# Modelling Coronavirus and Larvae Pyrausta Data: A Discrete Binomial Exponential II Distribution with Properties, Classical and Bayesian Estimation

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## Abstract

**:**

## 1. Introduction

## 2. The DBiExII Distribution

## 3. Distributional Statistics

#### 3.1. Mode

#### 3.2. Moments, Skewness, Kurtosis and Index of Dispersion

## 4. Parameter Estimation

#### 4.1. Point Estimation through Maximum Likelihood Approach

#### 4.2. Asymptotic Confidence Interval

#### 4.3. Bayesian Estimation

- Plug in with the initial values of p and $\theta $, as $({p}^{\left(0\right)},{\theta}^{\left(0\right)})$.
- Start with $j=1$.
- Generate ${p}^{\left(j\right)}$ from the conditional posterior distribution in Equation (36), through MH algorithm with normal proposal distribution.
- Generate ${\theta}^{\left(j\right)}$ from the conditional posterior density in Equation (37) using MH algorithm with normal proposal distribution.
- Set $j=j+1$.
- Repeat the steps 3–5, a large number of times, say N times, and obtain ${p}^{\left(j\right)}$ and ${\theta}^{\left(j\right)}$, $j=1,2,3,\cdots ,N$.

#### 4.4. Highest Posterior Density (HPD) Credible Interval

## 5. Numerical Illustration through Simulated Data

- Step 1.
- Generate 10,000 samples of size $n\in ${20, 40, 60, 100} from DBiExII distribution using Equation (6) with the arbitrary sets of parameters $(p,\theta )\in ${(0.3, 0.3), (0.3, 0.8), (0.8, 0.3), (0.5, 0.5), (0.8, 0.8)}.
- Step 2.
- Compute the MLEs, Bayes estimates (with NIPs and IPs), 95% asymptotic and HPD confidence intervals for each of the 10,000 samples.In the case of Bayesian estimation, it is important to note that we have calculated the Bayes estimates with IPs and NIPs under SELF. In IPs, the prior densities for the parameters p and $\theta $ are taken to be $Bet{a}_{1}({a}_{1},{b}_{1})$ and $Bet{a}_{1}({a}_{2},{b}_{2})$ distributions, respectively. The hyper-parameters in these prior densities have been selected in such a manner that the mean of a parameter’s prior density is almost equal to the corresponding assumed value of that parameter, whereas, all hyper-parameters are set to 1 for Bayes estimation under NIPs. Using the algorithm described in Section 4.3, we produced 21,000 realizations of the Markov chain of p and $\theta $ from their full conditional posterior distributions in order to calculate the required Bayesian quantities. To counteract the impact of the parameters beginning values, the first 1000 burn-in values for each chain have been eliminated. Additionally, we have stored every tenth observation to reduce the autocorrelation between draws. By plotting MCMC runs, posterior densities, and the autocorrelation function for each pair of true parameters, the convergence of the produced chains is investigated. For the sake of simplicity, we have only included these graphs for $(p,\theta )$ = (0.3, 0.3) in Figure 3. After the convergence testing, we have utilized simulated posterior samples to compute Bayes estimates and HPD intervals.
- Step 3.
- Compute the average estimate (AE), root mean squared error (RMSE), and average absolute bias (AB) for MLEs and Bayes estimates (with SELF under IPs and NIPs), whereas for 95% asymptotic and HPD confidence intervals, we calculate average lower confidence limit (ALCL), average upper confidence limit (AUCL), average width (AW), and coverage probability (CP).

- i.
- The RMSE of both estimators decreases with increasing sample size. This validates the consistency property of the estimators. Moreover, as the value of n increases, the absolute bias decreases toward zero.
- ii.
- Bayes estimators obtained under IPs show smaller RMSE as compared to the MLEs and Bayes estimators with NIPs.
- iii.
- Under NIPs, the Bayesian method becomes the first choice of estimation in the absence of prior information. This is due to the fact that the Bayes estimates under NIPs have the smallest estimation errors when compared to the MLEs.
- iv.
- The average width of classical and Bayesian intervals becomes smaller as the sample size increases. Moreover, the HPD interval with IPs outperforms asymptotic and HPD intervals under NIPs in terms of the width of the intervals. The CPs are close to the corresponding nominal levels in both classical and Bayesian intervals.
- v.
- In both estimation processes, the estimation of $\theta $ is more sensitive than the estimation of p since it results in more estimation error relative to the other parameter.

