NAG Library Function Document

nag_estim_gen_pareto (g07bfc)

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

y_{i}, i = 1, 2, \dots, n

be given by the generalized Pareto distribution:

F (y) = \{\begin{matrix} 1 - {(1 + \frac{ξ y}{β})}^{- 1 / ξ}, & ξ \neq 0 \\ 1 - e^{- \frac{y}{β}}, & ξ = 0; \end{matrix}

where

$β > 0$ and
$y \geq 0$ , when $ξ \geq 0$ ;
$0 \leq y \leq - \frac{β}{ξ}$ , when $ξ < 0$ .

Estimates

\hat{ξ}

and

\hat{β}

of the parameters

ξ

and

β

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:
$L = - n \ln \hat{β} - (1 + \frac{1}{\hat{ξ}}) \sum_{i = 1}^{n} \ln (1 + \frac{\hat{ξ} y_{i}}{\hat{β}}) .$

The variances and covariance of the asymptotic Normal distribution of parameter estimates

\hat{ξ}

and

\hat{β}

are returned if

\hat{ξ}

satisfies:

$\hat{ξ} < \frac{1}{4}$ for the MOM;
$\hat{ξ} < \frac{1}{2}$ for the PWM method;
$\hat{ξ} < - \frac{1}{2}$ for the MLE method.

If the MLE option is exercised, the observed variances and covariance of

\hat{ξ}

and

\hat{β}

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: n – IntegerInput

On entry: the number of observations.

Constraint:

n > 1

2: y[n] – const doubleInput

On entry: the

n

observations

y_{i}

, for

i = 1, 2, \dots, n

, assumed to follow a generalized Pareto distribution.

Constraints:

$y [i - 1] \geq 0.0$ ;
$\sum_{i = 1}^{n} y [i - 1] > 0.0$ .

3: optopt – Nag_OptimOptInput

On entry: determines the method of estimation, set:

$optopt = Nag_PWM$: For the method of probability-weighted moments.
$optopt = Nag_MOM$: For the method of moments.
$optopt = Nag_MOMMLE$: For maximum likelihood with starting values given by the method of moments estimates.
$optopt = Nag_PWMMLE$: For maximum likelihood with starting values given by the method of probability-weighted moments.

Constraint:

optopt = Nag_PWM

Nag_MOM

Nag_MOMMLE

Nag_PWMMLE

4: xi – double *Output

On exit: the parameter estimate

\hat{ξ}

5: beta – double *Output

On exit: the parameter estimate

\hat{β}

6: asvc[ $4$ ] – doubleOutput

On exit: the variance-covariance of the asymptotic Normal distribution of

\hat{ξ}

and

\hat{β}

asvc [0]

contains the variance of

\hat{ξ}

;

asvc [3]

contains the variance of

\hat{β}

;

asvc [1]

and

asvc [2]

contain the covariance of

\hat{ξ}

and

\hat{β}

7: obsvc[ $4$ ] – doubleOutput

On exit: if maximum likelihood estimates are requested, the observed variance-covariance of

\hat{ξ}

and

\hat{β}

obsvc [0]

contains the variance of

\hat{ξ}

;

obsvc [3]

contains the variance of

\hat{β}

;

obsvc [1]

and

obsvc [2]

contain the covariance of

\hat{ξ}

and

\hat{β}

8: ll – double *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: fail – NagError *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.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n > 1$ .
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 $y [i - 1] \leq 0.0$ : $i = ⟨value⟩$ , $y [i - 1] = ⟨value⟩$ .
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.

8 Parallelism and Performance

nag_estim_gen_pareto (g07bfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The search for maximum likelihood parameter estimates is further restricted by requiring

1 + \frac{\hat{ξ} y_{i}}{\hat{β}} > 0,

as this avoids the possibility of making the log-likelihood

L

arbitrarily high.

10 Example

This example calculates parameter estimates for

23

observations assumed to be drawn from a generalized Pareto distribution.

NAG Library Function Documentnag_estim_gen_pareto (g07bfc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_estim_gen_pareto (g07bfc)