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

Abstract

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.

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Acknowledgements

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.

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Appendices

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

$$\begin{aligned} f_t^{\prime }(x)= & {} - \frac{p^{\prime }(x)}{p^2(x)} \int _0^{x} \Big ( \mathbb {1}_{(0, t]} (s) - P(t) \Big ) p(s) \, \mathrm {d}s + \frac{1}{p(x)} \Big ( \mathbb {1}_{(0, t]} (x) - P(t) \Big ) p(x) \nonumber \\= & {} - \left( \frac{k - 1}{x} - \frac{1}{\lambda } \right) f_t(x) + \mathbb {1}_{(0, t]} (x) - P(t), \end{aligned}$$
(25)

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

$$\begin{aligned} f_t(x) = \frac{1}{p(x)} P(x) \big ( 1 - P(t) \big ) , \end{aligned}$$
(26)

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

$$\begin{aligned} \frac{1}{p(x)} \, P(x) = \int _0^x \left( \frac{s}{x} \right) ^{k - 1} e^{\lambda ^{-1} (x - s)} \, \mathrm {d}s = x e^{\lambda ^{-1} x} \int _0^1 z^{k - 1} e^{- \lambda ^{-1} x z} \, \mathrm {d}z \le \frac{x e^{\lambda ^{-1} x}}{k}, \end{aligned}$$
(27)

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. \(\square \)

Remark

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.

Proof

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

$$\begin{aligned} w_a\big ( {\widehat{\lambda }}_n^{-1} s \big ) - w_a(s) = w_a^{\prime }\big ( \xi _n(s) \big ) \Big ( {\widehat{\lambda }}_n^{-1} s - s \Big ), \end{aligned}$$
(28)

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. \(\square \)

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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

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Keywords

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