Hessian Matrix of the Negative Log-Likelihood for the Beta Distribution (gamma, delta+1 Parameterization)

Computes the analytic 2x2 Hessian matrix (matrix of second partial derivatives) of the negative log-likelihood function for the standard Beta distribution, using a parameterization common in generalized distribution families. The distribution is parameterized by gamma ($\gamma$) and delta ($\delta$), corresponding to the standard Beta distribution with shape parameters shape1 = gamma and shape2 = delta + 1. The Hessian is useful for estimating standard errors and in optimization algorithms.

Usage

hsbeta(par, data)

Arguments

par: A numeric vector of length 2 containing the distribution parameters in the order: gamma ($\gamma > 0$), delta ($\delta \ge 0$).
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 $\theta = (\gamma, \delta)$. 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 ($-\ell(\theta|\mathbf{x})$) for a Beta distribution with parameters shape1 = gamma ($\gamma$) and shape2 = delta + 1 ($\delta+1$). The components of the Hessian matrix ($-\mathbf{H}(\theta)$) are:

$$ -\frac{\partial^2 \ell}{\partial \gamma^2} = n[\psi'(\gamma) - \psi'(\gamma+\delta+1)] $$ $$ -\frac{\partial^2 \ell}{\partial \gamma \partial \delta} = -n\psi'(\gamma+\delta+1) $$ $$ -\frac{\partial^2 \ell}{\partial \delta^2} = n[\psi'(\delta+1) - \psi'(\gamma+\delta+1)] $$

where $\psi'(\cdot)$ is the trigamma function (trigamma). These formulas represent the second derivatives of $-\ell(\theta)$, consistent with minimizing the negative log-likelihood. They correspond to the relevant 2x2 submatrix of the general GKw Hessian (hsgkw) evaluated at $\alpha=1, \beta=1, \lambda=1$. Note the parameterization difference from the standard Beta distribution (shape2 = delta + 1).

The returned matrix is symmetric.

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd ed.). Wiley.

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

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

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rbeta_(1000, gamma = 2, delta = 3)
par <- c(gamma = 2, delta = 3)

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

## Agrees with a numerical Hessian of llbeta()
if (requireNamespace("numDeriv", quietly = TRUE))
  all.equal(H, numDeriv::hessian(llbeta, 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, llbeta, grbeta, data = x, method = "L-BFGS-B", lower = 1e-4)
sqrt(diag(solve(hsbeta(fit$par, x))))  # standard errors
#> [1] 0.08495446 0.17768971