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

Computes the analytic 2x2 Hessian matrix (matrix of second partial derivatives) of the negative log-likelihood function for the two-parameter Kumaraswamy (Kw) distribution with parameters alpha ($\alpha$) and beta ($\beta$). The Hessian is useful for estimating standard errors and in optimization algorithms.

Usage

hskw(par, data)

Arguments

par: A numeric vector of length 2 containing the distribution parameters in the order: alpha ($\alpha > 0$), beta ($\beta > 0$).
data: A numeric vector of observations. All values must be strictly between 0 and 1 (exclusive).

Value

Returns a 2x2 numeric matrix representing the Hessian matrix of the negative log-likelihood function, $-\partial^2 \ell / (\partial \theta_i \partial \theta_j)$, where $\theta = (\alpha, \beta)$. Returns a 2x2 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 ($-\ell(\theta|\mathbf{x})$). The components are the negative of the second derivatives of the log-likelihood $\ell$ (derived from the PDF in dkw).

Let $v_i = 1 - x_i^{\alpha}$. The second derivatives of the positive log-likelihood ($\ell$) are: $$ \frac{\partial^2 \ell}{\partial \alpha^2} = -\frac{n}{\alpha^2} - (\beta-1)\sum_{i=1}^{n}\frac{x_i^{\alpha}(\ln(x_i))^2}{v_i^2} $$ $$ \frac{\partial^2 \ell}{\partial \alpha \partial \beta} = - \sum_{i=1}^{n}\frac{x_i^{\alpha}\ln(x_i)}{v_i} $$ $$ \frac{\partial^2 \ell}{\partial \beta^2} = -\frac{n}{\beta^2} $$ The function returns the Hessian matrix containing the negative of these values.

Key properties of the returned matrix:

Dimensions: 2x2.
Symmetry: The matrix is symmetric.
Ordering: Rows and columns correspond to the parameters in the order $\alpha, \beta$.
Content: Analytic second derivatives of the negative log-likelihood.

This corresponds to the relevant 2x2 submatrix of the 5x5 GKw Hessian (hsgkw) evaluated at $\gamma=1, \delta=0, \lambda=1$.

References

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

Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), 70-81. doi:10.1016/j.stamet.2008.04.001

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

Author

Lopes, J. E.

Examples

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

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

## Agrees with a numerical Hessian of llkw()
if (requireNamespace("numDeriv", quietly = TRUE))
  all.equal(H, numDeriv::hessian(llkw, 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, llkw, grkw, data = x, method = "L-BFGS-B", lower = 1e-4)
sqrt(diag(solve(hskw(fit$par, x))))  # standard errors
#> [1] 0.06543864 0.15878554