[Paper Review] A right and left truncated gamma distribution with application to the stars

Lorenzo Zaninetti|arXiv (Cornell University)|Jan 1, 2014

Statistical Distribution Estimation and Applications2 references3 citations

TL;DR

This paper introduces a right- and left-truncated gamma distribution to model the initial mass function (IMF) of stars, accounting for physical lower and upper mass bounds (0.02 M☉ and 60 M☉). It derives normalization, moments, and random variate generation methods, and demonstrates via two stellar clusters (NGC 6611 and NGC 2362) that the truncated gamma provides a better fit than lognormal or four-power-law models, as shown by lower AIC and reduced χ² values.

ABSTRACT

The gamma density function is usually defined in interval between zero and infinity. This paper introduces an upper and a lower boundary to this distribution. The parameters which characterize the truncated gamma distribution are evaluated. A statistical test is performed on two samples of stars. A comparison with the lognormal and the four power law distribution is made.

Motivation & Objective

• To develop a physically motivated probability distribution for stellar masses bounded by minimum and maximum values.
• To derive analytical expressions for the normalization constant, mean, and cumulative distribution function of the doubly truncated gamma distribution.
• To apply the truncated gamma distribution to real astronomical data and compare its goodness-of-fit with standard IMF models.
• To evaluate the model's performance using statistical criteria such as AIC, reduced χ², and Kolmogorov-Smirnov test.

Proposed method

• Derives the probability density function (PDF) of the doubly truncated gamma distribution with lower (xₗ) and upper (xᵤ) bounds.
• Computes the normalization constant k using the upper and lower incomplete gamma functions, as shown in Equation (11).
• Derives the expected value of the truncated distribution via the upper incomplete gamma function, as in Equation (13).
• Formulates the cumulative distribution function (CDF) using the upper and lower incomplete gamma functions, as in Equation (15).
• Employs a numerical inversion method to generate random variates by solving DF(x) = R, where R is a uniform random number.
• Applies the model to two stellar clusters (NGC 6611 and NGC 2362), estimating parameters via moment matching and χ² minimization, and compares fits using AIC and K-S test.

Experimental results

Research questions

• RQ1 Can a doubly truncated gamma distribution provide a better statistical fit to observed stellar initial mass functions than standard models like lognormal or four-power-law distributions?
• RQ2 How do the parameters of the truncated gamma distribution (scale b, shape c, and truncation bounds) vary across different stellar clusters?
• RQ3 What is the performance of the truncated gamma model in terms of AIC, reduced χ², and Kolmogorov-Smirnov test statistics compared to alternative models?
• RQ4 Does the inclusion of physical lower and upper mass bounds (0.02 M☉ and 60 M☉) improve the realism and fit quality of the IMF model?
• RQ5 How does the truncated gamma distribution compare to other bounded distributions, such as the left-truncated beta distribution, in fitting the IMF?

Key findings

• For NGC 6611 (207 stars + brown dwarfs), the truncated gamma model achieved an AIC of 52.34, lower than the lognormal (71.24) and gamma (62.83), indicating a better fit.
• In NGC 6611, the reduced χ² was 2.77 for the truncated gamma, compared to 3.73 (lognormal) and 3.26 (gamma), showing improved goodness-of-fit.
• The Kolmogorov-Smirnov test yielded a higher p-value (P_KS = 0.061) for the truncated gamma model in NGC 6611, indicating better agreement with empirical data than the lognormal (P_KS = 0.04959).
• For NGC 2362 (272 stars), the truncated gamma model had an AIC of 33.88, lower than the lognormal (37.64) and gamma (34.28), again indicating superior model fit.
• The reduced χ² for the truncated gamma in NGC 2362 was 1.61, lower than the lognormal (1.86) and gamma (1.68), further supporting better fit quality.
• Despite improved fit, the left-truncated beta distribution outperforms the truncated gamma in both clusters, as reported in reference [21], suggesting potential for further model refinement.