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Getting Started with autorelevate
Introduction
The autorelevate package implements the autorelevated
family of probability distributions, obtained by applying the
autorelevation transform of Krakowski (1973) and Dileepkumar and
Sankaran (2022) to a baseline lifetime distribution. Ten baseline
distributions are supported: Weibull, Lomax, Burr XII, Gompertz,
Log-Logistic, Chen, Exponentiated Exponential, Power Lindley,
Log-normal, and Gamma. See ?autorelevate-package for the
transform’s derivation and full methodological references, and
citation("autorelevate") to cite the package itself.
A Real Dataset
The package bundles bladder_cancer, the remission times
(in months) of 128 bladder cancer patients (Lee & Wang, 2003), the
same dataset analyzed by Dileep Kumar, Shabeer, and Sankaran (2025, Sec.
8.1).
data(bladder_cancer)
summary(bladder_cancer)
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 0.080 3.348 6.395 9.366 11.838 79.050
Step 1: Diagnose the hazard shape with a TTT plot
Before fitting anything, a Total Time on Test (TTT) plot (Aarset,
1987) shows whether the hazard rate is likely increasing, decreasing,
bathtub, or upside-down bathtub (UBT), purely from the shape of the
empirical curve relative to the 45-degree line.
The curve is concave then convex, indicating a UBT-shaped hazard –
exactly the shape the Autorelevated Weibull was designed to capture (see
?haautorelevate for the theorem governing when the
Autorelevated Weibull hazard is increasing, decreasing, or UBT).
Step 2: Compare all ten baseline distributions
family_table <- compare_families(bladder_cancer)
print(family_table)
#>
#> Autorelevated Family Comparison Table (Ranked by AIC)
#> ----------------------------------------------------
#> Family p1 p2 LogLik AIC BIC CAIC HQIC KS_Stat
#> lomax 9.264e-02 3.709e+00 -410.9 825.7 831.4 825.8 828.0 0.03315
#> powerlindley 7.938e-01 6.252e-01 -411.1 826.3 832.0 826.4 828.6 0.05267
#> weibull 4.317e-01 7.250e-01 -411.4 826.8 832.5 826.9 829.1 0.05298
#> gamma 1.359e-01 5.439e-01 -413.9 831.8 837.5 831.9 834.1 0.07812
#> expexp 1.575e-01 5.279e-01 -415.1 834.1 839.8 834.2 836.4 0.08785
#> lognormal 6.287e-01 1.242e+00 -416.9 837.8 843.5 837.9 840.2 0.07087
#> chen 3.418e-01 2.981e-01 -419.2 842.4 848.1 842.5 844.7 0.09309
#> loglogistic 2.137e+00 1.487e+00 -420.4 844.8 850.5 844.9 847.1 0.06417
#> gompertz 4.414e+04 4.838e-06 -426.8 857.6 863.3 857.7 859.9 0.14139
#> burr 1.515e+00 7.020e-01 -431.9 867.8 873.5 867.9 870.1 0.16069
#> KS_Pval
#> 0.998963
#> 0.869647
#> 0.865056
#> 0.415419
#> 0.276668
#> 0.541129
#> 0.217228
#> 0.667571
#> 0.011976
#> 0.002693
Families are ranked by AIC; BIC, CAIC, and
HQIC are reported alongside for cross-checking, since they
penalize model complexity differently (though here all ten distributions
share the same two parameters, so the ranking is driven purely by
fit).
Step 3: Fit and diagnose the best distribution
best_dist <- family_table$Family[1]
fit_best <- fit_autorelevate(bladder_cancer, dist = best_dist, method = "mle")
summary(fit_best)
#>
#> Model Fit Summary (Distribution: lomax)
#> Estimation Method: MLE
#>
#> Estimate Std. Error
#> p1 0.092645 0.0326
#> p2 3.708918 0.9485
#>
#> --- Goodness-of-Fit & Model Selection ---
#> Log-Likelihood: -410.861
#> AIC: 825.722
#> BIC: 831.426
#> CAIC: 825.818
#> HQIC: 828.039
#> KS Statistic: 0.0331 (p-value: 0.9990)
plot(fit_best)
Comparing Estimation Methods
Beyond comparing baseline distributions, you can compare MLE against
Maximum Product of Spacings (MPS), Least Squares (LS), Weighted Least
Squares (WLS), and Cramer-von Mises (CvM) for a single distribution:
fit_all_methods(bladder_cancer, dist = best_dist)
#> Estimate_p1 Estimate_p2 KS_Statistic
#> MLE 0.09264470 3.708918 0.03314968
#> MPS 0.09264470 3.708918 0.03314968
#> LS 0.09224441 3.699603 0.02996296
#> WLS 0.09720036 3.555237 0.03065768
#> CVM 0.08642870 3.895910 0.02997489
MPS is a robust alternative to MLE when the likelihood is unbounded
or the density has a singularity; LS and WLS (Swain, Venkatraman, &
Wilson, 1988) are often preferred for small-to-moderate samples where
MLE can be unstable.
Working With Distributions Directly
All ten baseline distributions expose the same six functions:
dautorelevate(), pautorelevate(),
sautorelevate(), haautorelevate(),
qautorelevate(), and rautorelevate().
x <- seq(0.01, 5, by = 0.05)
plot(x, dautorelevate(x, dist = "weibull", p1 = 0.5, p2 = 1.5), type = "l",
ylab = "Density", main = "Autorelevated Weibull density")
plot(x, haautorelevate(x, dist = "weibull", p1 = 0.5, p2 = 0.8), type = "l",
ylab = "Hazard rate", main = "Upside-down bathtub hazard (beta = 0.8)")
References
Krakowski, M. (1973). The relevation transform and a generalization
of the gamma distribution function. Revue francaise d’automatique,
informatique, recherche operationnelle. Recherche operationnelle,
7(V2), 107-120.
Dileepkumar, M., & Sankaran, P. G. (2022). Some results of
auto-relevation transform in reliability analysis. Statistics and
Applications, 20(2), 251-263.
Dileep Kumar, M., Shabeer, A. M., & Sankaran, P. G. (2025).
Reliability properties and applications of autorelevated Weibull
distribution. American Journal of Mathematical and Management
Sciences, 44(3-4), 215-237.
Sharma, V. K., Pal, S., Bhardwaj, H., & Tyagi, V. (2026). The
autorelevated Lomax distribution: An upside-down bathtub hazard model
with properties and applications to cancer survival data. Submitted.
Aarset, M. V. (1987). How to identify a bathtub hazard rate. IEEE
Transactions on Reliability, R-36(1), 106-108.
Lee, E. T., & Wang, J. W. (2003). Statistical Methods for
Survival Data Analysis (3rd ed.). Wiley.
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.