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Fit Gompertz Distribution from Mode and Probability Interval

Source: R/fit_gompertz_from_ci.R
fit_gompertz_from_ci.Rd

This function estimates the parameters of a Gompertz distribution on [0, Inf) with pdf f(x; b, eta) = b * eta * exp(bx) * exp(-eta(exp(b*x) - 1)) so that its quantiles at probs match a target interval (default: the central 95 percent).

Usage

fit_gompertz_from_ci(
  mode_val,
  ci_lower,
  ci_upper,
  probs = c(0.025, 0.975),
  verbose = FALSE
)

Arguments

mode_val

Numeric >= 0. Reference mode, reported next to the fitted mode; it does not constrain the fit and need not lie inside the interval.

ci_lower

Numeric greater than or equal to 0. Lower bound of the target interval (e.g., 2.5 percent quantile).

ci_upper

Numeric greater than ci_lower. Upper bound of the target interval (e.g., 97.5 percent quantile).

probs

Numeric length-2 vector in (0, 1). Probability levels for the target bounds. Defaults to c(0.025, 0.975).

verbose

Logical. If TRUE, prints a diagnostic summary.

Value

A list containing:

  • b: Gompertz shape parameter

  • eta: Gompertz rate parameter

  • f0: Density at zero (finite and positive)

  • fitted_mode: The mode of the fitted density, -log(eta)/b (0 when eta >= 1)

  • fitted_ci: Named vector of fitted quantiles at probs

  • fitted_mean: Numerical estimate of the expected value via quadrature

  • fitted_sd: Numerical estimate of the standard deviation via quadrature

  • probs: The probability levels used

  • input_mode: Echo of mode_val

  • input_ci: Echo of c(lower = ci_lower, upper = ci_upper)

Details

The two quantiles determine the distribution: the ratio \(Q(p_2)/Q(p_1) = \log(1 + c_2/\eta) / \log(1 + c_1/\eta)\), with \(c_k = -\log(1 - p_k)\), depends on \(\eta\) alone and increases monotonically from 1 (\(\eta \to 0\)) to \(c_2/c_1\) (\(\eta \to \infty\), the exponential limit; about 146 for the central 95 percent), so \(\eta\) is solved from the target ratio and \(b\) from the scale. A ratio beyond that limit (or ci_lower = 0) is matched as closely as the family allows, anchored on ci_upper.

Setting the derivative of log f to zero gives the mode x* = -log(eta) / b, which is interior only when eta < 1; for eta >= 1 the density is monotone decreasing and the mode is 0. mode_val does not constrain the fit (a sample-based mode near zero is poorly determined) and may lie outside the interval, as it does for a monotone-decreasing target whose KDE mode falls below the lower quantile; the mode of the fitted density is returned as fitted_mode.

Examples

# Example: Fit Gompertz for small positive quantity
result <- fit_gompertz_from_ci(
  mode_val = 1e-8,
  ci_lower = 1e-9,
  ci_upper = 1e-6,
  probs = c(0.025, 0.975)
)
print(result)
#> $b
#> [1] 0.03688879
#> 
#> $eta
#> [1] 1e+08
#> 
#> $f0
#> [1] 3688879
#> 
#> $fitted_mode
#> [1] 0
#> 
#> $fitted_ci
#>        lower        upper 
#> 6.863279e-09 1.000000e-06 
#> 
#> $fitted_mean
#> [1] 2.71081e-07
#> 
#> $fitted_sd
#> [1] 2.710592e-07
#> 
#> $probs
#> [1] 0.025 0.975
#> 
#> $input_mode
#> [1] 1e-08
#> 
#> $input_ci
#> lower upper 
#> 1e-09 1e-06 
#>