## 6. Data Analysis

#### 6.1. Data Set I: COVID-19

#### 6.2. Data Set II: Larvae Pyrausta

#### 6.3. Bayesian Estimation for Data Sets I and II

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Parameter | Measure | |||||
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$\mathit{p}$ | $\mathit{\theta}$ | Mean | Variance | Skewness | Kurtosis | IxD |

$0.01$ | $0.01$ | $0.010337$ | $0.010443$ | $9.987279$ | $104.743575$ | $1.010326$ |

$0.1$ | $0.012573$ | $0.012719$ | $9.072254$ | $87.147341$ | $1.011601$ | |

$0.3$ | $0.018392$ | $0.018593$ | $7.493631$ | $60.338393$ | $1.010916$ | |

$0.5$ | $0.025763$ | $0.025936$ | $6.293304$ | $43.140116$ | $1.006720$ | |

$0.7$ | $0.035401$ | $0.035374$ | $5.310417$ | $31.192865$ | $0.999238$ | |

$0.9$ | $0.048544$ | $0.047945$ | $4.457520$ | $22.420924$ | $0.987655$ | |

$0.99$ | $0.056157$ | $0.055068$ | $4.100374$ | $19.193921$ | $0.980612$ | |

$0.1$ | $0.01$ | $0.112539$ | $0.125201$ | $3.462044$ | $16.984848$ | $1.112503$ |

$0.1$ | $0.126072$ | $0.141519$ | $3.308182$ | $15.870692$ | $1.122521$ | |

$0.3$ | $0.161276$ | $0.182253$ | $2.946320$ | $13.237950$ | $1.130068$ | |

$0.5$ | $0.205867$ | $0.230291$ | $2.575399$ | $10.759760$ | $1.118638$ | |

$0.7$ | $0.264179$ | $0.287110$ | $2.201012$ | $8.584148$ | $1.086800$ | |

$0.9$ | $0.343695$ | $0.353631$ | $1.819239$ | $6.743941$ | $1.028908$ | |

$0.99$ | $0.3897517$ | $0.386376$ | $0.386376$ | $6.031577$ | $0.991341$ | |

$0.3$ | $0.01$ | $0.432275$ | $0.619110$ | $2.369474$ | $10.613745$ | $1.432212$ |

$0.1$ | $0.467367$ | $0.682789$ | $2.322009$ | $10.336483$ | $1.460926$ | |

$0.3$ | $0.558652$ | $0.836902$ | $2.161759$ | $9.330162$ | $1.498074$ | |

$0.5$ | $0.674280$ | $1.008188$ | $1.954587$ | $8.141058$ | $1.495206$ | |

$0.7$ | $0.825485$ | $1.191830$ | $1.721058$ | $7.001209$ | $1.443793$ | |

$0.9$ | $1.031674$ | $1.368560$ | $1.478955$ | $6.065819$ | $1.326542$ | |

$0.99$ | $1.151101$ | $1.432036$ | $1.376903$ | $5.765703$ | $1.244058$ | |

$0.5$ | $0.01$ | $1.006966$ | $2.020850$ | $2.119947$ | $9.493563$ | $2.006869$ |

$0.1$ | $1.072962$ | $2.213565$ | $2.095548$ | $9.340058$ | $2.063039$ | |

$0.3$ | $1.244640$ | $2.674071$ | $1.982932$ | $8.611448$ | $2.148469$ | |

$0.5$ | $1.462098$ | $3.172759$ | $1.819516$ | $7.675631$ | $2.170004$ | |

$0.7$ | $1.746466$ | $3.682186$ | $1.630544$ | $6.767094$ | $2.108364$ | |

$0.9$ | $2.134240$ | $4.116220$ | $1.447103$ | $6.071141$ | $1.928658$ | |

$0.99$ | $2.358842$ | $4.230074$ | $1.386767$ | $5.901186$ | $1.793283$ | |

$0.9$ | $0.01$ | $9.047650$ | $90.90308$ | $2.002601$ | $9.009693$ | $10.04714$ |

$0.1$ | $9.499076$ | $99.23336$ | $1.988878$ | $8.906269$ | $10.44663$ | |

$0.3$ | $10.67337$ | $118.9939$ | $1.899065$ | $8.295979$ | $11.14866$ | |

$0.5$ | $12.16081$ | $140.0647$ | $1.757718$ | $7.474355$ | $11.51770$ | |

$0.7$ | $14.10593$ | $160.9421$ | $1.592996$ | $6.674835$ | $11.40953$ | |

$0.9$ | $16.75836$ | $177.2167$ | $1.445560$ | $6.099409$ | $10.57482$ | |

$0.99$ | $18.29467$ | $180.2078$ | $1.413765$ | $5.998139$ | $9.850289$ | |

$0.99$ | $0.01$ | $99.49999$ | $9999.247$ | $2.000707$ | $9.470644$ | $100.49496$ |

