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A Kolmogorov-Smirnov-Type Test for Dependently Double-Truncated Data

This paper proposes a Kolmogorov-Smirnov-type test for parametric distribution families of double-truncated lifespan data, utilizing copula models to account for dependence and demonstrating its effectiveness through simulations and an application to German enterprise lifespans that rejects the hypothesis of an age-homogeneous closure hazard.

Original authors: Anne-Marie Toparkus, Rafael Weissbach

Published 2026-07-22

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Anne-Marie Toparkus, Rafael Weissbach

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Mystery of the Missing Time

Imagine you are a detective trying to figure out how long a specific type of object usually lasts. Maybe it's a lightbulb, a smartphone, or even a business. In the world of statistics, this is called studying "lifespans." But there's a catch: you can't just wait for every single object to die to see how long it lived. You have to look at a snapshot of time.

This is where "truncation" comes in. Think of it like a camera that only takes photos of people who are already standing in a specific room. If you only photograph people who are currently in a waiting room, you miss everyone who hasn't arrived yet (left truncation) and everyone who has already left (right truncation). In real life, this happens all the time. For example, if you study businesses that closed down in 2014, you only see the ones that were already old enough to be around in 2014. You miss the ones that started in 2015, and you miss the ones that closed in 2010.

Usually, statisticians assume that when something started has nothing to do with how long it will last. They assume a business founded in 1990 is just as likely to close quickly as one founded in 2010. But in the real world, things are rarely that simple. Life expectancy changes over time, and the age at which you start something might actually influence how long it survives. This paper tackles the tricky problem of testing whether a lifespan follows a predictable pattern when the start time and the end time are secretly connected, like two dancers who are always moving in sync.

The Dance of the Double-Truncated Data

Anne-Marie Toparkus and Rafael Weißbach, statisticians from the University of Rostock, decided to build a new tool to solve this dance-floor mystery. They created a "Kolmogorov-Smirnov-Type Test," which is a fancy way of saying they built a super-sensitive ruler to measure how well a theoretical guess fits the messy reality of data.

Their main job was to test a specific guess: Do enterprise lifespans follow an "exponential distribution"? In plain English, an exponential distribution suggests that the risk of something closing down is constant. It's like a radioactive atom: it doesn't matter how old it is; the chance of it decaying in the next second is always the same. If this were true for businesses, it would mean a 20-year-old company is just as likely to go bust tomorrow as a 1-year-old company.

However, the authors suspected this wasn't the whole story. They knew that in the real world, the "age at truncation" (when the study starts) and the "lifespan" (how long the business lasts) are often dependent on each other. To model this connection, they used a mathematical tool called a "copula," specifically the Farlie-Gumbel-Morgenstern (FGM) copula. You can think of a copula as a glue that sticks two separate probability curves together, allowing them to wiggle in sync. The FGM copula is special because it can glue them together whether they are moving in the same direction or opposite directions.

The German Business Experiment

To put their new ruler to the test, the authors looked at a massive dataset: 55,279 German enterprises. These were companies founded between 1990 and 2013, and the study looked at which ones closed their doors between 2014 and 2016. Because the data only included companies that were alive in 2014 but closed by 2016, the data was "doubly truncated"—chopped off at both the beginning and the end.

The researchers ran their test twice. First, they assumed the start date and the lifespan were totally independent (the "easy" version). Then, they ran the test again using their FGM copula to account for the fact that older companies might behave differently than newer ones.

The Verdict: The Clock is Not Constant

The results were clear and decisive. When the authors compared their data against the "exponential distribution" (the idea that the risk of closing is constant over time), the test statistic was huge. In fact, it was so large that it far exceeded the critical values calculated through simulations.

The paper explicitly rejects the hypothesis that German enterprise lifespans follow an exponential distribution with a constant hazard rate. In other words, the data proves that the risk of a business closing is not the same at every age. The "age-homogeneous closure hazard" (the idea that age doesn't matter) was clearly wrong.

The authors also found that the dependence between the start date and the lifespan was real. They calculated a "Kendall's tau" of about 0.023, which sounds small, but the paper confirms it is statistically significant. This positive dependence suggests a negative time trend for life expectancy: as time went on, the expected lifespan of these businesses actually decreased.

How They Did It (The Magic Behind the Curtain)

You might wonder how they calculated this with such precision. The authors didn't just guess; they built a rigorous mathematical engine. They used a method called the "functional delta method" and "Donsker-class arguments." If that sounds like gibberish, think of it as a way to handle the fact that their data wasn't a perfect, smooth line but a jagged, bumpy mess of 55,000 individual points.

They developed an algorithm (Algorithm 1) that could crunch the numbers by checking specific "intersection points" and "projections" on a graph. Instead of checking every possible point in time (which would take forever), their clever math showed they only needed to check a finite number of spots to find the biggest difference between their guess and the reality.

To figure out if their results were actually significant or just a fluke, they had to simulate the "critical value." Since there was no simple formula for this complex, dependent data, they ran a computer simulation 1,000 times, generating random "Brownian bridges" (a type of random walk) to see what the test statistic should look like if the exponential guess were true. The real data was so far outside this simulated range that the rejection was undeniable.

The Takeaway

This paper doesn't just say "we found a pattern." It says, "We have mathematically proven that the simple idea of a constant risk of business failure is wrong for German enterprises." The authors suggest that the reason for this rejection is likely the changing nature of the business environment over time, rather than just the random distribution of when companies started.

While the paper focuses on German businesses, the tool they built is a powerful new way for statisticians to test theories about any kind of lifespan data where the start and end times are linked. It's a reminder that in the world of data, things are rarely independent, and sometimes, the best way to understand the future is to admit that the past and present are dancing together.

