Abstract
The one-parameter Bell family of distributions, introduced by Castellares et al. (Appl Math Model 56:172–185, 2018), is useful for modeling count data. This paper proposes and studies a goodness-of-fit test for this distribution, which is consistent against fixed alternatives. The finite sample performance of the proposed test is investigated by means of several Monte Carlo simulation experiments, and it is also compared with other related ones. Real data applications are considered for illustrative purposes.
Similar content being viewed by others
References
Baringhaus L, Henze N (1992) A goodness of fit test for the Poisson distribution based on the empirical generating function. Stat Probab Lett 13:269–274
Castellares F, Ferrari SLP, Lemonte AJ (2018) On the Bell distribution and its associated regression model for count data. Appl Math Model 56:172–185
Corless RM, Gonnet GH, Hare DEG, Jeffrey D, Knuth DE (1996) On the Lambert W function. Adv Comput Math 5:329–359
Epps TW (1995) A test of fit for lattice distributions. Commun Stat Theory Methods 24:1455–1479
Giacomini R, Politis DN, White H (2013) A warp-speed method for conducting Monte Carlo experiments involving bootstrap estimators. Econ Theory 29:567–589
Goerg GM (2016) LambertW: probabilistic models to analyze and Gaussianize heavy-tailed, skewed data. R library version 0.6.4
Gürtler N, Henze N (2000) Recent and classical goodness-of-fit test for the Poisson distribution. J Stat Plan Inference 90:207–225
Janssen A (2000) Global power functions of goodness of fit tests. Ann Stat 28:239–253
Jiménez-Gamero MD, Alba-Fernández MV (2019) Testing for the Poisson–Tweedie distribution. Math Comput Simul 164:146–162
Jiménez-Gamero MD, Batsidis A (2017) Minimum distance estimators for count data based on the probability generating function with applications. Metrika 80:503–545
Kemp AW (1992) Heine–Euler extensions of the Poisson distribution. Commun Stat Theory Methods 21:571–588
Kocherlakota S, Kocherlakota K (1986) Goodness of fit test for discrete distributions. Commun Stat Theory Methods 15:815–829
Klugman S, Panjer H, Willmot G (1998) Loss Models, From Data to Decisions. Wiley, New York
Kundu S, Majumdar S, Mukherjee K (2000) Central limit theorems revisited. Stat Probab Lett 47:265–275
Meintanis S, Bassiakos Y (2005) Goodness-of-fit tests for additively closed count models with an application to the generalized Hermite distribution. Sankhya A 67:538–552
Meintanis S (2008) New inference procedures for generalized Poisson distributions. J Appl Stat 35:751–762
Nakamura M, Pérez-Abreu V (1993a) Empirical probability generating function: an overview. Insur Math Econ 12:287–295
Nakamura M, Pérez-Abreu V (1993b) Use of an empirical probability generating function for testing a Poisson model. Can J Stat 21:149–156
Novoa Muñoz F, Jiménez-Gamero MD (2014) Testing for the bivariate Poisson distribution. Metrika 77:771–793
Novoa Muñoz F, Jiménez-Gamero MD (2016) A goodness-of-fit test for the multivariate Poisson distribution. Sort 40:1–26
Olver FWJ, Lozier DW, Boisvert RF, Clark CW (2010) NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge
R Core Team (2018) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna
Rashid A, Ahmad Z, Jan TR (2016) A new count data model with application in genetics and ecology. Electron J Appl Stat Anal 9:213–226
Rueda R, O’Reilly F (1999) Tests of fit for discrete distributions based on the probability generating function. Commun Stat Simul Comput 28:259–274
Rueda R, Pérez Abreu V, O’Reilly F (1991) Goodness of fit for the Poisson distribution based on the probability generating function. Commun Stat Theory Methods 20:3093–3110
Sichel HS (1951) The estimation of the parameters of a negative binomial distribution with special reference to psychological data. Psychometrika 16:107–127
Acknowledgements
The authors thank the Editor, the Associate Editor and two anonymous referees for their constructive comments and suggestions which helped to improve the presentation. M.D. Jiménez-Gamero has been partially supported by Grant MTM2017-89422-P of the Spanish Ministry of Economy, Industry and Competitiveness, the State Agency of Investigation, the European Regional Development Fund. Artur J. Lemonte acknowledges the financial support of the Brazilian agency CNPq (Grant 301808/2016–3).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that no conflict of interest (financial or otherwise) exists in the submission of this manuscript, and manuscript is approved by all authors for publication.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix
Proofs
Here we prove the results given in the previous sections.
