# Cumulant-Based Goodness-of-Fit Tests for the Tweedie, Bar-Lev and Enis Class of Distributions

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## Abstract

**:**

## 1. Introduction

## 2. TBE Cumulant Relationships among ${\mathrm{T}\mathrm{B}\mathrm{E}}_{\gamma}$ Models and Some Testing Tools

**Proposition**

**1.**

**Proposition**

**2.**

**Remark**

**1.**

**Theorem**

**1.**

**Remark**

**2.**

## 3. The Proposed Gof Test: Test Statistic, Asymptotic Null Distribution and Bootstrap Approximation

#### 3.1. Test Statistic

#### 3.2. Asymptotic Behavior of the Test Statistic

**Theorem**

**2.**

**Corollary**

**1.**

**Theorem**

**3.**

**Remark**

**3.**

**ν**—the vector of unknown parameters of ${TBE}_{{\gamma}_{0}}$. Hence, we write $M=M(\mathbf{\nu},{\gamma}_{0})$. Special cases of $M(\mathbf{\nu},{\gamma}_{0})$ can easily be obtained for each of the specific values of ${\gamma}_{0}$. For example, for ${\gamma}_{0}=2$, ${TBE}_{{\gamma}_{0}}$ is the family of gamma distributions with shape parameter a and rate parameter b, in which case, $\mathbf{\nu}=(a,b),$ ${\mu}_{i}=\Gamma (i+a){b}^{-i}\Gamma {\left(a\right)}^{-1}$; thus,

**ν**= $(a,\mathbf{\mu}).$ Let $X\sim {TBE}_{\gamma}$; then, the moment-generating function of X is derived (as shown in [3] (Equation 2.4)) as

**μ**can now be computed using ${\mu}_{i}={d}^{i}G\left(t\right)/d{t}^{i}{|}_{t=0}\doteq {g}_{i}\left(\mathbf{\nu}\right),i=1,...,6,$ where $\mathbf{\nu}=$$(a,\mathbf{\mu})$ and ${g}_{i}$ is some ${\mathbb{R}}^{2}\to \mathbb{R}$ mapping.

- 1.
- 2.
- Under the null hypothesis of ${\mathrm{T}\mathrm{B}\mathrm{E}}_{{\gamma}_{0}},$ compute the MLEs $\widehat{\mathbf{\nu}}$ and $\widehat{M}=M(\widehat{\mathbf{\nu}},{\gamma}_{0})$ of $\mathbf{\nu}$ and $M(\mathbf{\nu},{\gamma}_{0})$, respectively.
- 3.
- Approximate the p-value of the test using the relation$$\widehat{p}=1-{F}_{{\chi}_{1}^{2}}\left(n{\widehat{M}}^{-1}{\mathcal{S}}_{n}^{\mathrm{obs}}\left({\gamma}_{0}\right)\right),$$

#### 3.3. Bootstrap Approximation

- 1.
- Follow steps 1 and 2 of the pervious procedure.
- 2.
- For some large integer B, repeat the following steps for every $b\in \{1,...,B\}$:
- (a)
- Generate a bootstrap sample ${X}_{1}^{*b}$,...,${X}_{n}^{*b}$ from ${X}^{*}\sim {\mathrm{T}\mathrm{B}\mathrm{E}}_{{\gamma}_{0}}$ with parameter $\widehat{\mathbf{\nu}}$.
- (b)
- Based on the bootstrap sample, calculate the bootstrap ${\mathcal{S}}_{n}{\left({\gamma}_{0}\right)}^{*b}$ version of test statistic ${\mathcal{S}}_{n}\left({\gamma}_{0}\right).$

- 3.
- Approximate the p-value with $\widehat{p}=\frac{1}{B}{\sum}_{b=1}^{B}I\left\{{\mathcal{S}}_{n}{\left({\gamma}_{0}\right)}^{*b}\ge {\mathcal{S}}_{n}^{\mathrm{obs}}\left({\gamma}_{0}\right)\right\}$ and the critical point with ${S}_{c:\mathrm{B},n}{\left({\gamma}_{0}\right)}^{*}$, where $c=\lceil (1-\alpha )B\rceil $ and $\lceil \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\rceil $ is the ceiling function.

