loglogistic6_gradient_2 {drda}

6-parameter log-logistic function gradient and Hessian

Description

Evaluate at a particular set of parameters the gradient and Hessian of the 6-parameter log-logistic function.

Usage

loglogistic6_gradient_2(x, theta)

loglogistic6_hessian_2(x, theta)

loglogistic6_gradient_hessian_2(x, theta)

Arguments

x

numeric vector at which the function is to be evaluated.

theta

numeric vector with the six parameters in the form c(alpha, delta, eta, phi, nu, xi).

Details

The 6-parameter log-logistic function ⁠f(x; theta)⁠ is defined here as

⁠g(x; theta) = (x^eta / (xi * x^eta + nu * phi^eta))^(1 / nu)⁠ ⁠f(x; theta) = alpha + delta g(x; theta)⁠

where x >= 0, theta = c(alpha, delta, eta, phi, nu, xi), eta > 0, phi > 0, nu > 0, and xi > 0.

This set of functions use a different parameterization from link[drda]{loglogistic6_gradient}. To avoid the non-negative constraints of parameters, the gradient and Hessian computed here are for the function with eta2 = log(eta), phi2 = log(phi), nu2 = log(nu), and xi2 = log(xi).

Note that argument theta is on the original scale and not on the log scale.

Note: The 6-parameter log-logistic function is over-parameterized and non-identifiable from data. It is available only for theoretical research.

Value

Gradient or Hessian of the alternative parameterization evaluated at the specified point.