g07 Chapter Contents
g07 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_estim_gen_pareto (g07bfc)

## 1  Purpose

nag_estim_gen_pareto (g07bfc) estimates parameter values for the generalized Pareto distribution by using either moments or maximum likelihood.

## 2  Specification

 #include #include
 void nag_estim_gen_pareto (Integer n, const double y[], Nag_OptimOpt optopt, double *xi, double *beta, double asvc[], double obsvc[], double *ll, NagError *fail)

## 3  Description

Let the distribution function of a set of $n$ observations
 $yi , i=1,2,…,n$
be given by the generalized Pareto distribution:
 $Fy = 1- 1+ ξy β -1/ξ , ξ≠0 1-e-yβ , ξ=0;$
where
• $\beta >0$ and
• $y\ge 0$, when $\xi \ge 0$;
• $0\le y\le -\frac{\beta }{\xi }$, when $\xi <0$.
Estimates $\stackrel{^}{\xi }$ and $\stackrel{^}{\beta }$ of the parameters $\xi$ and $\beta$ are calculated by using one of:
• method of moments (MOM);
• probability-weighted moments (PWM);
• maximum likelihood estimates (MLE) that seek to maximise the log-likelihood:
 $L = -n ln⁡ β^ - 1+ 1ξ^ ∑ i=1 n ln 1+ ξ^yi β^ .$
The variances and covariance of the asymptotic Normal distribution of parameter estimates $\stackrel{^}{\xi }$ and $\stackrel{^}{\beta }$ are returned if $\stackrel{^}{\xi }$ satisfies:
• $\stackrel{^}{\xi }<\frac{1}{4}$ for the MOM;
• $\stackrel{^}{\xi }<\frac{1}{2}$ for the PWM method;
• $\stackrel{^}{\xi }<-\frac{1}{2}$ for the MLE method.
If the MLE option is exercised, the observed variances and covariance of $\stackrel{^}{\xi }$ and $\stackrel{^}{\beta }$ is returned, given by the negative inverse Hessian of $L$.

## 4  References

Hosking J R M and Wallis J R (1987) Parameter and quantile estimation for the generalized Pareto distribution Technometrics 29(3)
McNeil A J, Frey R and Embrechts P (2005) Quantitative Risk Management Princeton University Press

## 5  Arguments

1:     nIntegerInput
On entry: the number of observations.
Constraint: ${\mathbf{n}}>1$.
2:     y[n]const doubleInput
On entry: the $n$ observations ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, assumed to follow a generalized Pareto distribution.
Constraints:
• ${\mathbf{y}}\left[i-1\right]\ge 0.0$;
• $\sum _{\mathit{i}=1}^{n}{\mathbf{y}}\left[i-1\right]>0.0$.
3:     optoptNag_OptimOptInput
On entry: determines the method of estimation, set:
${\mathbf{optopt}}=\mathrm{Nag_PWM}$
For the method of probability-weighted moments.
${\mathbf{optopt}}=\mathrm{Nag_MOM}$
For the method of moments.
${\mathbf{optopt}}=\mathrm{Nag_MOMMLE}$
For maximum likelihood with starting values given by the method of moments estimates.
${\mathbf{optopt}}=\mathrm{Nag_PWMMLE}$
For maximum likelihood with starting values given by the method of probability-weighted moments.
Constraint: ${\mathbf{optopt}}=\mathrm{Nag_PWM}$, $\mathrm{Nag_MOM}$, $\mathrm{Nag_MOMMLE}$ or $\mathrm{Nag_PWMMLE}$.
4:     xidouble *Output
On exit: the parameter estimate $\stackrel{^}{\xi }$.
On exit: the parameter estimate $\stackrel{^}{\beta }$.
6:     asvc[$4$]doubleOutput
On exit: the variance-covariance of the asymptotic Normal distribution of $\stackrel{^}{\xi }$ and $\stackrel{^}{\beta }$. ${\mathbf{asvc}}\left[0\right]$ contains the variance of $\stackrel{^}{\xi }$; ${\mathbf{asvc}}\left[3\right]$ contains the variance of $\stackrel{^}{\beta }$; ${\mathbf{asvc}}\left[1\right]$ and ${\mathbf{asvc}}\left[2\right]$ contain the covariance of $\stackrel{^}{\xi }$ and $\stackrel{^}{\beta }$.
7:     obsvc[$4$]doubleOutput
On exit: if maximum likelihood estimates are requested, the observed variance-covariance of $\stackrel{^}{\xi }$ and $\stackrel{^}{\beta }$. ${\mathbf{obsvc}}\left[0\right]$ contains the variance of $\stackrel{^}{\xi }$; ${\mathbf{obsvc}}\left[3\right]$ contains the variance of $\stackrel{^}{\beta }$; ${\mathbf{obsvc}}\left[1\right]$ and ${\mathbf{obsvc}}\left[2\right]$ contain the covariance of $\stackrel{^}{\xi }$ and $\stackrel{^}{\beta }$.
8:     lldouble *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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>1$.
NE_INTERNAL_ERROR
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.
NE_OPTIMIZE
The maximum likelihood optimization failed; try a different starting point by selecting the other maximum likelihood estimation option in argument optopt.
Variance of data in y is too low for method of moments optimization.
NE_REAL_ARRAY
On entry, at least one ${\mathbf{y}}\left[i-1\right]\le 0.0$: $i=〈\mathit{\text{value}}〉$, ${\mathbf{y}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
NE_ZERO_SUM
The sum of y is zero within machine precision.
NW_PARAM_DIST
The distribution of maximum likelihood estimates cannot be calculated and the asymptotic distribution is not available for the returned parameter estimates.
NW_PARAM_DIST_ASYM
The asymptotic distribution is not available for the returned parameter estimates.
NW_PARAM_DIST_OBS
The distribution of maximum likelihood estimates cannot be calculated for the returned parameter estimates because the Hessian matrix could not be inverted.

## 7  Accuracy

Not applicable.

The search for maximum likelihood parameter estimates is further restricted by requiring
 $1+ ξ^yi β^ > 0 ,$
as this avoids the possibility of making the log-likelihood $L$ arbitrarily high.

## 9  Example

This example calculates parameter estimates for $23$ observations assumed to be drawn from a generalized Pareto distribution.

### 9.1  Program Text

Program Text (g07bfce.c)

### 9.2  Program Data

Program Data (g07bfce.d)

### 9.3  Program Results

Program Results (g07bfce.r)