## 4. Numerical Studies of Gamma and Modified Bessel-Type NEFs

#### 4.1. A Simulation Study of the Gamma NEF

- The inverse Gaussian distribution, denoted with $\mathrm{I}\mathrm{G}(\mu ,\lambda ),$ with density$$f(x;\mu ,\lambda )={\left(\frac{\lambda}{2\pi {x}^{3}}\right)}^{\frac{1}{2}}exp\left(-\frac{\lambda {(x-\mu )}^{2}}{2{\mu}^{2}x}\right),\phantom{\rule{0.166667em}{0ex}}x>0,$$
- Lognormal distribution $\mathrm{L}\mathrm{N}(\mu ,\varphi )$ with density$$f(x;\mu ,\varphi )=exp\left(-{(ln\left(x\right)-\mu )}^{2}/\left(2{\varphi}^{2}\right)\right)/\left(\varphi x\sqrt{2\pi}\right),\phantom{\rule{0.166667em}{0ex}}\mu \in R,\varphi >0,\phantom{\rule{0.166667em}{0ex}}x>0,$$Data from $\mathrm{L}\mathrm{N}(\mu ,\varphi )$ with $(\mu ,\varphi )=(0,0.5)$, $(0,0.6)$, $(0,1)$, $(0,1.4)$, $(0,2)$, $(0,3)$, $(0,5)$, $(0.5,1)$ were considered, where the last setting corresponds to the lognormal distribution with mean e and variance ${e}^{3}-{e}^{2}$.
- Half-Cauchy distribution $\mathrm{H}\mathrm{C}(0,1)$ with density$$f\left(x\right)=\frac{2}{\pi}\frac{1}{1+{x}^{2}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}x>0.$$
- Beta distribution $\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(a,b)$ with density$$f(x;a,b)=\frac{{x}^{a-1}{(1-x)}^{b-1}}{B(a,b)},\phantom{\rule{0.166667em}{0ex}}0<x<1,\phantom{\rule{0.166667em}{0ex}}a>0,\phantom{\rule{0.166667em}{0ex}}b>0,$$
- Pareto distribution $\mathrm{P}\mathrm{a}(a,b)$ with density$$f(x;a,b)=a{b}^{a}{x}^{-(a+1)},\phantom{\rule{0.166667em}{0ex}}b>0,\phantom{\rule{0.166667em}{0ex}}a>0,\phantom{\rule{0.166667em}{0ex}}x>0,$$
- Shifted-Pareto distribution $\mathrm{S}\mathrm{P}\left(\nu \right)$ with density $\nu /{(1+x)}^{1+\nu}.$

- Test statistic ${G}_{n,a}$, where n is the sample size and $a>0$ is a tuning parameter. This test statistic was recently proposed by [28]. The corresponding test belongs to a class of weighted L2-type tests of fit to the gamma distribution. They are based on a fixed point property of a transformation connected to a Steinian characterization of the family of gamma distributions.
- Test statistics ${T}_{n,a}^{\left(1\right)}$ and ${T}_{n,a}^{\left(2\right)}$, where n is the sample size and $a>0$ is a tuning parameter, proposed by [29]. The corresponding tests belong to a class of gof tests for the gamma distribution that utilizes the empirical Laplace transform.
- The test statistic proposed by [30], which is based on the ratio of two variance estimators. It is denoted in the sequel with ${V}_{n}$.

`gamma_test`in the R package

**goft**. Finally, each of the tests under discussion was implemented using parametric bootstrap with B = 1000 bootstrap samples.

- The empirical size of the test proposed in this paper got closer to the nominal level of 0.05 as the sample size increased.
- The empirical power of the tests increased, as expected, as the sample size increased.
- No test yielded the highest power against all alternatives analyzed, i.e., no test showed uniform superiority over the others, as indeed was expected according to the theoretical results in [31].
- For the $\mathrm{I}\mathrm{G}(1,\lambda )$ model, the larger $\lambda $ was, the better the performance of our gof test was with respect to the other tests considered in the simulation study. However, for the remaining alternative distributions, tests ${G}_{n,0.5}$ and ${T}_{n,1}^{\left(1\right)}$ performed better than the proposed test.

#### 4.2. Numerical Examples for the Modified Bessel-Type NEF

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

gof | Goodness-of-fit |

NEF | Natural exponential family |

PVF | Power variance functions |

TBE | Tweedie, Bar-Lev and Enis |

VF | Variance function |

## Appendix A. Proofs

**Proof**

**of**

**Theorem**

**2.**

**Proof**

**of**

**Corollary**

**1.**

**Lemma**

**A1.**

**Proof.**

**Proof**

**of**

**Theorem**

**3.**

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**Figure 1.**Power curves of the test statistic for 21 alternative distributions, where the null distribution is gamma. The variance M was estimated using the MLE. This figure corresponds to Table 1 in the main text ($n=50$; $\alpha =0.05$).

