The function may be called by the names: g07bfc, nag_univar_estim_genpareto or nag_estim_gen_pareto.
Let the distribution function of a set of observations
be given by the generalized Pareto distribution:
, when ;
, when .
Estimates and of the parameters and are calculated by using one of:
method of moments (MOM);
probability-weighted moments (PWM);
maximum likelihood estimates (MLE) that seek to maximize the log-likelihood:
The variances and covariance of the asymptotic Normal distribution of parameter estimates and are returned if satisfies:
for the MOM;
for the PWM method;
for the MLE method.
If the MLE option is exercised, the observed variances and covariance of and is returned, given by the negative inverse Hessian of .
Hosking J R M and Wallis J R (1987) Parameter and quantile estimation for the generalized Pareto distribution Technometrics29(3)
McNeil A J, Frey R and Embrechts P (2005) Quantitative Risk Management Princeton University Press
1: – IntegerInput
On entry: the number of observations.
2: – const doubleInput
On entry: the observations
, for , assumed to follow a generalized Pareto distribution.
3: – Nag_OptimOptInput
On entry: determines the method of estimation, set:
For the method of probability-weighted moments.
For the method of moments.
For maximum likelihood with starting values given by the method of moments estimates.
For maximum likelihood with starting values given by the method of probability-weighted moments.
, , or .
4: – double *Output
On exit: the parameter estimate .
5: – double *Output
On exit: the parameter estimate .
6: – doubleOutput
On exit: the variance-covariance of the asymptotic Normal distribution of and . contains the variance of ; contains the variance of ; and contain the covariance of and .
7: – doubleOutput
On exit: if maximum likelihood estimates are requested, the observed variance-covariance of and . contains the variance of ; contains the variance of ; and contain the covariance of and .
8: – double *Output
On exit: if maximum likelihood estimates are requested, ll contains the log-likelihood value at the end of the optimization; otherwise ll is set to .
9: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
The optimization of log-likelihood failed to converge; no maximum likelihood estimates are returned. Try using the other maximum likelihood option by resetting optopt. If this also fails, moments-based estimates can be returned by an appropriate setting of optopt.
Variance of data in y is too low for method of moments optimization.
The asymptotic distribution of parameter estimates is invalid and the distribution of maximum likelihood estimates cannot be calculated for the returned parameter estimates because the Hessian matrix could not be inverted.
The asymptotic distribution is not available for the returned parameter estimates.
The distribution of maximum likelihood estimates cannot be calculated for the returned parameter estimates because the Hessian matrix could not be inverted.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g07bfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g07bfc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The search for maximum likelihood parameter estimates is further restricted by requiring
as this avoids the possibility of making the log-likelihood arbitrarily high.
This example calculates parameter estimates for observations assumed to be drawn from a generalized Pareto distribution.