Proof of Proposition 1
It can be checked that the PGF of \(X \sim \mathrm{Bell}(\theta )\) given in (1) satisfies the differential equation given in (2). Next, we proof that it is the only PGF in G satisfying such differential equation. It is well-known that the solution of the linear differential equation of order one of the form \(y^{\prime }+p(t) y=0\), where \(y=y(t)\), \(y'=\frac{\partial }{\partial t}y(t)\) and p(t) is a continuous function in t, is given by \(y=C \exp (-\int p(t) dt)\), where C is an arbitrary constant. Since the differential equation (2) is of this form, we have that
Taking into account that g is a PGF, it must satisfy \(g(1)=1\), implying that \(C=\exp \left( -e^{\theta }\right) \) and hence the desired result is obtained. \(\square \)
Let \(\phi (x;\theta )=(\phi (x; 0,\theta ),\phi (x; 1,\theta ), \ldots )\) and
We have the following lemmas.
Lemma 1
Let \(X_1, \ldots , X_n\) be independent and identically distributed from X, a random variable taking values in \(\mathbb {N}_{0}\) with probability mass function \(p(k)=\Pr (X=k)\), \(k\in \mathbb {N}_{0}\), so that \(E(X^2)<\infty \), then
Proof
By definition,
and thus
Taking into account that
the result follows.\(\square \)
Lemma 2
Let \(X_1, \ldots , X_n\) be independent and identically distributed from X, a random variable taking values in \(\mathbb {N}_{0}\), then
Proof
We have that
By interchanging the order of the sums, one gets
Taking into account that
the result follows.\(\square \)
Lemma 3
Let \(X_1, \ldots , X_n\) be independent and identically distributed from X, a random variable taking values in \(\mathbb {N}_{0}\). Assume that \(\widehat{\theta } {\mathop {\longrightarrow }\limits ^{a.s.(P)}} \theta \), for some \(\theta >0\). For each \(k\in \mathbb {N}_{0}\), let \(\theta _k=\alpha _{k}\theta +(1-\alpha _{k})\widehat{\theta }\), for some \(\alpha _{k} \in [0,1]\). Then,
Proof
Let \(\widetilde{\theta }=\max \{ \widehat{\theta }, \theta \}\). Proceeding as in the proof of Lemma 2, we get that
Since \(f_1(\widetilde{\theta })^2\) is a continuous function of \(\widehat{\theta }\), we have that
and the result follows.\(\square \)
Lemma 4
Let \(X_1, \ldots , X_n\) be independent and identically distributed from X, a random variable taking values in \(\mathbb {N}_{0}\). Assume that \(\widehat{\theta } {\mathop {\longrightarrow }\limits ^{a.s.(P)}} \theta \), for some \(\theta >0\). Given the data, let \(X_1^*,\ldots , X_n^*\) be independent and identically distributed from \(X^* \sim \mathrm{Bell}(\widehat{\theta })\). Let \(\widehat{d}^*(k;\theta )\) be defined as \(\widehat{d}(k;\theta )\) with \(\widehat{p}(k)\) replaced by
Then,
- (I)
\(\displaystyle \sum _{k \ge 0} \left[ \frac{\partial }{\partial \theta }\widehat{d}^*(k;\widehat{\theta })-\mu (k; \widehat{\theta }) \right] ^2 {\mathop {\longrightarrow }\limits ^{P_*}} 0\), a.s.(P),
- (II)
\(\displaystyle \sum _{k \ge 0} \left[ \mu (k; {\theta })-\mu (k; \widehat{\theta }) \right] ^2 \rightarrow 0\), a.s.(P).
Proof
(I) We have that
Since \(f_1(\widehat{\theta })^2\) is a continuous function of \(\widehat{\theta }\), we have that
We also have that
Therefore,
and the result in part (I) follows.