**Figure 2.**Power curves of the test statistic for 21 alternative distributions, where the null distribution is gamma. The variance M was estimated using the MLE. This figure corresponds to Table 2 in the main text ($n=100$; $\alpha =0.05$).

**Table 1.**Percentage of 10,000 Monte Carlo samples declared to be significant by various tests for the gamma distribution ($n=50,\alpha =0.05$). MLE $\widehat{\mathbf{\nu}}$ was used.

Alternative | Asym | Btstr | ${\mathit{G}}_{\mathit{n},0.5}$ | ${\mathit{T}}_{\mathit{n},1}^{\left(1\right)}$ | ${\mathit{T}}_{\mathit{n},4}^{\left(2\right)}$ | Goft |
---|---|---|---|---|---|---|

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(2,3)$ | 3.15 | 4.00 | 5.12 | 5.28 | 4.84 | 1.56 |

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(2,2)$ | 3.28 | 4.28 | 5.34 | 5.33 | 5.24 | 1.46 |

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(2,1)$ | 3.23 | 4.17 | 5.12 | 5.31 | 5.31 | 1.80 |

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(0.5,1)$ | 1.78 | 2.90 | 4.56 | 5.37 | 4.69 | 1.27 |

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(0.5,2)$ | 1.57 | 2.82 | 4.21 | 5.05 | 4.56 | 1.12 |

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(2,5)$ | 3.16 | 4.04 | 5.64 | 5.52 | 5.56 | 1.62 |

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(1,1)$ | 2.57 | 3.59 | 5.00 | 5.50 | 5.31 | 1.32 |

$\mathrm{I}\mathrm{G}(1,0.5)$ | 30.80 | 35.29 | 83.47 | 88.62 | 84.22 | 49.31 |

$\mathrm{I}\mathrm{G}(1,1)$ | 27.18 | 30.89 | 64.42 | 66.77 | 64.51 | 34.00 |

$\mathrm{I}\mathrm{G}(1,2)$ | 23.40 | 26.31 | 41.09 | 42.08 | 32.27 | 22.32 |

$\mathrm{I}\mathrm{G}(1,4)$ | 19.15 | 20.69 | 20.85 | 22.40 | 6.68 | 13.19 |

$\mathrm{I}\mathrm{G}(1,8)$ | 15.05 | 15.73 | 7.76 | 12.75 | 0.04 | 8.07 |

$\mathrm{I}\mathrm{G}(1,10)$ | 13.87 | 14.21 | 5.05 | 10.54 | 0.01 | 6.76 |

$\mathrm{L}\mathrm{N}(0,0.5)$ | 22.45 | 24.10 | 21.82 | 24.18 | 8.14 | 16.03 |

$\mathrm{L}\mathrm{N}(0,0.6)$ | 25.48 | 27.98 | 31.16 | 32.31 | 19.24 | 21.47 |

$\mathrm{L}\mathrm{N}(0,1)$ | 33.39 | 37.68 | 61.40 | 64.00 | 62.02 | 40.58 |

$\mathrm{L}\mathrm{N}(0,1.4)$ | 35.75 | 40.74 | 76.55 | 82.31 | 78.36 | 53.79 |

$\mathrm{L}\mathrm{N}(0,2)$ | 37.94 | 43.83 | 72.60 | 92.81 | 88.46 | 63.86 |

$\mathrm{L}\mathrm{N}(0,3)$ | 77.09 | 80.80 | 1.24 | 95.38 | 91.10 | 66.60 |

$\mathrm{L}\mathrm{N}(0,5)$ | 93.02 | 94.62 | 0.00 | 94.43 | 89.16 | 56.50 |

$\mathrm{L}\mathrm{N}(0.5,1)$ | 33.51 | 37.93 | 61.50 | 63.96 | 62.18 | 40.60 |

$\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(2,0.5)$ | 78.96 | 81.64 | 99.63 | 99.82 | 48.00 | 99.72 |

$\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(2,2)$ | 0.00 | 0.01 | 64.88 | 64.49 | 45.64 | 44.75 |

$\mathrm{P}\mathrm{a}(1,1)$ | 81.80 | 84.91 | 96.43 | 99.98 | 70.99 | 98.43 |

$\mathrm{P}\mathrm{a}(1,2)$ | 95.97 | 96.44 | 99.94 | 99.89 | 6.60 | 98.17 |

$\mathrm{H}\mathrm{C}(0,1)$ | 57.89 | 63.01 | 83.22 | 90.19 | 87.04 | 79.14 |

$\mathrm{S}\mathrm{P}\left(1\right)$ | 51.95 | 57.19 | 77.58 | 92.49 | 91.34 | 76.93 |

$\mathrm{S}\mathrm{P}\left(2\right)$ | 33.99 | 38.51 | 58.67 | 59.07 | 59.05 | 44.37 |

**Table 2.**Percentage of 10,000 Monte Carlo samples declared to be significant by various tests for the gamma distribution ($n=100,\alpha =0.05$). MLE $\widehat{\mathbf{\nu}}$ was used.