$0.1$ | $104.23675$ | $10914.68$ | $1.985322$ | $8.289761$ | $104.71048$ | |

$0.3$ | $116.55852$ | $13085.84$ | $1.896061$ | $8.306480$ | $112.26846$ | |

$0.5$ | $132.16610$ | $15400.08$ | $15400.08$ | $6.596502$ | $116.52070$ | |

$0.7$ | $152.57602$ | $17691.24$ | $1.593731$ | $9.967532$ | $115.95033$ | |

$0.9$ | $180.40772$ | $19472.97$ | $1.444991$ | $17.831995$ | $107.93868$ | |

$0.99$ | $196.52806$ | $19796.40$ | $1.409224$ | $-9.710393$ | $100.73070$ |

$(\mathit{p},\mathit{\theta})$ | n | AE$\left(\mathit{p}\right)$ | RMSE$\left(\mathit{p}\right)$ | AB$\left(\mathit{p}\right)$ | AE$\left(\mathit{\theta}\right)$ | RMSE$\left(\mathit{\theta}\right)$ | AB$\left(\mathit{\theta}\right)$ |
---|---|---|---|---|---|---|---|

(0.3, 0.3) | 25 | 0.2504 | 0.1169 | 0.0978 | 0.4972 | 0.4472 | 0.3982 |

50 | 0.2673 | 0.0960 | 0.0791 | 0.4282 | 0.3878 | 0.3420 | |

100 | 0.2812 | 0.0774 | 0.0643 | 0.3693 | 0.3278 | 0.2900 | |

200 | 0.2916 | 0.0620 | 0.0523 | 0.3271 | 0.2755 | 0.2440 | |

(0.3, 0.8) | 25 | 0.3072 | 0.0924 | 0.0734 | 0.7318 | 0.3242 | 0.2404 |

50 | 0.3065 | 0.0741 | 0.0578 | 0.7522 | 0.2702 | 0.1951 | |

100 | 0.3033 | 0.0557 | 0.0432 | 0.7772 | 0.1987 | 0.1435 | |

200 | 0.3014 | 0.0393 | 0.0308 | 0.7899 | 0.1342 | 0.1003 | |

(0.8, 0.3) | 25 | 0.7696 | 0.0682 | 0.0521 | 0.4364 | 0.3751 | 0.3348 |

50 | 0.7806 | 0.0511 | 0.0400 | 0.3852 | 0.3287 | 0.2949 | |

100 | 0.7900 | 0.0380 | 0.0308 | 0.3361 | 0.2816 | 0.2536 | |

200 | 0.7958 | 0.0297 | 0.0247 | 0.3001 | 0.2447 | 0.2183 | |

(0.5, 0.5) | 25 | 0.4734 | 0.1072 | 0.0874 | 0.5325 | 0.3600 | 0.3194 |

50 | 0.4895 | 0.0844 | 0.0699 | 0.4624 | 0.3235 | 0.2780 | |

100 | 0.4985 | 0.0663 | 0.0546 | 0.4691 | 0.2802 | 0.2297 | |

200 | 0.5031 | 0.0533 | 0.0431 | 0.4907 | 0.2389 | 0.1857 | |

(0.8, 0.8) | 25 | 0.7991 | 0.0419 | 0.0330 | 0.7361 | 0.2762 | 0.1899 |

50 | 0.7991 | 0.0315 | 0.0248 | 0.7667 | 0.2127 | 0.1435 | |

100 | 0.7990 | 0.0223 | 0.0174 | 0.7863 | 0.1376 | 0.0958 | |

200 | 0.8001 | 0.0153 | 0.0121 | 0.7906 | 0.0894 | 0.0663 |

$(\mathit{p},\mathit{\theta})$ | n | ALCL$\left(\mathit{p}\right)$ | AUCL$\left(\mathit{p}\right)$ | AW$\left(\mathit{p}\right)$ | CP$\left(\mathit{p}\right)$ | ALCL$\left(\mathit{\theta}\right)$ | AUCL$\left(\mathit{\theta}\right)$ | AW$\left(\mathit{\theta}\right)$ | CP$\left(\mathit{\theta}\right)$ |
---|---|---|---|---|---|---|---|---|---|

(0.3, 0.3) | 25 | 0.0182 | 0.5754 | 0.5572 | 0.9014 | 0.0463 | 0.9963 | 0.9501 | 0.9107 |