(II) We have that
where
We first deal with \(\varDelta _1\). We have that
To deal with the second term on the right-hand of (13), note that since
there exists \(k_1=k_1(\theta ) \ge 1\) such that \(\frac{\partial }{\partial \theta } p(k; \theta ) >0\), \(\forall \, k \ge k_1\). We are assuming that \(\widehat{\theta } {\mathop {\longrightarrow }\limits ^{a.s.(P)}} \theta \), for some \(\theta >0\), and therefore for any \(\varepsilon >0\) there exists \(n_0 \in \mathbb {N}\) (depending on \(\theta \), \(\varepsilon \) and the sequence \(X_1, X_2, \ldots \)) such that \(\widehat{\theta } \le \theta +\varepsilon \)a.s.(P) \(\forall \ n \ge n_0\). For any \(k_0 \ge k_1\), we have that
Since \(\delta _1(\widehat{\theta })\) is a continuous function of \(\widehat{\theta }\), it follows that
As for \(\delta _2(\widehat{\theta })\), because \(k_0 \ge k_1\), we have that
and the right-hand side of the above expression is as small as desired for large enough \(k_0\). Therefore, we have shown that
This fact together with (12) shows that \(\varDelta _1 {\mathop {\longrightarrow }\limits ^{a.s.(P)}} 0\).
We have that
with
Notice that \(0 \le |M_2(u,v)| \le 1\), \(\forall u,\, v \ge 0\). By applying the mean value theorem,
where \(\widetilde{\theta }_v=\alpha _v\widehat{\theta }+(1-\alpha _v)\theta \), for some \(\alpha _v \in (0,1)\). As in the proof of Lemma 3, let \(\widetilde{\theta }=\max \{\theta , \widehat{\theta }\}\). Note that \(\widehat{\theta }_v \le \widetilde{\theta }\), \(\forall \, v \ge 1\). From the above considerations, we have that
Since the right-hand side of the above expression is a continuous function of \(\theta \), it follows that
and thus \(\varDelta _2 {\mathop {\longrightarrow }\limits ^{a.s.(P)}} 0\).
Finally,
with
By applying the mean value theorem (as done when studying \(\varDelta _2\)), we get
Since the right-hand side of the above expression is a continuous function of \(\theta \), it follows that
and thus \(\varDelta _3 {\mathop {\longrightarrow }\limits ^{a.s.(P)}} 0\).\(\square \)
Lemma 5
Let \(X_1, \ldots , X_n\) be independent and identically distributed from X, a random variable taking values in \(\mathbb {N}_{0}\). Assume that \(\widehat{\theta } {\mathop {\longrightarrow }\limits ^{a.s.(P)}} \theta \), for some \(\theta >0\). For each \(k\in \mathbb {N}_{0}\), let \(\theta _k= \alpha _{k}\theta +(1-\alpha _{k})\widehat{\theta }\), for some \(\alpha _{k} \in [0,1]\). Then,
Proof
The proof is parallel to that of \(\varDelta _3 {\mathop {\longrightarrow }\limits ^{a.s.(P)}} 0\) in the proof of Lemma 4.\(\square \)
Proof of Theorem 1
By applying the mean value theorem, we get, for each \(k \in \mathbb {N}_{0}\), that
with \(\theta _k=\alpha _{k}\theta +(1-\alpha _{k})\widehat{\theta }\), for some \(\alpha _{k} \in (0,1)\). From Lemma 1, \(E(\Vert \phi (X;\theta )\Vert _2^2)<\infty \) and thus by the SLLN in Hilbert spaces and the continuous mapping theorem, it follows that
Finally, the result follows from (14), (15) and Lemma 3. \(\square \)
Proof of Theorem 2
From expansion (14),
with \(\theta _k=\alpha _{k}\theta +(1-\alpha _{k})\widehat{\theta }\), for some \(\alpha _{k} \in (0,1)\). Assumption 1 and Lemmas 2 and 4 imply that
with \(\Vert r_1\Vert _2=o_P(1)\). Now, by applying the SLLN in Hilbert spaces and Assumption 1, we get
with \(\Vert r_2\Vert _2=o_P(1)\). By the central limit theorem in Hilbert spaces,
where \(Y(X; \cdot , \theta ) =(Y(X;0, \theta ), Y(X;1, \theta ), \ldots )\). The result follows from (16)–(18) and the continuous mapping theorem. \(\square \)
Proof of Theorem 3
Proceeding as in the proof of Theorem 2, we have that
with \(\Vert r^*_1\Vert _2=o_{P_*}(1)\) a.s.(P). Let
By applying Lemma 4 and Assumption 2, we get
with \(\Vert r_2^*\Vert _2=o_{P_*}(1)\)a.s.(P). To prove the result we derive the asymptotic distribution of \(Y_n^*\), showing that it coincides with the asymptotic distribution of \(S_{n}(\widehat{\theta } )\) when the data come from \(X\sim \mathrm{Bell}(\theta )\). With this aim, we apply Theorem 1.1 in Kundu et al. (2000). So, we will show that conditions (i)–(iii) in that theorem hold. For \(k \ge 0\), let \(e_k(j)= I(k=j)\). \(\{e_k\}_{k\ge 0}\) is an orthonormal basis of \(l^2\). We have that \(E_*\{\langle Y(X_i^*;\cdot , \widehat{\theta }),e_k \rangle _2\}= E_*\{Y(X^*;k, \widehat{\theta })\}=0\), \(\forall \, k \ge 0\), and by Lemma 1 and Assumption 2, \(E_*\{\Vert Y(X^*;\cdot , \widehat{\theta })\Vert _2^2\}<\infty \).
