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# gplike

Generalized Pareto negative log-likelihood

## Syntax

nlogL = gplike(params,data)
[nlogL,ACOV] = gplike(params,data)

## Description

nlogL = gplike(params,data) returns the negative of the log-likelihood nlogL for the two-parameter generalized Pareto (GP) distribution, evaluated at parameters params. params(1) is the tail index (shape) parameter, K, params(2) is the scale parameter, sigma, and params(3) is the threshold (location) parameter, mu.

[nlogL,ACOV] = gplike(params,data) returns the inverse of Fisher's information matrix, ACOV. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of ACOV are their asymptotic variances. ACOV is based on the observed Fisher's information, not the expected information.

When K = 0 and theta = 0, the GP is equivalent to the exponential distribution. When K > 0 and theta = sigma/K, the GP is equivalent to the Pareto distribution. The mean of the GP is not finite when K ≥ 1, and the variance is not finite when K1/2. When K0, the GP has positive density for

X > theta, or, when

K < 0, $0\le \text{\hspace{0.17em}}\frac{x-\theta }{\sigma }\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}-\frac{1}{k}$.