50 | 0.0422 | 0.5385 | 0.4963 | 0.9493 | 0.0775 | 0.9892 | 0.9117 | 0.9431 | |

100 | 0.0785 | 0.5095 | 0.4310 | 0.9577 | 0.0808 | 0.9779 | 0.8971 | 0.9555 | |

200 | 0.1175 | 0.4815 | 0.3640 | 0.9584 | 0.0746 | 0.9644 | 0.8898 | 0.9632 | |

(0.3, 0.8) | 25 | 0.0739 | 0.5565 | 0.4827 | 0.9237 | 0.2095 | 0.9997 | 0.7902 | 0.9495 |

50 | 0.1343 | 0.4814 | 0.3471 | 0.9386 | 0.3320 | 0.9993 | 0.6673 | 0.9586 | |

100 | 0.1856 | 0.4215 | 0.2358 | 0.9323 | 0.4398 | 0.9997 | 0.5598 | 0.9585 | |

200 | 0.2225 | 0.3804 | 0.1579 | 0.9561 | 0.5386 | 0.9950 | 0.4564 | 0.9462 | |

(0.8, 0.3) | 25 | 0.5989 | 0.9095 | 0.3106 | 0.9033 | 0.0966 | 0.9739 | 0.8773 | 0.9487 |

50 | 0.6464 | 0.8962 | 0.2498 | 0.9305 | 0.1191 | 0.9883 | 0.8691 | 0.9383 | |

100 | 0.6818 | 0.8863 | 0.2045 | 0.9446 | 0.1378 | 0.9954 | 0.8577 | 0.9682 | |

200 | 0.7093 | 0.8757 | 0.1664 | 0.9645 | 0.0836 | 0.9180 | 0.8344 | 0.9783 | |

(0.5, 0.5) | 25 | 0.2060 | 0.7288 | 0.5228 | 0.9302 | 0.1591 | 0.9978 | 0.8387 | 0.9542 |

50 | 0.2715 | 0.6981 | 0.4266 | 0.9231 | 0.1609 | 0.9947 | 0.8337 | 0.9681 | |

100 | 0.3275 | 0.6642 | 0.3367 | 0.9425 | 0.1619 | 0.9905 | 0.8286 | 0.9592 | |

200 | 0.3735 | 0.6298 | 0.2563 | 0.9592 | 0.1796 | 0.9499 | 0.7703 | 0.9536 | |

(0.8, 0.8) | 25 | 0.6981 | 0.8908 | 0.1927 | 0.9655 | 0.3656 | 0.9997 | 0.6341 | 0.9359 |

50 | 0.7332 | 0.8624 | 0.1292 | 0.9571 | 0.4697 | 0.9996 | 0.5298 | 0.9448 | |

100 | 0.7552 | 0.8423 | 0.0871 | 0.9564 | 0.5557 | 0.9972 | 0.4415 | 0.9569 | |

200 | 0.7701 | 0.8301 | 0.0600 | 0.9546 | 0.6251 | 0.9559 | 0.3309 | 0.9607 |

$(\mathit{p},\mathit{\theta})$ | n | AE$\left(\mathit{p}\right)$ | RMSE$\left(\mathit{p}\right)$ | AB$\left(\mathit{p}\right)$ | AE$\left(\mathit{\theta}\right)$ | RMSE$\left(\mathit{\theta}\right)$ | AB$\left(\mathit{\theta}\right)$ |
---|---|---|---|---|---|---|---|

