A new characterization of the Gamma distribution and associated goodness-of-fit tests


We propose a class of weighted \(L^2\)-type tests of fit to the Gamma distribution. Our novel procedure is based on a fixed point property of a new transformation connected to a Steinian characterization of the family of Gamma distributions. We derive the weak limits of the statistic under the null hypothesis and under contiguous alternatives. The result on the limit null distribution is used to prove the asymptotic validity of the parametric bootstrap that is implemented to run the tests. Further, we establish the global consistency of our tests in this bootstrap setting, and conduct a Monte Carlo simulation study to show the competitiveness to existing test procedures.

This is a preview of subscription content, log in to check access.


  1. Abramowitz M, Stegun IA (1965) Handbook of mathematical functions. Dover Publ., Mineola

    Google Scholar 

  2. Allison JS, Santana L (2015) On a data-dependent choice of the tuning parameter appearing in certain goodness-of-fit tests. J Stat Comput Simul 85(16):3276–3288

    MathSciNet  Article  Google Scholar 

  3. Allison JS, Santana L, Smit N, Visagie IJH (2017) An ‘apples to apples’ comparison of various tests for exponentiality. Comput Stat 32(4):1241–1283

    MathSciNet  MATH  Article  Google Scholar 

  4. Baringhaus L, Gaigall D (2015) On an independence test approach to the goodness-of-fit problem. J Multivar Anal 140(Supplement C):193–208

    MathSciNet  MATH  Article  Google Scholar 

  5. Baringhaus L, Henze N (2000) Tests of fit for exponentiality based on a characterization via the mean residual life function. Stat Pap 41(2):225–236

    MathSciNet  MATH  Article  Google Scholar 

  6. Baringhaus L, Henze N (2008) A new weighted integral goodness-of-fit statistic for exponentiality. Stat Probab Lett 78(8):1006–1016

    MathSciNet  MATH  Article  Google Scholar 

  7. Baringhaus L, Henze N (2017) Cramér–von Mises distance: probabilistic interpretation, confidence intervals, and neighbourhood-of-model validation. J Nonparametr Stat 29(2):167–188

    MathSciNet  MATH  Article  Google Scholar 

  8. Baringhaus L, Ebner B, Henze N (2017) The limit distribution of weighted \(L^2\)-goodness-of-fit statistics under fixed alternatives, with applications. Ann Inst Stat Math 69(5):969–995

    MATH  MathSciNet  Article  Google Scholar 

  9. Betsch S, Ebner B (2018) Testing normality via a distributional fixed point property in the Stein characterization. ArXiv e-prints arXiv:1803.07069

  10. Bhattacharya B (2001) Testing equality of scale parameters against restricted alternatives for \(m\ge 3\) gamma distributions with unknown common shape parameter. J Stat Comput Simul 69(4):353–368

    MATH  Article  Google Scholar 

  11. Chen X, White H (1998) Central limit and functional central limit theorems for Hilbert-valued dependent heterogeneous arrays with applications. Econom Theory 14(2):260–284

    MathSciNet  Article  Google Scholar 

  12. Chen LHY, Goldstein L, Shao QM (2011) Normal approximation by Steins method. Springer, Berlin

    Google Scholar 

  13. del Barrio E, Cuesta-Albertos JA, Matrán C, Csörgő S, Cuadras CM, de Wet T, Giné E, Lockhart R, Munk A, Stute W (2000) Contributions of empirical and quantile processes to the asymptotic theory of goodness-of-fit tests. TEST 9(1):1–96

    MathSciNet  Article  Google Scholar 

  14. Döbler C (2017) Distributional transformations without orthogonality relations. J Theor Probab 30(1):85–116

    MathSciNet  MATH  Article  Google Scholar 

  15. Goldstein L, Reinert G (1997) Stein’s method and the zero bias transformation with application to simple random sampling. Ann Appl Probab 7(4):935–952

