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 maximize 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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_OPTIMIZE: 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.
NE_REAL_ARRAY: On entry, $y [〈value〉] = 〈value〉$ .
Constraint: $y [i - 1] \geq 0.0$ for all $i$ .
NE_ZERO_SUM: The sum of y is zero within machine precision.
NW_PARAM_DIST: 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.
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.

nag_estim_gen_pareto (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.

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 Document

nag_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

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