(0.3, 0.3) | 25 | 0.2998 | 0.0118 | 0.0095 | 0.2998 | 0.0089 | 0.0072 |

50 | 0.2999 | 0.0110 | 0.0090 | 0.2998 | 0.0078 | 0.0063 | |

100 | 0.3002 | 0.0093 | 0.0074 | 0.3001 | 0.0065 | 0.0052 | |

200 | 0.3006 | 0.0076 | 0.0061 | 0.3005 | 0.0051 | 0.0041 | |

(0.3, 0.8) | 25 | 0.2981 | 0.0117 | 0.0094 | 0.8064 | 0.0161 | 0.0126 |

50 | 0.2987 | 0.0114 | 0.0091 | 0.8057 | 0.0141 | 0.0111 | |

100 | 0.2988 | 0.0106 | 0.0085 | 0.8062 | 0.0124 | 0.0099 | |

200 | 0.2994 | 0.0086 | 0.0069 | 0.8055 | 0.0107 | 0.0086 | |

(0.8, 0.3) | 25 | 0.7986 | 0.0122 | 0.0097 | 0.2962 | 0.0064 | 0.0051 |

50 | 0.7989 | 0.0116 | 0.0094 | 0.2963 | 0.0061 | 0.0049 | |

100 | 0.7992 | 0.0116 | 0.0093 | 0.2998 | 0.0057 | 0.0046 | |

200 | 0.7993 | 0.0098 | 0.0078 | 0.3012 | 0.0056 | 0.0046 | |

(0.5, 0.5) | 25 | 0.4995 | 0.0115 | 0.0092 | 0.4999 | 0.0115 | 0.0090 |

50 | 0.4997 | 0.0115 | 0.0091 | 0.4999 | 0.0099 | 0.0079 | |

100 | 0.5000 | 0.0098 | 0.0079 | 0.5002 | 0.0084 | 0.0067 | |

200 | 0.5005 | 0.0081 | 0.0065 | 0.5004 | 0.0065 | 0.0053 | |

(0.8, 0.8) | 25 | 0.7988 | 0.0123 | 0.0099 | 0.8054 | 0.0190 | 0.0151 |

50 | 0.7990 | 0.0123 | 0.0099 | 0.8051 | 0.0154 | 0.0123 | |

100 | 0.7992 | 0.0107 | 0.0086 | 0.8035 | 0.0124 | 0.0106 | |

200 | 0.7992 | 0.0090 | 0.0073 | 0.8033 | 0.0097 | 0.0086 |

$(\mathit{p},\mathit{\theta})$ | n | ALCL$\left(\mathit{p}\right)$ | AUCL$\left(\mathit{p}\right)$ | AW$\left(\mathit{p}\right)$ | CP$\left(\mathit{p}\right)$ | ALCL$\left(\mathit{\theta}\right)$ | AUCL$\left(\mathit{\theta}\right)$ | AW$\left(\mathit{\theta}\right)$ | CP$\left(\mathit{\theta}\right)$ |
---|---|---|---|---|---|---|---|---|---|

(0.3, 0.3) | 25 | 0.2543 | 0.3462 | 0.0919 | 0.9231 | 0.2132 | 0.3868 | 0.1736 | 0.9061 |

50 | 0.2561 | 0.3438 | 0.0877 | 0.9389 | 0.2133 | 0.3859 | 0.1725 | 0.9394 | |

100 | 0.2601 | 0.3411 | 0.0809 | 0.9469 | 0.2151 | 0.3860 | 0.1709 | 0.9482 | |

200 | 0.2640 | 0.3358 | 0.0718 | 0.9604 | 0.2156 | 0.3843 | 0.1687 | 0.9597 | |

(0.3, 0.8) | 25 | 0.2552 | 0.3451 | 0.0899 | 0.8974 | 0.7150 | 0.8856 | 0.1706 | 0.8968 |

50 | 0.2574 | 0.3420 | 0.0846 | 0.9151 | 0.7162 | 0.8833 | 0.1671 | 0.9289 | |

100 | 0.2615 | 0.3383 | 0.0768 | 0.9394 | 0.7179 | 0.8803 | 0.1624 | 0.949 | |

200 | 0.2662 | 0.3336 | 0.0673 | 0.9451 | 0.7216 | 0.8772 | 0.1556 | 0.9679 | |

(0.8, 0.3) | 25 | 0.7596 | 0.8386 | 0.0789 | 0.9341 | 0.2133 | 0.3868 | 0.1736 | 0.9281 |

50 | 0.7647 | 0.8333 | 0.0686 | 0.9409 | 0.2132 | 0.3863 | 0.1731 | 0.9326 | |

100 | 0.7703 | 0.8271 | 0.0569 | 0.9498 | 0.2136 | 0.3863 | 0.1727 | 0.9649 | |

200 | 0.7758 | 0.8212 | 0.0454 | 0.9562 | 0.2142 | 0.3856 | 0.1714 | 0.9585 | |

(0.5, 0.5) | 25 | 0.4545 | 0.5454 | 0.0909 | 0.9457 | 0.4139 | 0.5865 | 0.1726 | 0.9149 |

50 | 0.4568 | 0.5427 | 0.0859 | 0.9462 | 0.4145 | 0.5854 | 0.1709 | 0.9446 | |

100 | 0.4614 | 0.5396 | 0.0783 | 0.9574 | 0.4162 | 0.5845 | 0.1683 | 0.9527 | |

200 | 0.4652 | 0.5338 | 0.0687 | 0.9656 | 0.4173 | 0.5820 | 0.1648 | 0.9533 | |

(0.8, 0.8) | 25 | 0.7620 | 0.8361 | 0.0742 | 0.9346 | 0.7153 | 0.8848 | 0.1694 | 0.9353 |

