Loglikelihood adjustment for model fits

This function is generic. It performs adjustment of the loglikelihood associated with fitted model objects, following Chandler and Bate (2007). Certain classes of extreme value model objects are supported automatically. For details see the alogLik help pages for the packages: evd, evir, extRemes, fExtremes, ismev, mev, POT, texmex. User-supplied objects can also be supported: the requirements for these objects are explained in Details.

alogLik(
  x,
  cluster = NULL,
  use_vcov = TRUE,
  binom = FALSE,
  k,
  inc_cens = TRUE,
  ...
)

Arguments

x

A fitted model object with certain associated S3 methods. See Details.

cluster

A vector or factor indicating from which cluster the respective log-likelihood contributions from loglik originate. The length of cluster must be consistent with the estfun method to be used in the estimation of the 'meat' V of the sandwich estimator of the covariance matrix of the parameters to be passed to adjust_loglik. In most cases, cluster must have length equal to the number of observations in data. The exception is the GP (only) model (binom = FALSE), where the cluster may either contain a value for each observation in the raw data, or for each threshold exceedance in the data.

If cluster is not supplied (is NULL) then it is assumed that each observation forms its own cluster. See Details for further details.

use_vcov

A logical scalar. Should we use the vcov S3 method for x (if this exists) to estimate the Hessian of the independence loglikelihood to be passed as the argument H to adjust_loglik? Otherwise, H is estimated inside adjust_loglik using optimHess.

binom

A logical scalar. This option is only relevant to GP models and is only available in the stationary (no covariates) case. If binom = FALSE then loglikelihood adjustment is only performed using the GP model. If binom = TRUE then loglikelihood adjustment is also performed for inferences about the probability of threshold exceedance, using a Bernoulli model for the instances of threshold exceedance.

k

A non-negative integer scalar. This option is only relevant to GP models and is only available in the stationary (no covariates) case. If k is supplied then it is passed as the run parameter \(K\) to kgaps for making inferences about the extremal index \(\theta\) using the \(K\)-gaps model of Suveges and Davison (2010).

inc_cens

A logical scalar. This argument is only relevant if k is supplied. Passed to kgaps to indicate whether or not to include censored inter-exceedance times, relating to the first and last observations.

...

Further arguments to be passed to the functions in the sandwich package meat (if cluster = NULL), or meatCL (if cluster is not NULL).

Value

An object inheriting from class "chandwich". See

adjust_loglik.

The original fitted model object is available as an attribute named

"original_fit", accessible using attr(name, "original_fit"), where name is the name of the object to which the object returned from alogLik is assigned.

If binom = TRUE then the returned object has an extra attribute named pu_aloglik that contains an object inheriting from class

"chandwich" relating specifically to inferences about the probability of threshold exceedance. Also, the 4th component of the class of the returned object becomes "bin-gpd".

If k is supplied then the returned object has an extra attribute named theta that contains an object inheriting from class

c("kgaps", "exdex") relating specifically to inferences about the extremal index \(\theta\). See the Value section in

kgaps.

If x is one of the supported models then the class of the returned object is a vector of length 5. The first 3 components are

c("lax", "chandwich", "name_of_package"), where

"name_of_package" is the name of the package from which the input object x originated. The remaining 2 components depend on the model that was fitted. See the documentation of the relevant package for details:

evd,

evir,

extRemes,

fExtremes,

ismev,

mev,

POT,

texmex.

Otherwise, the class of the returned object is

c("lax", "chandwich", class(x)).

Objects returned from `aloglik` have `anova`, `coef`, `confint`, `logLik`, `nobs`, `plot`, `print`, `summary` and `vcov` methods.

Details

Object x must have the following S3 methods:

logLikVec: returns a vector of the contributions to the independence loglikelihood from individual observations;
coef: returns a vector of model coefficients, see coef;
nobs: returns the number of (non-missing) observations used in a model fit, see nobs;

and may have the following S3 methods

vcov: returns the estimated variance-covariance matrix of the (main) parameters of a fitted model, see vcov;
estfun: returns an \(n\) by \(k\) matrix, in which each column gives the derivative of the loglikelihood at each of \(n\) observation with respect to the \(k\) parameters of the model, see estfun.

Loglikelihood adjustment is performed using the adjust_loglik function in the chandwich package. The relevant arguments to adjust_loglik, namely loglik, mle, H and V, are created based on the class of the object x.

If a vcov method is not available, or if use_vcov = FALSE, then the variance-covariance matrix of the MLE (from which H is calculated) is estimated inside adjust_loglik using optimHess.

The sandwich package is used to estimate the variance matrix V of the score vector: meat is used if cluster = NULL; meatCL is used if cluster is not NULL. If cluster is NULL then any arguments of meatCL present in ... will be ignored. Similarly, if cluster is not NULL then any arguments of meat present in ... will be ignored. meat and meatCL require an estfun method to be available, which, in the current context, provides matrix of score contributions. If a bespoke estfun method is not provided then this is constructed by estimating the score contributions using jacobian.

Examples

See the (package-specific) examples in evd, evir, extRemes,fExtremes, ismev, mev, POT and texmex.

References

Chandler, R. E. and Bate, S. (2007). Inference for clustered data using the independence loglikelihood. Biometrika, 94(1), 167-183. doi:10.1093/biomet/asm015

Suveges, M. and Davison, A. C. (2010) Model misspecification in peaks over threshold analysis, The Annals of Applied Statistics, 4(1), 203-221. doi:10.1214/09-AOAS292

Zeileis (2006) Object-Oriented Computation and Sandwich Estimators. Journal of Statistical Software, 16, 1-16. doi:10.18637/jss.v016.i09