    MathSciNet  MATH  Article  Google Scholar 

  16. Goldstein L, Reinert G (2005) Distributional transformations, orthogonal polynomials, and Stein characterizations. J Theor Probab 18(1):237–260

    MathSciNet  MATH  Article  Google Scholar 

  17. Gürtler N, Henze N (2000) Recent and classical goodness-of-fit tests for the Poisson distribution. J Stat Plan Inference 90(2):207–225

    MathSciNet  MATH  Article  Google Scholar 

  18. Hájek J, Šidák Z, Sen PK (1999) Theory of rank tests. Probability and mathematical statistics. Academic Press, Cambridge

    Google Scholar 

  19. Henze N (1996) Empirical-distribution-function goodness-of-fit tests for discrete models. Can J Stat 24(1):81–93

    MathSciNet  MATH  Article  Google Scholar 

  20. Henze N, Wagner T (1997) A new approach to the BHEP tests for multivariate normality. J Multivar Anal 62(1):1–23

    MathSciNet  MATH  Article  Google Scholar 

  21. Henze N, Meintanis SG, Ebner B (2012) Goodness-of-fit tests for the gamma distribution based on the empirical Laplace transform. Commun Stat Theory Methods 41(9):1543–1556

    MathSciNet  MATH  Article  Google Scholar 

  22. Husak GJ, Michaelsen J, Funk C (2007) Use of the gamma distribution to represent monthly rainfall in Africa for drought monitoring applications. Int J Climatol 27(7):935–944

    Article  Google Scholar 

  23. Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1, 2nd edn. Wiley, New York

    Google Scholar 

  24. Kallioras AG, Koutrouvelis IA, Canavos GC (2006) Testing the fit of gamma distributions using the empirical moment generating function. Commun Stat Theory Methods 35(3):527–540

    MathSciNet  MATH  Article  Google Scholar 

  25. Ley C, Swan Y (2013) Stein’s density approach and information inequalities. Electron Commun Probab 18:14

    MathSciNet  MATH  Article  Google Scholar 

  26. Plubin B, Siripanich P (2017) An alternative goodness-of-fit test for a gamma distribution based on the independence property. Chiang Mai J Sci 44(3):1180–1190

    Google Scholar 

  27. R Development Core Team (2018) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna

    Google Scholar 

  28. Vakhania NN (1981) Probability distributions on linear spaces. North Holland, New York

    Google Scholar 

  29. Villaseñor JA, González-Estrada E (2015) A variance ratio test of fit for gamma distributions. Stat Probab Lett 96(Supplement C):281–286

    MathSciNet  MATH  Article  Google Scholar 

  30. Wilding GE, Mudholkar GS (2008) A gamma goodness-of-fit test based on characteristic independence of the mean and coefficient of variation. J Stat Plan Inference 138(12):3813–3821

    MathSciNet  MATH  Article  Google Scholar 

Download references


The authors thank Norbert Henze for fruitful discussions and helpful comments on the presentation of the material. They also want to express their gratitude to an anonymous referee, an associate editor, and the journal editor, for their insights during the revision process, which led to a major improvement of the article.

Author information



Corresponding author

Correspondence to Bruno Ebner.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Proof of Theorem 1

Note that if \(X \sim \varGamma (k, \lambda )\), X has Lebesgue density \(p = p(\cdot \, ; k, \lambda )\), and the fundamental theorem of calculus (utilizing the boundedness conditions for functions in \(\mathcal {F}\)) implies

$$\begin{aligned} \mathbb {E}\left[ f^{\prime }(X) + \left( \frac{k-1}{X} - \frac{1}{\lambda } \right) f(X) \right]&= \mathbb {E}\left[ f^{\prime }(X) + \frac{p^{\prime }(X)}{p(X)} \, f(X) \right] \\&= \int _0^{\infty } \frac{\mathrm {d}}{\mathrm {d}x} \Big ( f(x) p(x) \Big ) \, \mathrm {d}x \\&= \lim _{x \, \rightarrow \, \infty } f(x) \, p(x) - \lim _{x \, \searrow \, 0} f(x) \, p(x) \\&= 0 \end{aligned}$$

