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On the Discrete Past Geometric Vitality Function and Its Properties

This paper introduces the Discrete Past Geometric Vitality Function (PGVF) as a multiplicative analogue of the classical past vitality function for discrete lifetimes, establishing its theoretical properties, proving its ability to uniquely characterize distributions, and proposing a nonparametric estimator validated through simulations and real-data applications.

Original authors: Sulfi S. A., Abdul Sathar E. I.

Published 2026-10-01

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CC BY 4.0

Original authors: Sulfi S. A., Abdul Sathar E. I.

Original paper licensed under CC BY 4.0 (https://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

In the world of reliability and survival analysis, scientists often ask how long a machine, a biological organism, or a system will last. Traditionally, the focus has been on the future: if a component has survived up to a certain point, how much longer is it expected to live? This forward-looking view is useful for planning and maintenance. However, there is another, equally important perspective that looks backward. When a system fails, or when we analyze a group of items that have already stopped working by a specific time, we are left with a collection of "past lives." Understanding the patterns within these truncated histories helps researchers understand how systems age, how they deteriorate, and whether they fail early or late. For decades, statisticians have measured these past lifetimes using simple averages, adding up the durations and dividing by the count to find a typical value. This method works well for many situations, but it treats every unit of time as a straight line, adding them up in a way that might miss subtle, proportional shifts in how things wear out.

A team of researchers from the University of Kerala has introduced a new way to look at these past histories, specifically for data that comes in whole numbers, like the count of days a machine ran or the number of cycles a part completed. They developed a tool called the Discrete Past Geometric Vitality Function. Instead of adding up the lifetimes to find an average, this new method multiplies them together in a specific mathematical way to find a "geometric mean." Think of it as shifting from counting steps to measuring the volume of space covered; it captures a different kind of central tendency that is less swayed by extreme outliers. While the traditional average might be dragged upward by a single, unusually long-lasting item, this new geometric measure stays more stable, offering a clearer picture of the typical behavior of the group that failed. The researchers proved that this new function is not just a different way of calculating a number; it is a unique fingerprint. If you know this geometric measure for every possible point in time, you can mathematically reconstruct the entire original distribution of lifetimes, meaning no information is lost.

The study also explored how this new measure behaves compared to the old one. The researchers showed that for systems where lifetimes are skewed—where a few items last much longer than the rest—the traditional average tends to inflate, making the group look more robust than it really is. The geometric measure, by contrast, remains steady, reflecting the reality that most items in the group failed relatively early. They demonstrated that this new function is directly linked to the "reversed hazard rate," a concept that describes the likelihood of failure at a specific moment given that the item has already failed by that time. By connecting these two ideas, the authors established that the new function can classify how a system ages, telling us if it is becoming more or less likely to fail as time passes. They also showed that the order of these measures remains consistent even if the time units are scaled up or down, making the tool robust for different types of data.

To ensure this new tool could be used in the real world, the team created a method to estimate it from actual data without assuming a specific theoretical model beforehand. They used a technique called discrete kernel smoothing, which is a way of drawing a smooth curve through scattered data points, but with a special correction for the fact that time cannot be negative. In computer simulations, they tested this new estimator against older methods using thousands of fake datasets. The results showed that their new approach strikes a better balance between accuracy and stability. It avoided the wild swings seen in simple counting methods and the systematic errors found in other smoothing techniques, providing a reliable estimate of the geometric vitality even with moderate amounts of data.

Finally, the researchers applied their method to a real-world dataset involving insurance policies, looking at how long customers stayed with a company before leaving. This data, measured in months, represents a classic discrete lifetime problem. When they calculated the new geometric vitality function for this data, they found that it tracked the underlying patterns of customer retention closely. While the new method did not completely eliminate the differences between the raw data and standard statistical models, it provided a stable and consistent view of how the "past lives" of these customers were distributed. The study concludes that this geometric approach offers a valuable, complementary lens for understanding discrete reliability data, capturing proportional aging behaviors that traditional averages might overlook. It provides a flexible framework for analyzing how systems fail in the past, opening the door for more nuanced assessments in fields ranging from engineering to actuarial science.

Technical Summary: On the Discrete Past Geometric Vitality Function and Its Properties

Problem Statement
In reliability theory and survival analysis, measures based on past lifetime information are essential for retrospective ageing analysis. While classical measures such as the Past Vitality Function (PVF) and Mean Past Lifetime (MPL) summarize truncated lifetimes through arithmetic averaging, they are inherently additive. The authors argue that these additive measures may fail to adequately capture proportional or multiplicative ageing behaviors, which are common in biological systems and reliability contexts. Furthermore, while geometric vitality measures have been developed for continuous distributions (e.g., Nair and Rajesh [2000]; Abdul Sathar and Gayathri [2021]), a formal geometric vitality measure based on past lifetimes for discrete distributions has not been developed, despite the prevalence of discrete or count-based reliability data.

