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.

ttt_plot(bladder_cancer)

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.