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

Computes the analytic 3x3 Hessian matrix (matrix of second partial derivatives) of the negative log-likelihood function for the Exponentiated Kumaraswamy (EKw) distribution with parameters alpha ($\alpha$), beta ($\beta$), and lambda ($\lambda$). This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where $\gamma = 1$ and $\delta = 0$. The Hessian is useful for estimating standard errors and in optimization algorithms.

Usage

hsekw(par, data)

Arguments

par: A numeric vector of length 3 containing the distribution parameters in the order: alpha ($\alpha > 0$), beta ($\beta > 0$), lambda ($\lambda > 0$).
data: A numeric vector of observations. All values must be strictly between 0 and 1 (exclusive).

Value

Returns a 3x3 numeric matrix representing the Hessian matrix of the negative log-likelihood function, $-\partial^2 \ell / (\partial \theta_i \partial \theta_j)$, where $\theta = (\alpha, \beta, \lambda)$. Returns a 3x3 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 EKw log-likelihood ($\gamma=1, \delta=0$ case of GKw, see llekw): $$ \ell(\theta | \mathbf{x}) = n[\ln(\lambda) + \ln(\alpha) + \ln(\beta)] + \sum_{i=1}^{n} [(\alpha-1)\ln(x_i) + (\beta-1)\ln(v_i) + (\lambda-1)\ln(w_i)] $$ where $\theta = (\alpha, \beta, \lambda)$ and intermediate terms are:

$v_i = 1 - x_i^{\alpha}$
$w_i = 1 - v_i^{\beta} = 1 - (1-x_i^{\alpha})^{\beta}$

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

Key properties of the returned matrix:

Dimensions: 3x3.
Symmetry: The matrix is symmetric.
Ordering: Rows and columns correspond to the parameters in the order $\alpha, \beta, \lambda$.
Content: Analytic second derivatives of the negative log-likelihood.

This corresponds to the relevant 3x3 submatrix of the 5x5 GKw Hessian (hsgkw) evaluated at $\gamma=1, \delta=0$. The exact analytical formulas are implemented directly.

References

Nadarajah, S., Cordeiro, G. M., & Ortega, E. M. (2012). The exponentiated Kumaraswamy distribution. Journal of the Franklin Institute, 349(3),

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

(Note: Specific Hessian formulas might be derived or sourced from additional references).

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rekw(1000, alpha = 2, beta = 3, lambda = 1.2)
par <- c(alpha = 2, beta = 3, lambda = 1.2)

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

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

## At the MLE it is the observed information; its inverse estimates the
## covariance of the estimates
fit <- optim(par, llekw, grekw, data = x, method = "L-BFGS-B", lower = 1e-4)
sqrt(diag(solve(hsekw(fit$par, x))))  # standard errors
#> [1] 0.4924578 0.3495889 0.3400468