Healthy longevity distributions derived via maximum entropy from statistical moments reveal that for most health measures females have longer healthy life expectancy and males have more dispersed distributions with a lower mode, while the probability of a man outliving a woman in healthy lifespan varies substantially by health indicator.
Key Findings
Results
For most health measures, probabilities of health loss at younger ages were higher for males than for females, and females had a longer healthy life expectancy.
Data came from the Survey of Health, Ageing and Retirement in Europe (SHARE) combined with the Human Mortality Database.
A Markov chain model was used to estimate the first three statistical moments of healthy longevity distributions starting at age 60.
The maximum entropy method was applied to derive full healthy longevity distributions from those moments.
This pattern held for most but not all health measures examined.
Results
Males had more dispersed healthy longevity distributions with a lower mode compared to females.
Dispersion was captured by the standard deviation of the healthy longevity distribution.
The mode represents the age with the highest probability of health loss.
Males showing a lower mode indicates that the most common age of health loss occurred earlier for men than for women.
This pattern of greater male dispersion was observed across most health measures.
Results
For most health measures, healthy longevity distributions were negatively skewed, with a mode age higher than the healthy life expectancy age.
Negative skewness means the distribution has a longer left tail, indicating that while most people remain healthy until relatively late ages, some experience early health loss.
The mode (age with highest probability of health loss) exceeded the mean healthy life expectancy age.
This finding was consistent across most health indicators examined.
The first three statistical moments (mean, variance, skewness) were estimated via the Markov chain model and used as inputs to the maximum entropy method.
Results
The probability of a man having a longer healthy lifespan than a woman was below 50% for most health measures and was lowest for living free of cardiovascular disease.
The healthy lifespan outsurvival statistic was used to formally compare male and female distributions.
This statistic represents the probability that a randomly selected man will have a longer healthy lifespan than a randomly selected woman.
Living free of cardiovascular disease showed the lowest probability of male outliving female in healthy lifespan.
The healthy lifespan outsurvival statistic is applied for the first time in the healthy longevity field in this study.
Results
The probability of a man living free of arthritis or rheumatism for longer than a woman was above 50%.
This was an exception to the general pattern where males had lower probability of longer healthy lifespan than females.
The result reflects that arthritis and rheumatism are conditions where females have higher prevalence or earlier onset.
The healthy lifespan outsurvival statistic exceeded 0.50 for this specific condition in males versus females.
Results
The most similar healthy longevity distributions between males and females were observed for life free of any chronic conditions and life with no more than one chronic condition.
The Hellinger distance was used to quantify similarity between male and female distributions.
The Hellinger distance is applied for the first time in the healthy longevity field in this study.
Smallest Hellinger distances between sexes were found for the two chronic condition-based health measures.
Different health measures produced substantially different levels of distributional similarity between sexes.
Methods
The study applied the maximum entropy method to derive full healthy longevity distributions from the first three statistical moments estimated via a Markov chain model.
Input data were from the Survey of Health, Ageing and Retirement in Europe (SHARE) and the Human Mortality Database.
The Markov chain model estimated mean, variance, and skewness of healthy longevity distributions at age 60.
The maximum entropy approach reconstructs the full probability distribution consistent with the known statistical moments.
Multiple health measures were examined, including cardiovascular disease, arthritis/rheumatism, chronic conditions count, and others.
Methods
The healthy lifespan outsurvival statistic and the Hellinger distance were used for the first time in the healthy longevity field to formally compare distributions.
The healthy lifespan outsurvival statistic quantifies the probability that one group (e.g., males) will have a longer healthy lifespan than another (e.g., females).
The Hellinger distance measures the statistical distance between two probability distributions.
Prior healthy longevity research had typically focused on average values such as healthy life expectancy.
Recent studies had begun examining standard deviation of healthy longevity, but formal distributional comparison methods had not been applied.
Cosulich R, di Lego V, Zarulli V. (2026). Deriving and comparing healthy longevity distributions by gender and health prevalence measures: a statistical moments and maximum entropy approach.. Population health metrics. https://doi.org/10.1186/s12963-026-00470-9