logistic6_gradient_2 {drda}

6-parameter logistic function gradient and Hessian

Description

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

Usage

logistic6_gradient_2(x, theta)

logistic6_hessian_2(x, theta)

logistic6_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 logistic function ⁠f(x; theta)⁠ is defined here as

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

where theta = c(alpha, delta, eta, phi, nu, xi), eta > 0, nu > 0, and xi > 0. When delta is positive (negative) the curve is monotonically increasing (decreasing).

This set of functions use a different parameterization from link[drda]{logistic6_gradient}. To avoid the non-negative constraints of parameters, the gradient and Hessian computed here are for the function with eta2 = log(eta), 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 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.