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Goodness-of-Fit Testing for Location-Scale
Distributions via Lorenz Curve
Shikhar Tyagi, Arvind Pandey, Bhupendra Singh,
Vrijesh Tripathi
Introduction
The gofLorenz package implements
goodness-of-fit (GoF) test statistics and graphical methods for
location-scale distributions under progressive Type-II censoring, based
on the research by Lee (2024). The testing approach
uses the modified Lorenz curve (\(mLC\)) and ratio modified sample Lorenz
curve (\(rLC\)) to assess how
effectively observed failure times align with a target theoretical
location-scale distribution.
In addition to Lee (2024)’s test statistics (\(L^+, L^-, L^{(1)}, L^{(2)}, L^{(3)},
L^{(4)}\)), the package computes order statistics distance test
statistics (\(C^+, C^-, C_m, K_m, T^{(1)},
T^{(2)}\)) proposed by Pakyari and Balakrishnan
(2013) for comparison.
Progressive Type-II Censoring Scheme
In progressive Type-II censoring, \(n\) units are placed on test. Upon
observing the 1st failure (\(X_{1:m:n}\)), \(R_1\) surviving units are randomly removed.
Upon observing the 2nd failure (\(X_{2:m:n}\)), \(R_2\) surviving units are randomly removed.
Finally, upon observing the \(m\)-th
failure (\(X_{m:m:n}\)), all remaining
\(R_m = n - m - \sum_{i=1}^{m-1} R_i\)
units are removed.
Example 1: Breaking Strength Data (Normal Distribution)
Consider the breaking strength data (\(n =
20\), \(m = 8\)) with
progressive censoring scheme \(R = (0, 4, 1,
3, 0, 2, 0, 2)\):
library(gofLorenz)
data(breaking_strength)
x1 <- breaking_strength$x
n1 <- breaking_strength$n
m1 <- breaking_strength$m
R1 <- breaking_strength$R
# Perform Goodness-of-Fit Test for Normal Distribution
fit1 <- gof_lorenz(x = x1, n = n1, m = m1, R = R1, dist = "norm", mc_rep = 200)
print(fit1)
#>
#> =========================================================
#> Goodness-of-Fit Test Based on Lorenz Curve (Lee, 2024)
#> =========================================================
#> Hypothesized Distribution : norm
#> Sample Size (n) : 20
#> Observed Failures (m) : 8
#> Censoring Scheme (R) : (0, 4, 1, 3, 0, 2, 0, 2)
#> Monte Carlo Replicates : 200
#> Confidence Level : 95.00%
#> ---------------------------------------------------------
#>
#> 1. Lorenz Curve Test Statistics (Lee, 2024):
#> Statistic Value p.value Crit.Value
#> L+ 0.00866 0.655 0.12590
#> L- 0.01021 0.600 0.10322
#> L(1) 0.01021 1.000 0.12685
#> L(2) 0.01887 1.000 0.13040
#> L(3) 0.00004 1.000 0.00584
#> L(4) 0.00487 1.000 0.06214
#>
#> 2. Order Statistics Distance Statistics (Pakyari & Balakrishnan, 2013):
#> Statistic Value p.value Crit.Value
#> C+ 0.07908 0.865 0.18491
#> C- 0.03699 0.975 0.13975
#> Cm 0.07908 0.905 0.18605
#> Km 0.11607 0.955 0.30502
#> T(1) 0.00148 0.955 0.00932
#> T(2) 0.03278 0.930 0.07681
#> =========================================================
Visual Diagnostic: L-plot
The lorenz_plot() function graphs \(L\text{-plot}(p_{j:m:n}) = |1 -
rLC(p_{j:m:n})|\) versus \(p_{j:m:n}\). Convergence near 0 supports
the hypothesized distribution.
Example 2: Insulating Fluid Data (Gumbel Distribution)
Consider log-transformed insulating fluid test data (\(n = 19\), \(m =
8\)) with scheme \(R = (0, 0, 3, 0, 3,
0, 0, 5)\):
data(insulating_fluid)
x2 <- insulating_fluid$x
n2 <- insulating_fluid$n
m2 <- insulating_fluid$m
R2 <- insulating_fluid$R
# Perform Goodness-of-Fit Test for Gumbel Distribution
fit2 <- gof_lorenz(x = x2, n = n2, m = m2, R = R2, dist = "gumbel", mc_rep = 200)
summary(fit2)
#>
#> =========================================================
#> Goodness-of-Fit Test Based on Lorenz Curve (Lee, 2024)
#> =========================================================
#> Hypothesized Distribution : gumbel
#> Sample Size (n) : 19
#> Observed Failures (m) : 8
#> Censoring Scheme (R) : (0, 0, 3, 0, 3, 0, 0, 5)
#> Monte Carlo Replicates : 200
#> Confidence Level : 95.00%
#> ---------------------------------------------------------
#>
#> 1. Lorenz Curve Test Statistics (Lee, 2024):
#> Statistic Value p.value Crit.Value
#> L+ 0.01812 0.515 0.14738
#> L- 0.01666 0.525 0.08108
#> L(1) 0.01812 0.975 0.14738
#> L(2) 0.03479 0.840 0.14738
#> L(3) 0.00018 0.895 0.00663
#> L(4) 0.01141 0.835 0.06294
#>
#> 2. Order Statistics Distance Statistics (Pakyari & Balakrishnan, 2013):
#> Statistic Value p.value Crit.Value
#> C+ 0.07234 0.505 0.13553
#> C- 0.06692 0.560 0.12636
#> Cm 0.07234 0.705 0.14830
#> Km 0.13925 0.595 0.22586
#> T(1) 0.00208 0.600 0.00556
#> T(2) 0.03936 0.565 0.06201
#> =========================================================
References
- Lee, K. (2024). A New Test Statistic to Assess the Goodness of Fit
of Location-Scale Distribution Based on Progressive Censored Data.
Symmetry, 16(2), 202. https://doi.org/10.3390/sym16020202
- Pakyari, R., & Balakrishnan, N. (2013). Goodness-of-fit tests
for progressively Type II censored data from location-scale
distribution. Journal of Statistical Computation and
Simulation, 83(1), 167–178. https://doi.org/10.1080/00949655.2011.625424
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