50 | 0.7671 | 0.8303 | 0.0632 | 0.9451 | 0.7171 | 0.8825 | 0.1654 | 0.9492 | |

100 | 0.7730 | 0.8247 | 0.0516 | 0.9563 | 0.7200 | 0.8788 | 0.1587 | 0.9560 | |

200 | 0.7787 | 0.8196 | 0.0409 | 0.9512 | 0.7259 | 0.8745 | 0.1486 | 0.9512 |

$(\mathit{p},\mathit{\theta})$ | n | AE$\left(\mathit{p}\right)$ | RMSE$\left(\mathit{p}\right)$ | AB$\left(\mathit{p}\right)$ | AE$\left(\mathit{\theta}\right)$ | RMSE$\left(\mathit{\theta}\right)$ | AB$\left(\mathit{\theta}\right)$ |
---|---|---|---|---|---|---|---|

(0.3, 0.3) | 25 | 0.2982 | 0.0110 | 0.0093 | 0.2963 | 0.0088 | 0.0071 |

50 | 0.2989 | 0.0107 | 0.0086 | 0.2966 | 0.0076 | 0.0063 | |

100 | 0.2991 | 0.0091 | 0.0073 | 0.2967 | 0.0060 | 0.0047 | |

200 | 0.2992 | 0.0073 | 0.0059 | 0.2968 | 0.0049 | 0.0038 | |

(0.3, 0.8) | 25 | 0.2997 | 0.0111 | 0.0089 | 0.7993 | 0.0153 | 0.0122 |

50 | 0.2997 | 0.0111 | 0.0089 | 0.7996 | 0.0136 | 0.0109 | |

100 | 0.2999 | 0.0104 | 0.0084 | 0.7998 | 0.0116 | 0.0093 | |

200 | 0.3001 | 0.0084 | 0.0068 | 0.8003 | 0.0089 | 0.0072 | |

(0.8, 0.3) | 25 | 0.7996 | 0.0115 | 0.0091 | 0.2999 | 0.0058 | 0.0046 |

50 | 0.7998 | 0.0112 | 0.0089 | 0.3001 | 0.0050 | 0.0040 | |

100 | 0.8005 | 0.0111 | 0.0089 | 0.3005 | 0.0049 | 0.0039 | |

200 | 0.8007 | 0.0096 | 0.0077 | 0.3002 | 0.0044 | 0.0035 | |

(0.5, 0.5) | 25 | 0.4998 | 0.0110 | 0.0084 | 0.4997 | 0.0110 | 0.0088 |

50 | 0.5120 | 0.0107 | 0.0075 | 0.4991 | 0.0092 | 0.0073 | |

100 | 0.4995 | 0.0091 | 0.0069 | 0.5001 | 0.0080 | 0.0065 | |

200 | 0.5022 | 0.0074 | 0.0060 | 0.5002 | 0.0065 | 0.0051 | |

(0.8, 0.8) | 25 | 0.7991 | 0.0120 | 0.0096 | 0.7998 | 0.0184 | 0.0146 |

50 | 0.7992 | 0.0120 | 0.0096 | 0.7999 | 0.0151 | 0.0120 | |

100 | 0.8056 | 0.0106 | 0.0084 | 0.8002 | 0.0112 | 0.0098 | |

200 | 0.8009 | 0.0088 | 0.0070 | 0.8008 | 0.0091 | 0.0077 |

$(\mathit{p},\mathit{\theta})$ | n | ALCL$\left(\mathit{p}\right)$ | AUCL$\left(\mathit{p}\right)$ | AW$\left(\mathit{p}\right)$ | CP$\left(\mathit{p}\right)$ | ALCL$\left(\mathit{\theta}\right)$ | AUCL$\left(\mathit{\theta}\right)$ | AW$\left(\mathit{\theta}\right)$ | CP$\left(\mathit{\theta}\right)$ |
---|---|---|---|---|---|---|---|---|---|

(0.3, 0.3) | 25 | 0.2534 | 0.3445 | 0.0911 | 0.9172 | 0.2129 | 0.3807 | 0.1678 | 0.9179 |

50 | 0.2548 | 0.3418 | 0.0870 | 0.9320 | 0.2129 | 0.3796 | 0.1666 | 0.9460 | |

100 | 0.2591 | 0.3394 | 0.0803 | 0.9501 | 0.2143 | 0.3794 | 0.1651 | 0.9548 | |

200 | 0.2636 | 0.3349 | 0.0713 | 0.9486 | 0.2152 | 0.3785 | 0.1632 | 0.9665 | |

(0.3, 0.8) | 25 | 0.2543 | 0.3435 | 0.0892 | 0.9061 | 0.7245 | 0.8883 | 0.1639 | 0.9082 |