Alternative | Asym | Btstr | ${\mathit{G}}_{\mathit{n},0.5}$ | ${\mathit{T}}_{\mathit{n},1}^{\left(1\right)}$ | ${\mathit{T}}_{\mathit{n},4}^{\left(2\right)}$ | Goft |
---|---|---|---|---|---|---|

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(2,3)$ | 3.54 | 4.31 | 5.14 | 5.03 | 4.94 | 2.06 |

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(2,2)$ | 3.73 | 4.62 | 4.96 | 5.11 | 5.47 | 2.04 |

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(2,1)$ | 4.17 | 5.18 | 5.21 | 5.16 | 5.11 | 2.33 |

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(0.5,1)$ | 2.25 | 3.69 | 4.84 | 5.38 | 5.24 | 1.97 |

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(0.5,2)$ | 2.62 | 4.15 | 4.52 | 5.04 | 4.68 | 2.07 |

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(2,5)$ | 4.05 | 4.74 | 5.53 | 5.50 | 5.53 | 2.31 |

$\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(1,1)$ | 3.07 | 4.18 | 4.78 | 5.34 | 5.48 | 2.01 |

$\mathrm{I}\mathrm{G}(1,0.5)$ | 47.21 | 52.27 | 99.01 | 99.67 | 92.70 | 84.76 |

$\mathrm{I}\mathrm{G}(1,1)$ | 41.67 | 46.02 | 92.50 | 94.77 | 84.11 | 67.27 |

$\mathrm{I}\mathrm{G}(1,2)$ | 37.00 | 40.15 | 71.20 | 74.37 | 50.81 | 47.06 |

$\mathrm{I}\mathrm{G}(1,4)$ | 30.66 | 32.73 | 41.35 | 45.33 | 5.08 | 29.07 |

$\mathrm{I}\mathrm{G}(1,8)$ | 23.39 | 24.15 | 17.39 | 23.86 | 0.01 | 16.69 |

$\mathrm{I}\mathrm{G}(1,10)$ | 21.10 | 21.68 | 12.27 | 19.75 | 0.00 | 13.89 |

$\mathrm{L}\mathrm{N}(0,0.5)$ | 35.94 | 38.17 | 40.71 | 45.37 | 6.72 | 33.24 |

$\mathrm{L}\mathrm{N}(0,0.6)$ | 42.02 | 44.92 | 55.53 | 59.12 | 27.88 | 44.23 |

$\mathrm{L}\mathrm{N}(0,1)$ | 54.04 | 57.73 | 89.24 | 91.49 | 77.23 | 73.54 |

$\mathrm{L}\mathrm{N}(0,1.4)$ | 57.54 | 62.26 | 97.10 | 98.49 | 87.61 | 86.47 |

$\mathrm{L}\mathrm{N}(0,2)$ | 58.01 | 64.14 | 77.64 | 99.77 | 91.71 | 93.12 |

$\mathrm{L}\mathrm{N}(0,3)$ | 92.18 | 93.59 | 0.05 | 99.97 | 99.51 | 95.82 |

$\mathrm{L}\mathrm{N}(0,5)$ | 98.36 | 98.77 | 0.00 | 99.96 | 99.61 | 93.74 |

$\mathrm{L}\mathrm{N}(0.5,1)$ | 54.00 | 58.51 | 89.12 | 91.41 | 77.58 | 73.87 |

$\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(2,0.5)$ | 99.03 | 99.25 | 100.00 | 100.00 | 54.70 | 100.00 |

$\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(2,2)$ | 0.63 | 5.42 | 92.12 | 92.43 | 78.20 | 90.08 |

