Gradient of the Negative Log-Likelihood for the McDonald (Mc)/Beta Power Distribution

Computes the gradient vector (vector of first partial derivatives) of the negative log-likelihood function for the McDonald (Mc) distribution (also known as Beta Power) with parameters gamma ($\gamma$), delta ($\delta$), and lambda ($\lambda$). This distribution is the special case of the Generalized Kumaraswamy (GKw) distribution where $\alpha = 1$ and $\beta = 1$. The gradient is useful for optimization.

Usage

grmc(par, data)

Arguments

par: A numeric vector of length 3 containing the distribution parameters in the order: 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 numeric vector of length 3 containing the partial derivatives of the negative log-likelihood function $-\ell(\theta | \mathbf{x})$ with respect to each parameter: $(-\partial \ell/\partial \gamma, -\partial \ell/\partial \delta, -\partial \ell/\partial \lambda)$. Returns a vector of 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

The components of the gradient vector of the negative log-likelihood ($-\nabla \ell(\theta | \mathbf{x})$) for the Mc ($\alpha=1, \beta=1$) model are:

$$ -\frac{\partial \ell}{\partial \gamma} = n[\psi(\gamma) - \psi(\gamma+\delta+1)] - \lambda\sum_{i=1}^{n}\ln(x_i) $$ $$ -\frac{\partial \ell}{\partial \delta} = n[\psi(\delta+1) - \psi(\gamma+\delta+1)] - \sum_{i=1}^{n}\ln(1-x_i^{\lambda}) $$ $$ -\frac{\partial \ell}{\partial \lambda} = -\frac{n}{\lambda} - \gamma\sum_{i=1}^{n}\ln(x_i) + \delta\sum_{i=1}^{n}\frac{x_i^{\lambda}\ln(x_i)}{1-x_i^{\lambda}} $$

where $\psi(\cdot)$ is the digamma function (digamma). These formulas represent the derivatives of $-\ell(\theta)$, consistent with minimizing the negative log-likelihood. They correspond to the relevant components of the general GKw gradient (grgkw) evaluated at $\alpha=1, \beta=1$.

References

McDonald, J. B. (1984). Some generalized functions for the size distribution of income. Econometrica, 52(3), 647-663. doi:10.2307/1913469

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 gradient formulas might be derived or sourced from additional references).

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rmc(200, gamma = 0.5, delta = 5, lambda = 3)
par <- c(gamma = 0.5, delta = 5, lambda = 3)

## Gradient of the negative log-likelihood llmc(), not of the log-likelihood
g <- grmc(par, x)
g
#> [1]  4.494737  2.750941 -4.223447

## A small step against the gradient lowers llmc()
llmc(par - 1e-4 * g, x) < llmc(par, x)
#> [1] TRUE

## Agrees with a numerical derivative of llmc()
if (requireNamespace("numDeriv", quietly = TRUE))
  all.equal(g, numDeriv::grad(llmc, par, data = x), tolerance = 1e-6)
#> [1] TRUE