Pareto distribution log-likelihood — loglik

Computes the log-likelihood for i.i.d. Pareto observations. L(α, xₘ | x) = nlog(α) + nα*log(xₘ) - (α+1)*Σlog(xᵢ)

Usage

loglik_pareto(alpha, x_min, x)

Arguments

alpha: Shape parameter α (value object), must be positive
x_min: Minimum/scale parameter xₘ (fixed positive number). All observations must be >= x_min.
x: Numeric vector of observations (must be >= x_min)

Value

A value object representing the log-likelihood

Details

The Pareto distribution is used to model heavy-tailed phenomena like income distributions, city sizes, etc. Here x_min is typically known (e.g., min(x)) and alpha is estimated.

Examples

if (FALSE) { # \dontrun{
# Generate Pareto data
alpha_true <- 2
x_min <- 1
u <- runif(100)
x <- x_min * (1 - u)^(-1/alpha_true)

# Fit (alpha only, x_min = min(x) is fixed)
result <- fit(
  function(log_alpha) {
    alpha <- exp(log_alpha)
    loglik_pareto(alpha, x_min = min(x), x)
  },
  params = c(log_alpha = 0)
)
} # }