Let \(\mathcal {C}\) denote the operator defined in (11) and let \(\mathcal {C}_n\) be similarly defined by replacing \(\varrho (k,r)=Cov_{\theta }\{Y(X;k,\theta ),Y(X;r,\theta )\}\) with \(\varrho _n(k,r)=Cov_*\{\{Y(X^*; k,\widehat{\theta })Y(X^*; r,\widehat{\theta }) \}\), \(k\in \mathbb {N}_{0}\), \(r \in \mathbb {N}_{0}\). Assumption 2 and Lemma 4 imply that
Setting \(a_{k,r}=\langle \mathcal {C} e_k,e_r\rangle _2\) in the aforementioned Theorem 1.1, this proves that condition (i) holds. Similarly, condition (ii) holds since
and \(\sum _{k\ge 0}a_{kk}<\infty \). Finally, condition (iii) readily follows from Assumption 2. \(\square \)
Practical issues
Next, we describe some computational issues related to the calculation of the test statistics considered in the simulation study of Sect. 4. The test statistics \(S_{n}(\widehat{\theta })\), \(R_{n,w}(\widehat{\theta })\) and \(M_{n,w}(\widehat{\theta })\) are defined by means of infinite sums. However, these sums have to be truncated at some finite value, say M; that is,
From the numerical results, we have noted that taking \(M=20\) yields sufficiently precise values of these statistics. Finally, note that
with
and, therefore,
can be recursively calculated as follows: \({\textit{coef}}(0;\theta )=\theta \), and \({\textit{coef}}(u;\theta )={\textit{coef}}(u-1;\theta )\theta /u\) for \(u\ge 1\).
Calculation of the bootstrap p-value
Let T denote any of the three test statistics and let \(T_{obs}\) stand for the observed value of such statistic. The bootstrap p-value, \(\hat{p}=P_*(T \ge T_{obs})\) cannot be exactly calculated. Nevertheless, it can be approximated as follows.
- 1.
Calculate the observed value of the test statistics for the available dataset \(X_1,\ldots , X_n\), say \(S_{obs}(\widehat{\theta })\), \(M_{obs}(\widehat{\theta })\) and \(R_{obs}(\widehat{\theta })\).
- 2.
Generate B bootstrap samples \(X_1^{*b},\ldots , X_n^{*b}\) from \(X^*\sim \mathrm{Bell}(\widehat{\theta })\), for \(b = 1,\ldots , B\).
- 3.
Calculate the test statistics \(S_n(\widehat{\theta })\), \(M_{n,w}(\widehat{\theta })\) and \(R_{n,w}(\widehat{\theta })\) for each bootstrap sample and denote them, respectively, by \(S_b^*\), \(M_b^*\) and \(R_b^*\) for \(b = 1,\ldots , B\).
- 4.
Compute the p-values of the tests based on the statistics \(S_n(\widehat{\theta })\), \(M_{n,w}(\widehat{\theta })\) and \(R_{n,w}(\widehat{\theta })\) by means, respectively, of the expressions
$$\begin{aligned} \widehat{p}_S =\frac{\#\{S_b^*\ge S_{obs}(\widehat{\theta })\}}{B},\quad \widehat{p}_M =\frac{\#\{M_b^*\ge M_{obs}(\widehat{\theta })\}}{B},\quad \widehat{p}_R =\frac{\#\{R_b^*\ge R_{obs}(\widehat{\theta })\}}{B}. \end{aligned}$$
Rights and permissions
About this article
Cite this article
Batsidis, A., Jiménez-Gamero, M.D. & Lemonte, A.J. On goodness-of-fit tests for the Bell distribution. Metrika 83, 297–319 (2020). https://doi.org/10.1007/s00184-019-00733-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-019-00733-6