$\mathrm{P}\mathrm{a}(1,1)$ | 95.05 | 95.93 | 94.74 | 100.00 | 44.17 | 100.00 |

$\mathrm{P}\mathrm{a}(1,2)$ | 99.91 | 99.92 | 100.00 | 100.00 | 0.49 | 100.00 |

$\mathrm{H}\mathrm{C}(0,1)$ | 81.95 | 84.71 | 88.58 | 99.31 | 71.40 | 98.00 |

$\mathrm{S}\mathrm{P}\left(1\right)$ | 75.69 | 79.69 | 78.12 | 99.77 | 80.91 | 97.55 |

$\mathrm{S}\mathrm{P}\left(2\right)$ | 57.81 | 62.56 | 85.02 | 84.67 | 72.94 | 76.30 |

**Table 3.**Percentage of 10,000 Monte Carlo samples declared to be significant by various tests for the modified Bessel distribution $\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}(\mu ,\varphi )$. MLE $\widehat{\mathbf{\nu}}$ was used, and $\alpha =0.05$.

Alternative | Asym | Btstr | ||
---|---|---|---|---|

$\mathbf{n}=\mathbf{20}$ | $\mathbf{n}=\mathbf{50}$ | $\mathbf{n}=\mathbf{20}$ | $\mathbf{n}=\mathbf{50}$ | |

$\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}(1,1)$ | 1.15 | 2.70 | 2.10 | 3.70 |

$\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}(2,1)$ | 0.61 | 1.71 | 1.10 | 2.70 |

$\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}(0.5,1)$ | 1.63 | 2.64 | 2.50 | 3.20 |

$\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}(1,2)$ | 0.35 | 3.26 | 0.40 | 3.70 |

$\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}(0.5,2)$ | 0.66 | 2.09 | 1.20 | 3.10 |

$\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}(1,0.5)$ | 2.30 | 3.28 | 3.60 | 4.10 |

**Table 4.**IIT Kanpur and vinyl chloride data sets investigated in Section 4.2.

IIT Kanpur Data | Vinyl Chloride Data | ||||||||
---|---|---|---|---|---|---|---|---|---|

29 | 25 | 50 | 15 | 13 | 5.1 | 1.2 | 1.3 | 0.6 | 0.5 |

27 | 15 | 18 | 7 | 7 | 2.4 | 0.5 | 1.1 | 8 | 0.8 |

8 | 19 | 12 | 18 | 5 | 0.4 | 0.6 | 0.9 | 0.4 | 2 |

21 | 15 | 86 | 21 | 15 | 0.5 | 5.3 | 3.2 | 2.7 | 2.9 |

14 | 39 | 15 | 14 | 70 | 2.5 | 2.3 | 1 | 0.2 | 0.1 |

44 | 6 | 23 | 58 | 19 | 0.1 | 1.8 | 0.9 | 2 | 4 |

50 | 23 | 11 | 6 | 34 | 6.8 | 1.2 | 0.4 | 0.2 | |

18 | 28 | 34 | 12 | 37 | |||||

4 | 60 | 20 | 23 | 40 | |||||

65 | 19 | 31 |

IIT Kanpur Data | Vinyl Chloride Data | |
---|---|---|

Sample size | 48 | 34 |

Mean | 25.90 | 1.88 |

Median | 19.5 | 1.15 |

Standard deviation | 18.60 | 1.95 |

Inter-quartile range | 20 | 1.98 |

**Table 6.**Results of p-values for the test statistic in the special case of ${\gamma}_{0}=2.5$ for the two data sets.

IIT Kanpur Data | Vinyl Chloride Data | |
---|---|---|

Asym | 0.767 | 0.849 |

Btstr | 0.493 | 0.431 |

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## Share and Cite

**MDPI and ACS Style**

Bar-Lev, S.K.; Batsidis, A.; Einbeck, J.; Liu, X.; Ren, P.
Cumulant-Based Goodness-of-Fit Tests for the Tweedie, Bar-Lev and Enis Class of Distributions. *Mathematics* **2023**, *11*, 1603.
https://doi.org/10.3390/math11071603

**AMA Style**

Bar-Lev SK, Batsidis A, Einbeck J, Liu X, Ren P.
Cumulant-Based Goodness-of-Fit Tests for the Tweedie, Bar-Lev and Enis Class of Distributions. *Mathematics*. 2023; 11(7):1603.
https://doi.org/10.3390/math11071603

**Chicago/Turabian Style**

Bar-Lev, Shaul K., Apostolos Batsidis, Jochen Einbeck, Xu Liu, and Panpan Ren.
2023. "Cumulant-Based Goodness-of-Fit Tests for the Tweedie, Bar-Lev and Enis Class of Distributions" *Mathematics* 11, no. 7: 1603.
https://doi.org/10.3390/math11071603