for any \(f \in \mathcal {F}\). For the converse, define \(f_t : (0, \infty ) \rightarrow \mathbb {R}\) by

$$\begin{aligned} f_t(x) = \frac{1}{p(x)} \int _0^{x} \Big ( \mathbb {1}_{(0, t]} (s) - P(t) \Big ) p(s) \, \mathrm {d}s,\quad t>0, \end{aligned}$$

where \(P(t) = \int _0^t p(s) \, \mathrm {d}s\) is the distribution function of the \(\varGamma (k, \lambda )\)-law. Apparently, \(f_t\) is differentiable with

$$\begin{aligned} f_t^{\prime }(x)= & {} - \frac{p^{\prime }(x)}{p^2(x)} \int _0^{x} \Big ( \mathbb {1}_{(0, t]} (s) - P(t) \Big ) p(s) \, \mathrm {d}s + \frac{1}{p(x)} \Big ( \mathbb {1}_{(0, t]} (x) - P(t) \Big ) p(x) \nonumber \\= & {} - \left( \frac{k - 1}{x} - \frac{1}{\lambda } \right) f_t(x) + \mathbb {1}_{(0, t]} (x) - P(t), \end{aligned}$$

and we have

$$\begin{aligned} \lim _{x \, \rightarrow \, \infty } f_t(x) \, p(x) = \int _0^{\infty } \Big ( \mathbb {1}_{(0, t]} (s) - P(t) \Big ) p(s) \, \mathrm {d}s = P(t) - P(t) = 0 . \end{aligned}$$

Noting that, for \(x < t\), the function takes the form

$$\begin{aligned} f_t(x) = \frac{1}{p(x)} P(x) \big ( 1 - P(t) \big ) , \end{aligned}$$

we infer \(\lim _{x \searrow 0} f_t(x) \, p(x) = \lim _{x \searrow 0} P(x) \big ( 1 - P(t) \big ) = 0\), and with the estimate

$$\begin{aligned} \frac{1}{p(x)} \, P(x) = \int _0^x \left( \frac{s}{x} \right) ^{k - 1} e^{\lambda ^{-1} (x - s)} \, \mathrm {d}s = x e^{\lambda ^{-1} x} \int _0^1 z^{k - 1} e^{- \lambda ^{-1} x z} \, \mathrm {d}z \le \frac{x e^{\lambda ^{-1} x}}{k}, \end{aligned}$$

Eq. (26) also implies \(\lim _{x \searrow 0} f_t(x) = 0\). Next, note that

$$\begin{aligned} \left| \left( \frac{k - 1}{x} - \frac{1}{\lambda } \right) f_t(x) \right| \le 2 \left| \left( \frac{k - 1}{x} - \frac{1}{\lambda } \right) \right| \frac{\min \big \{ P(x), \, 1 - P(x) \big \}}{p(x)}, \end{aligned}$$

where we denote the right-hand side by \(\kappa (x)\). With (27) we have

$$\begin{aligned} \lim _{x \, \searrow \, 0} \kappa (x) \le 2 \lim _{x \, \searrow \, 0} \left| \left( \frac{k - 1}{x} - \frac{1}{\lambda } \right) \right| \frac{x e^{\lambda ^{-1} x}}{k} = \frac{|k - 1|}{k}, \end{aligned}$$