50 | 0.2563 | 0.3402 | 0.0838 | 0.9282 | 0.7250 | 0.8859 | 0.1610 | 0.9222 | |

100 | 0.2607 | 0.3368 | 0.0761 | 0.9584 | 0.7277 | 0.8842 | 0.1565 | 0.9444 | |

200 | 0.2662 | 0.3328 | 0.0666 | 0.9696 | 0.7300 | 0.8800 | 0.1501 | 0.9589 | |

(0.8, 0.3) | 25 | 0.7613 | 0.8396 | 0.0783 | 0.9321 | 0.2125 | 0.3802 | 0.1678 | 0.9292 |

50 | 0.7666 | 0.8346 | 0.0680 | 0.9401 | 0.2127 | 0.3799 | 0.1672 | 0.9431 | |

100 | 0.7713 | 0.8277 | 0.0564 | 0.9562 | 0.2131 | 0.3796 | 0.1665 | 0.9448 | |

200 | 0.7772 | 0.8222 | 0.0450 | 0.9574 | 0.2136 | 0.3792 | 0.1657 | 0.9620 | |

(0.5, 0.5) | 25 | 0.4546 | 0.5448 | 0.0902 | 0.9437 | 0.4163 | 0.5833 | 0.1670 | 0.9264 |

50 | 0.4563 | 0.5416 | 0.0853 | 0.9500 | 0.4162 | 0.5819 | 0.1657 | 0.9364 | |

100 | 0.4608 | 0.5385 | 0.0778 | 0.9699 | 0.4184 | 0.5817 | 0.1634 | 0.9549 | |

200 | 0.4657 | 0.5338 | 0.0682 | 0.9662 | 0.4200 | 0.5802 | 0.1601 | 0.9686 | |

(0.8, 0.8) | 25 | 0.7626 | 0.8362 | 0.0737 | 0.9112 | 0.7238 | 0.8864 | 0.1626 | 0.9394 |

50 | 0.7678 | 0.8305 | 0.0627 | 0.9422 | 0.7255 | 0.8842 | 0.1587 | 0.9459 | |

100 | 0.7734 | 0.8247 | 0.0513 | 0.9570 | 0.7267 | 0.8793 | 0.1526 | 0.9417 | |

200 | 0.7790 | 0.8196 | 0.0406 | 0.9760 | 0.7311 | 0.8745 | 0.1434 | 0.9639 |

Distribution | Abbreviation | Author(s) |
---|---|---|

Discrete Exponential (Geometric) | DEx | - |

Generalized Discrete Exponential | GDEx | Gómez-Déniz [25] |

Discrete generalized exponential type II | DGExII | Nekoukhou et al. [26] |

Discrete Rayleigh | DR | Roy [5] |

Discrete inverse Rayleigh | DIR | Hussain and Ahmad [27] |

Discrete Bilal | DBe | Eliwa et al. [28] |

Discrete Burr–Hatke | DBH | El-Morshedy et al. [15] |

Discrete Pareto | DPa | Krishna and Pundir [9] |

Discrete inverse Weibull | DIW | Jazi et al. [29] |

Discrete Burr type II | DBX-II | Para and Jan [30] |

Discrete log-logistic | DLogL | Para and Jan [31] |

Parameter → | p | $\mathit{\theta}$ | ||||
---|---|---|---|---|---|---|

Model ↓ | MLE | Std-er | C. I | MLE | Std-er | C. I |

DBiExII | $0.5966$ | $0.0429$ | $[0.5123,0.6808]$ | $0.6436$ | $0.1674$ | $[0.3155,0.9718]$ |

DEx | $0.7039$ | $0.0225$ | $[0.6598,0.7479]$ | − | − | − |

GDEx | $0.6449$ | $0.0439$ | $[0.5588,0.7309]$ | $1.6124$ | $0.5023$ | $[0.6274,2.5969]$ |

DGExII | $0.6739$ | $0.0336$ | $[0.6079,0.7398]$ | $1.2149$ | $0.1947$ | $[0.8333,1.5966]$ |

DR | $0.9306$ | $0.0061$ | $[0.9186,0.9426]$ | − | − | − |

DIR | $0.1768$ | $0.0329$ | $[0.1122,0.2414]$ | − | − | − |

DBe | $0.7487$ | $0.0143$ | $[0.7206,0.7767]$ | − | − | − |

DBH | $0.9315$ | $0.0269$ | $[0.8789,0.9842]$ | − | − | − |

DPa | $0.4152$ | $0.0332$ | $[0.3500,0.4803]$ | − | − | − |

DIW | $0.2338$ | $0.0381$ | $[0.1591,0.3086]$ | $1.2658$ | $0.1134$ | $[1.0436,1.4879]$ |

DB-XII | $0.6225$ | $0.0487$ | $[0.5271,0.7179]$ | $2.3359$ | $0.3772$ | $[1.5967,3.0751]$ |

DLogL | $2.0210$ | $0.1890$ | $[1.6505,2.3915]$ | $1.7457$ | $0.1523$ | $[1.4472,2.0443]$ |

