

family

"kuma" (default) or "beta".

taus

Numeric vector of quantile levels to fit.

tol_ll

Absolute log-likelihood tolerance for the invariance check (default 1; fits whose log-likelihood differs from the median by more than this are flagged).

verbose

Logical; print the diagnostic table.

Value

A list named by tau, each element with components par, loglik, fonte ("padrao" or "reinicializado": standard fit or restarted) and aprovado (logical, passed both checks), with a "diagnostico" attribute (a data frame) summarizing the checks for every level.

Details

1. Log-likelihood invariance. The tau-reparametrization describes the same distribution family for every tau; only the parametrization changes, not the model. Under correct convergence, the maximized log-likelihood should therefore be (approximately) the same across all levels. A fit whose log-likelihood departs from the others by more than tol_ll has not found the global maximum.

2. Quantile monotonicity. mu_t^{(tau)}, the conditional tau-quantile, must be non-decreasing in tau for every t, by definition of a quantile. This catches spurious optima the invariance check alone can miss (the likelihood surface can be nearly flat near a spurious optimum).

Fits flagged by either check are retried once from the parameter estimates of each already-accepted level (not only the nearest one, since the spurious basin can be reached from several "reasonable" starting points), keeping the restart with the highest log-likelihood among those that restore monotonicity with the levels already accepted. This function does not cascade restarts further: an earlier investigation found that a restart cascade can converge to a degenerate region of the parameter space that, despite a nominally higher log-likelihood, violates the invariance property itself (an unambiguous sign of a spurious local optimum) – instead, unresolved cases are flagged for manual inspection.

See also

fit_copula()