and since L’Hospital’s rule gives \(\lim _{x \rightarrow \infty } \tfrac{1 - P(x)}{p(x)} = - \lim _{x \rightarrow \infty } \left( \tfrac{k - 1}{x} - \tfrac{1}{\lambda } \right) ^{-1} = \lambda \), we have \(\lim _{x \rightarrow \infty } \kappa (x) \le 2\). These limit relations, combined with the continuity of \(\kappa \) on \((0, \infty )\), imply the boundedness of \(x \mapsto \kappa (x)\), and thus the boundedness of \(x \mapsto \left( \tfrac{k - 1}{x} - \tfrac{1}{\lambda } \right) f_t(x)\). Therefore, (25) yields \(f_t \in \mathcal {F}\). By the assumption and (25),

$$\begin{aligned} 0 = \mathbb {E}\left[ f_t^{\prime }(X) + \left( \frac{k-1}{X} - \frac{1}{\lambda } \right) f_t(X) \right] = \mathbb {P}(X \le t) - P(t). \end{aligned}$$

As t was arbitrary, X follows the \(\varGamma (k, \lambda )\)-law. \(\square \)


The requirement \(\lim _{x \, \searrow \, 0} f(x) = 0\) for functions in \(\mathcal {F}\) is not yet needed in this proof. Still, as the last step in the proof of Theorem 2 relies on this assumption, we had to include it in the characterization given in Theorem 1.

On the weight functions

With the following lemma, we ensure that the density function of an exponential distribution is an admissible weight function.

Lemma 2

The functions \(w_a(s) = e^{- a s}\), \(s > 0\), \(a > 0\), satisfy the weight function conditions (4) and (5) as stated in Sect. 3.


The function \(w_a\) clearly satisfies (4). In order to show (5), let \(0< \varepsilon < 1/6\) be arbitrary. In the case \(|{\widehat{\lambda }}_n^{-1} - 1| \le \varepsilon \), a Taylor expansion gives

$$\begin{aligned} w_a\big ( {\widehat{\lambda }}_n^{-1} s \big ) - w_a(s) = w_a^{\prime }\big ( \xi _n(s) \big ) \Big ( {\widehat{\lambda }}_n^{-1} s - s \Big ), \end{aligned}$$

where \(\big |\xi _n(s) - s\big | \le \big | {\widehat{\lambda }}_n^{-1} s - s \big | \le s / 6\). Therefore, \(\xi _n(s) - s \ge - s / 6\) which implies

$$\begin{aligned} \frac{\big | w_a^{\prime }\big ( \xi _n(s) \big ) \big |^3}{\big (w_a(s)\big )^2} = a^3 \exp \big ( - 3a \xi _n(s) + 2a s \big ) \le a^3 e^{- a s / 2}. \end{aligned}$$

Together with (28), we get

$$\begin{aligned} n \int _{0}^\infty \left| w_a\big ( {\widehat{\lambda }}_n^{-1} s \big ) - w_a(s) \right| ^{3} \big ( w_a(s) \big )^{-2} \mathrm {d}s&= n \int _0^\infty \Big | {\widehat{\lambda }}_n^{-1} s - s \Big |^3 \, \frac{\big | w_a^{\prime }\big ( \xi _n(s) \big ) \big |^3}{\big (w_a(s)\big )^2} \, \mathrm {d}s \\&\le \varepsilon a^3 \Big ( \sqrt{n} \big ( {\widehat{\lambda }}_n^{-1} - 1 \big ) \Big )^2 \int _0^\infty s^3 e^{- a s / 2} \, \mathrm {d}s. \end{aligned}$$

As \(\varepsilon \) was arbitrary, the tightness of \(\big \{ \sqrt{n} ({\widehat{\lambda }}_n^{-1} - 1) \big \}_{n \, \in \, \mathbb {N}}\) finished the proof. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Betsch, S., Ebner, B. A new characterization of the Gamma distribution and associated goodness-of-fit tests. Metrika 82, 779–806 (2019). https://doi.org/10.1007/s00184-019-00708-7

Download citation


  • Bootstrap procedure
  • Contiguous alternatives
  • Density approach
  • Gamma distribution
  • Goodness-of-fit tests
  • Stein’s method