NAG Library Routine Document

G07BFF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{N}$ – INTEGERInput

On entry: the number of observations.

Constraint: $\mathbf{N}>1$.

2:     $\mathrm{Y}\left(\mathbf{N}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the $n$ observations $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, assumed to follow a generalized Pareto distribution.

Constraints:

• $\mathbf{Y}\left(i\right)\ge 0.0$;
• ${\displaystyle \sum _{\mathit{i}=1}^n}\mathbf{Y}\left(i\right)>0.0$.

3:     $\mathrm{OPTOPT}$ – INTEGERInput

On entry: determines the method of estimation, set:

$\mathbf{OPTOPT}=-2$

For the method of probability-weighted moments.

$\mathbf{OPTOPT}=-1$

For the method of moments.

$\mathbf{OPTOPT}=1$

For maximum likelihood with starting values given by the method of moments estimates.

$\mathbf{OPTOPT}=2$

For maximum likelihood with starting values given by the method of probability-weighted moments.

Constraint: $\mathbf{OPTOPT}=-2$, $-1$, $1$ or $2$.

4:     $\mathrm{XI}$ – REAL (KIND=nag_wp)Output

On exit: the parameter estimate $\hat{\xi }$.

5:     $\mathrm{BETA}$ – REAL (KIND=nag_wp)Output

On exit: the parameter estimate $\hat{\beta }$.

6:     $\mathrm{ASVC}\left(4\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the variance-covariance of the asymptotic Normal distribution of $\hat{\xi }$ and $\hat{\beta }$. $\mathbf{ASVC}\left(1\right)$ contains the variance of $\hat{\xi }$; $\mathbf{ASVC}\left(4\right)$ contains the variance of $\hat{\beta }$; $\mathbf{ASVC}\left(2\right)$ and $\mathbf{ASVC}\left(3\right)$ contain the covariance of $\hat{\xi }$ and $\hat{\beta }$.

7:     $\mathrm{OBSVC}\left(4\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: if maximum likelihood estimates are requested, the observed variance-covariance of $\hat{\xi }$ and $\hat{\beta }$. $\mathbf{OBSVC}\left(1\right)$ contains the variance of $\hat{\xi }$; $\mathbf{OBSVC}\left(4\right)$ contains the variance of $\hat{\beta }$; $\mathbf{OBSVC}\left(2\right)$ and $\mathbf{OBSVC}\left(3\right)$ contain the covariance of $\hat{\xi }$ and $\hat{\beta }$.

8:     $\mathrm{LL}$ – REAL (KIND=nag_wp)Output

On exit: if maximum likelihood estimates are requested, LL contains the log-likelihood value $L$ at the end of the optimization; otherwise LL is set to $-1.0$.

9:     $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

On entry, $\mathbf{N}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{N}>1$.

$\mathbf{IFAIL}=2$

On entry, $\mathbf{Y}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{Y}\left(i\right)\ge 0.0$ for all $i$.

$\mathbf{IFAIL}=3$

On entry, $\mathbf{OPTOPT}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{OPTOPT}=-2$, $-1$, $1$ or $2$.

$\mathbf{IFAIL}=6$

The asymptotic distribution is not available for the returned parameter estimates.

$\mathbf{IFAIL}=7$

The distribution of maximum likelihood estimates cannot be calculated for the returned parameter estimates because the Hessian matrix could not be inverted.

$\mathbf{IFAIL}=8$

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.

$\mathbf{IFAIL}=9$

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.

$\mathbf{IFAIL}=10$

Variance of data in Y is too low for method of moments optimization.

$\mathbf{IFAIL}=11$

The sum of Y is zero within machine precision.

$\mathbf{IFAIL}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.8 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.7 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-999$

Dynamic memory allocation failed.

See Section 3.6 in the Essential Introduction for further information.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (g07bffe.f90)

10.2  Program Data

Program Data (g07bffe.d)

10.3  Program Results

Program Results (g07bffe.r)