Hessian Matrix of the Negative Log-Likelihood for the GKw Distribution

Computes the analytic Hessian matrix (matrix of second partial derivatives) of the negative log-likelihood function for the five-parameter Generalized Kumaraswamy (GKw) distribution. This is typically used to estimate standard errors of maximum likelihood estimates or in optimization algorithms.

Usage

hsgkw(par, data)

Arguments

par: A numeric vector of length 5 containing the distribution parameters in the order: alpha ($\alpha > 0$), beta ($\beta > 0$), gamma ($\gamma > 0$), delta ($\delta \ge 0$), lambda ($\lambda > 0$).
data: A numeric vector of observations. All values must be strictly between 0 and 1 (exclusive).

Value

Returns a 5x5 numeric matrix representing the Hessian matrix of the negative log-likelihood function, i.e., the matrix of second partial derivatives $-\partial^2 \ell / (\partial \theta_i \partial \theta_j)$. Returns a 5x5 matrix populated with NaN if any parameter values are invalid according to their constraints, or if any value in data is not in the interval (0, 1).

Details

This function calculates the analytic second partial derivatives of the negative log-likelihood function based on the GKw PDF (see dgkw). The log-likelihood function $\ell(\theta | \mathbf{x})$ is given by: $$ \ell(\theta) = n \ln(\lambda\alpha\beta) - n \ln B(\gamma, \delta+1) + \sum_{i=1}^{n} [(\alpha-1) \ln(x_i) + (\beta-1) \ln(v_i) + (\gamma\lambda - 1) \ln(w_i) + \delta \ln(z_i)] $$ where $\theta = (\alpha, \beta, \gamma, \delta, \lambda)$, $B(a,b)$ is the Beta function (beta), and intermediate terms are:

$v_i = 1 - x_i^{\alpha}$
$w_i = 1 - v_i^{\beta} = 1 - (1-x_i^{\alpha})^{\beta}$
$z_i = 1 - w_i^{\lambda} = 1 - [1-(1-x_i^{\alpha})^{\beta}]^{\lambda}$

The Hessian matrix returned contains the elements $- \frac{\partial^2 \ell(\theta | \mathbf{x})}{\partial \theta_i \partial \theta_j}$.

Key properties of the returned matrix:

Dimensions: 5x5.
Symmetry: The matrix is symmetric.
Ordering: Rows and columns correspond to the parameters in the order $\alpha, \beta, \gamma, \delta, \lambda$.
Content: Analytic second derivatives of the negative log-likelihood.

The exact analytical formulas for the second derivatives are implemented directly (often derived using symbolic differentiation) for accuracy and efficiency, typically using C++.

References

Carrasco, J. M. F., Ferrari, S. L. P., & Cordeiro, G. M. (2010). A new generalized Kumaraswamy distribution. arXiv preprint arXiv:1004.0911. doi:10.48550/arXiv.1004.0911

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81(7), 883-898. doi:10.1080/00949650903530745

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88. doi:10.1016/0022-1694(80)90036-0

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rgkw(1000, alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, lambda = 1.2)
par <- c(alpha = 2, beta = 3, gamma = 1.5, delta = 0.5, lambda = 1.2)

## Hessian of the negative log-likelihood llgkw()
H <- hsgkw(par, x)
isSymmetric(H)
#> [1] TRUE

## Agrees with a numerical Hessian of llgkw()
if (requireNamespace("numDeriv", quietly = TRUE))
  all.equal(H, numDeriv::hessian(llgkw, par, data = x), tolerance = 1e-8)
#> [1] TRUE