Goodness-of-Fit Tests and Diagrams Based on Mellin Log-Cumulants

A family of three complementary goodness-of-fit tests based on an adaptation of Hotelling's T-squared statistic applied to vectors of sample log-cumulants (Mellin statistics) for positive-support reliability data. The package provides the asymptotic chi-squared reference and parametric bootstrap p-values for reliable finite-sample inference, covering the Weibull, Frechet, Gamma, Inverse-Gamma, Log-Normal, and Log-Logistic families. It also provides three diagnostic diagrams (log-cumulant, kurtosis-skewness, and coefficient-of-variation) with bootstrap concentration ellipses, in the spirit of moment-ratio diagrams. Methods are described in Santos, Ospina, Espinheira and Oliveira (2025).

logcumulant

logcumulant provides goodness-of-fit tests and diagnostic diagrams for positive-support reliability data, built on Mellin log-cumulants. It implements a family of three complementary Hotelling-type \eqn{T^2} statistics together with three moment-ratio-style diagrams, all calibrated by a parametric bootstrap for reliable finite-sample inference.

The package is built around a few design choices:

Three nested tests. T2_(2,3) targets log-scale shape (dispersion and skewness); T2_(1,2,3) adds the entropy-related first log-cumulant; and T2_(1,...,6) exploits higher-order log-cumulants for tail discrimination.
Bootstrap-first inference. The discrepancy covariance is typically ill-conditioned, so the asymptotic chi-squared reference is unreliable in finite samples; the parametric bootstrap restores correct size.
Diagnostic diagrams. Log-cumulant, kurtosis-skewness, and coefficient-of-variation diagrams, each overlaying the theoretical loci of six reliability families with a bootstrap cloud and a 95% concentration ellipse.
Six reliability families. Weibull, Frechet, Gamma, Inverse-Gamma, Log-Normal, and Log-Logistic, all fitted by maximum likelihood with a fast C++ core.

Installation

The package depends on a small set of CRAN packages. Install them first if needed:

install.packages(c("Rcpp", "RcppArmadillo", "MASS", "VGAM", "actuar",
                   "numDeriv", "goftest", "ggplot2", "gridExtra"))

Then install the development version from GitHub:

# install.packages("remotes")
remotes::install_github("raydonal/logcumulant")

remotes (or devtools) installs the required dependencies automatically.

A 30-second tour

library(logcumulant)
data(reliability_datasets)

bb <- reliability_datasets$BallBearing

# Quick log-cumulant diagram with bootstrap cloud
plot_lc(bb, B = 100)

# The three diagnostic diagrams
three_diagrams(bb, "Ball Bearing")

# Compare all six families (T2, AD, CvM, AIC with bootstrap p-values)
gof_compare_all(bb, use_bootstrap = TRUE)

When to use which tool

Goal	Function
Quick log-cumulant diagram	`plot_lc()`
Log-cumulant diagram (full control)	`log_cumulant_diagram()`
Kurtosis-skewness diagram	`kurtosis_diagram()`
Coefficient-of-variation diagram	`cv_diagram()`
All three diagrams together	`three_diagrams()`
Several datasets on one diagram	`multi_lc_diagram()`
The three T-squared statistics	`T2_all()`
Bootstrap p-values	`T2_bootstrap()`
Full model comparison	`gof_compare_all()`
Size / power simulation	`size_study()`, `power_study()`