Technical Summary: A Kolmogorov-Smirnov-Type Test for Dependent Double-Truncated Data

Problem Statement
The paper addresses the challenge of testing the goodness-of-fit for a parametric distribution family of lifespans ( $X$ ) when data is subject to double truncation (both left and right) and where the truncation mechanism is dependent on the lifespan. In standard survival analysis, independence between the variable of interest and the truncation time is often assumed. However, in demographic and economic contexts (such as enterprise lifespans), life expectancy often exhibits non-stationary behavior, creating a dependence between the lifespan and the age at which the unit enters the study (truncation age, $T$ ). Existing literature largely covers nonparametric contributions for independent truncation or specific parametric models, but a rigorous nonparametric test for the distribution of $X$ under dependent double truncation was lacking.

Methodology
The authors propose a Kolmogorov-Smirnov (KS) type test statistic adapted for the bivariate setting of dependent double truncation.

Model Framework:
- The population consists of units born within a time window of length $G$ .
- Lifespan $X$ and truncation age $T$ are modeled with a joint distribution $F_\theta(x, t) = C_\vartheta(F^X_\theta(x), F_T(t))$ , where $C_\vartheta$ is a copula (specifically the Farlie-Gumbel-Morgenstern (FGM) copula in the main example) capturing the dependence structure via parameter $\vartheta$ .
- Observations are restricted to a parallelogram $D = \{(x, t) : 0 < t \le x \le t+s, t \le G\}$ , representing the study period $s$ following the birth period.
Test Statistic Construction:
- The test compares the empirical distribution of the observed truncated sample against the theoretical conditional distribution $F^{obs}_\theta$ .
- The statistic is defined as:
  $\sqrt{M_n} \frac{1}{\sqrt{\hat{\alpha}_{\hat{\theta}_n}}} \sup_{(x,t) \in S} \left| \hat{P}_{n,D}([0, x] \times [0, t]) - \hat{\alpha}_{\hat{\theta}_n} F^{obs}_{\hat{\theta}_n}(x, t) \right|$
  where $M_n$ is the random number of observed units, $\hat{\theta}_n$ is a consistent estimator for the parameters (including the dependence parameter), and $\hat{\alpha}$ is the estimated probability of observation.
- Computation: The authors derive an algorithm (Algorithm 1) that reduces the supremum calculation to a finite number of operations. The maximum deviation is evaluated at the observed points, intersection points of the empirical and theoretical boundaries, projections onto the edges of the support $D$ , and specific corners.
Asymptotic Theory:
- The paper utilizes Donsker-class arguments and the functional delta method to derive the asymptotic distribution of the test statistic under the null hypothesis ( $H_0$ ).
- The limit process is shown to be a Gaussian process $H_{P_{\theta_0}}$ , which is a modification of a Brownian bridge. This process accounts for the estimation of both the distribution parameters ( $\theta$ ) and the observation probability (which effectively handles the random sample size $M_n$ ).
- The covariance function of this limiting process is derived explicitly, allowing for the simulation of critical values.
Critical Value Determination:
- Since the asymptotic distribution depends on the unknown parameters, critical values are computed via simulation (Algorithm 2). This involves generating Gaussian vectors based on the derived covariance matrix over a grid of the support $D$ .

Key Contributions

Dependent Truncation Test: The paper provides the first KS-type goodness-of-fit test specifically designed for dependent double-truncated data, moving beyond the independence assumption prevalent in prior literature.
Algorithmic Implementation: It offers a computationally feasible algorithm to calculate the test statistic by identifying a finite set of points where the supremum occurs, adapting previous univariate and independent bivariate methods to the parallelogram support of double truncation.
Asymptotic Derivation: The authors rigorously derive the limiting distribution of the composite test statistic (where parameters are estimated), establishing the convergence to a specific Gaussian process and providing the necessary covariance structure for inference.
Application to Real Data: The methodology is applied to a large dataset of 55,279 German enterprise lifespans (founded 1990–2013, closed 2014–2016).

Results

Simulation and Application: The authors applied the test to German enterprise data to check if lifespans follow an exponential distribution.
- Under the assumption of independence (Product Copula), the test statistic was 1.651, and under dependence (FGM Copula), it was 1.6917.
- The simulated critical values at the 99% level were approximately 0.3558 (independence) and 0.2914 (dependence).
Rejection of Hypothesis: In both cases, the test statistic significantly exceeded the critical values, leading to a clear rejection of the exponential distribution hypothesis.
Interpretation: The authors note that while the rejection could theoretically stem from the assumption of a uniform distribution for the foundation dates (births), prior literature suggests that a homogeneous Poisson process for foundations yields a uniform $T$ . Therefore, the rejection is likely driven by the inappropriateness of the exponential distribution for enterprise lifespans, particularly given the observed negative time trend in life expectancy (positive dependence between age at study start and lifespan).

Significance and Claims
The paper claims that its primary contribution is enabling a rigorous goodness-of-fit test for parametric families in the presence of dependent double truncation, a scenario common in demography and economics but previously difficult to test nonparametrically.

Theoretical Modesty: The authors note that the theoretical requirements are minimal, primarily adding the finiteness of the influence function $\phi_{\theta_0}$ (Assumption A5) to standard parametric analysis assumptions.
Practical Utility: The application demonstrates that ignoring dependence (as done in previous studies of this dataset) can lead to misleading conclusions. The test successfully detects that the "age-homogeneous closure hazard" (exponential distribution) is rejected when dependence is properly modeled.
Limitations: The authors acknowledge that for small samples, the test's power and actual level would require further simulation study. They also note that the current computational method (Cholesky decomposition) has cubic complexity, though faster alternatives exist. The paper does not claim to solve all issues of truncation (e.g., censoring is not fully integrated) but focuses on the specific gap of dependent double truncation.