Methodology
The paper introduces the Discrete Past Geometric Vitality Function (PGVF), denoted as $G(t)$ , for a discrete random variable $X$ with cumulative distribution function (CDF) $F(t)$ . It is defined as the geometric mean of the truncated lifetime distribution:
$\log G(t) = E[\log X \mid X \le t] = \frac{1}{F(t)} \sum_{x=1}^{t} (\log x) p(x)$
where $p(x)$ is the probability mass function.

The study proceeds through three main methodological pillars:

Structural Analysis: The authors derive fundamental properties of the PGVF, including its relationship with the reversed hazard rate ( $\delta_X(t)$ ), monotonicity conditions, stochastic ordering, and transformation properties under scaling.
Nonparametric Estimation: A nonparametric estimator for the PGVF is proposed using discrete kernel smoothing with boundary correction. This addresses the issue of kernel mass falling outside the support of discrete data (specifically near $x=1$ ). The estimator utilizes a renormalized kernel to ensure validity on the support $\mathbb{N}$ .
Empirical Validation: The performance of the proposed estimator is evaluated against an empirical estimator and an asymmetric discrete triangular kernel estimator through Monte Carlo simulations (using a zero-truncated Poisson distribution) and a real-data application involving insurance policy durations.

Key Contributions

Definition of Discrete PGVF: The paper formally defines the Discrete PGVF as a multiplicative analogue to the classical PVF, providing a geometric summary of truncated lifetimes.
Characterization Results:
- It is established that the Discrete PGVF uniquely determines the underlying distribution.
- A direct functional relationship is derived between the PGVF and the reversed hazard rate: $\delta_X(t) = \frac{\nabla \log G(t)}{\log(t/G(t-1))}$ .
- Characterization theorems link the PGVF to the "reversed lack-of-memory" property and specific distributional forms (e.g., reversed geometric-type distributions).
Ageing Classes: The paper defines and characterizes classes of distributions based on the monotonicity of the PGVF, specifically Decreasing Past Geometric Vitality Function (DPGVF) and Increasing Past Geometric Vitality Function (IPGVF).
Stochastic Ordering: A new partial ordering ( $Z \ge_{PGVF} X$ ) is introduced based on the PGVF, and its invariance under common scaling transformations is proven.
Estimation Framework: A boundary-corrected kernel estimator is developed. Theoretical properties, including consistency, asymptotic normality, and the optimal bandwidth order ( $h_{opt} \asymp n^{-1/3}$ ), are derived.

Results

Structural Comparison: Numerical and graphical comparisons with the classical PVF (using a right-skewed Negative Binomial model) demonstrate that the PVF is more sensitive to large observations due to arithmetic aggregation, whereas the PGVF (geometric mean) remains more stable. The paper shows that two distributions can have identical PVF values but distinct PGVF values, highlighting the PGVF's ability to distinguish multiplicative structures.
Simulation Studies:
- In simulations with sample sizes $n=50$ and $n=200$ , the proposed boundary-corrected kernel estimator achieved a favorable bias-variance trade-off.
- While the empirical estimator was nearly unbiased, it exhibited high variance. The asymmetric kernel estimator reduced variance but introduced significant bias. The proposed estimator maintained small bias while effectively reducing variance, resulting in lower Mean Squared Error (MSE) for moderate and large truncation levels.
Real-Data Application: Applied to insurance claim duration data, the estimators were compared against a fitted Negative Binomial reference. The results indicated that kernel smoothing had a limited impact on this specific dataset due to its wide support and dense distribution. However, the proposed estimator performed competitively, with differences in bias and MSE among the methods being relatively small.

Significance
The paper claims that the Discrete PGVF offers a structurally distinct and flexible framework for analyzing past lifetime characteristics in discrete settings. By shifting from additive to multiplicative summaries, it captures proportional ageing behaviors that may be obscured by traditional arithmetic measures. The work contributes to reliability theory by:

Filling the gap in geometric vitality measures for discrete distributions.
Providing characterization results that link geometric vitality to reversed hazard rates and memory properties.
Offering a robust nonparametric estimation tool with boundary correction, suitable for practical applications in fields such as actuarial science and reliability engineering where discrete count data is prevalent.

The authors conclude that the Discrete PGVF serves as a complementary perspective for ageing analysis, particularly in scenarios involving cumulative wear, replication processes, or highly skewed count data. Future work is suggested to focus on censored data extensions and adaptive bandwidth selection.

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