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Kolmogorov-Smirnov Goodness-of-Fit Test for Dependently Double-Truncated Durations

Shikhar Tyagi, Arvind Pandey, Bhupendra Singh, Vrijesh Tripathi

2026-07-31

Introduction

The DepDoubleTruncKS package provides methods for performing the two-dimensional Kolmogorov-Smirnov-type goodness-of-fit test for exponentially distributed durations subject to double truncation (left and right truncation), accounting for stochastic dependence between duration and truncation age via copulas.

This methodology was developed by Toparkus & Weißbach (2026) in Lifetime Data Analysis.

Method Overview

Given double-truncated observations \((X_j^{\text{obs}}, T_j^{\text{obs}})_{j=1}^{m_n}\) falling inside the truncation parallelogram: \[D = \{ (x, t)^T \mid 0 < t \le x \le t + s, t \le G \}\]

  1. Parameter Estimation: Profile maximum likelihood / Z-estimation estimates rate \(\hat{\theta}_n\) and copula parameter \(\hat{\vartheta}_n\).
  2. 2D KS Test Statistic: Evaluated over observations, boundary projections, and discordant intersection points (Algorithm 1).
  3. Asymptotic Limit Process: Critical values and p-values are obtained via Gaussian process field simulation (Algorithm 2).

Quick Start Example

set.seed(2026)

# Load sample dataset
data("enterprise_data")

# Perform KS test under FGM copula dependent truncation
res <- ks_dep_trunc(
  x = enterprise_data$x[1:100],
  t = enterprise_data$t[1:100],
  s = 3,
  G = 24,
  model = "fgm",
  grid_dim = 15,
  n_sim = 100
)

# Print results summary
print(res)
#> 
#> =========================================================
#>   2D Kolmogorov-Smirnov Test for Double-Truncated Data
#> =========================================================
#> 
#> Model Family:           Exponential Lifespan (Exp(theta))
#> Truncation Dependence:  FGM Copula 
#> Study Duration (s):      3 
#> Max Age Bound (G):       24 
#> Observed Sample (m_n):   100 
#> Est. Latent Sample (n):  1028.7 
#> 
#> --- Parameter Estimates ---
#> Rate (theta_hat):        0.09039 
#> Dependence (vtheta_hat): -0.15656 
#> Obs. Prob. (alpha_hat):  0.09721 
#> 
#> --- Goodness-of-Fit Test Results ---
#> KS Test Statistic (D_mn): 1.15837 
#> P-value:                 0 
#> 
#> Critical Values (Algorithm 2):
#>   10% (alpha = 0.10):     0.80327 
#>   5%  (alpha = 0.05):     0.85279 
#>   1%  (alpha = 0.01):     0.99565 
#> 
#> Decision (alpha = 0.05): Reject H0: Parametric model assumption rejected. 
#> =========================================================

# Plot observations and truncation boundaries
plot(res)

Reference

Toparkus, A.-M. and Weißbach, R. (2026). Kolmogorov-Smirnov-type test for dependently double-truncated durations: A copula approach. Lifetime Data Analysis, 32, 41. doi:10.1007/s10985-026-09722-0.

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.