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
for any \(f \in \mathcal {F}\). For the converse, define \(f_t : (0, \infty ) \rightarrow \mathbb {R}\) by
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
and we have
Noting that, for \(x < t\), the function takes the form
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
Eq. (26) also implies \(\lim _{x \searrow 0} f_t(x) = 0\). Next, note that
where we denote the right-hand side by \(\kappa (x)\). With (27) we have
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),
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
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
Together with (28), we get
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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DOI: https://doi.org/10.1007/s00184-019-00708-7