No. | Observed | Expected Frequency | ||||||
---|---|---|---|---|---|---|---|---|

X | Frequency | DBiExII | DEx | DR | DIR | DBe | DBH | DPa |

$\mathbf{0}$ | 32 | $31.3801$ | $36.1263$ | $8.4645$ | $21.5662$ | $19.2478$ | $65.1759$ | $55.6679$ |

$\mathbf{1}$ | 27 | $25.9175$ | $25.4287$ | $22.0300$ | $57.5405$ | $30.7382$ | $21.5347$ | $19.8889$ |

$\mathbf{2}$ | 17 | $19.7549$ | $17.8988$ | $27.6341$ | $21.5260$ | $25.5994$ | $10.6342$ | $10.3779$ |

$\mathbf{3}$ | 14 | $14.3464$ | $12.5987$ | $25.2606$ | $8.8444$ | $17.8606$ | $6.2813$ | $6.4241$ |

$\mathbf{4}$ | 8 | $10.0864$ | $8.8679$ | $18.3967$ | $4.3528$ | $11.4844$ | $4.1105$ | $4.3897$ |

$\mathbf{5}$ | 7 | $6.9287$ | $6.2420$ | $11.0489$ | $2.4366$ | $7.0553$ | $2.8746$ | $3.2002$ |

$\mathbf{6}$ | 6 | $4.6771$ | $4.3937$ | $5.5663$ | $1.4943$ | $4.2138$ | $2.1058$ | $2.4424$ |

$\mathbf{7}$ | 5 | $3.1145$ | $3.0926$ | $2.3750$ | $0.9801$ | $2.4707$ | $1.5963$ | $1.9288$ |

$\mathbf{8}$ | 5 | $2.0515$ | $2.1768$ | $0.8635$ | $0.6767$ | $1.4305$ | $1.2423$ | $1.5640$ |

$\mathbf{9}$ | 1 | $3.7429$ | $5.1745$ | $0.3604$ | $2.5824$ | $1.8993$ | $6.4444$ | $16.1161$ |

Total | $\mathbf{122}$ | $\mathbf{122}$ | $\mathbf{122}$ | $\mathbf{122}$ | $\mathbf{122}$ | $\mathbf{122}$ | $\mathbf{122}$ | $\mathbf{122}$ |

$-l$ | $\mathbf{249}.\mathbf{0626}$ | $250.3056$ | $279.9239$ | $278.0657$ | $255.5355$ | $277.0495$ | $279.8059$ | |

AIC | $\mathbf{502}.\mathbf{1251}$ | $502.6112$ | $561.8477$ | $558.1313$ | $513.0710$ | $556.0990$ | $561.6119$ | |

CAIC | $\mathbf{502}.\mathbf{2260}$ | $502.6445$ | $561.8811$ | $558.1647$ | $513.1043$ | $556.1323$ | $561.6452$ | |

${\chi}^{2}$ | $\mathbf{2}.\mathbf{2108}$ | $8.4698$ | $89.7303$ | $57.2829$ | $18.5571$ | $43.5311$ | $64.2454$ | |

p-value | $\mathbf{0}.\mathbf{8193}$ | $0.2057$ | <0.0001 | <0.0001 | $0.0023$ | <0.0001 | <0.0001 |

No. | Observed | Expected Frequency | |||||
---|---|---|---|---|---|---|---|

X | Frequency | DBiExII | GDEx | DGExII | DIW | DB-XII | DLogL |

$\mathbf{0}$ | 32 | $31.3801$ | $31.0604$ | $31.2734$ | $28.5245$ | $34.1655$ | $27.6305$ |

$\mathbf{1}$ | 27 | $25.9175$ | $25.7405$ | $27.2030$ | $38.1383$ | $35.8539$ | $32.81328$ |

$\mathbf{2}$ | 17 | $19.7549$ | $19.8882$ | $19.7995$ | $18.3045$ | $17.0802$ | $20.7934$ |

$\mathbf{3}$ | 14 | $14.3464$ | $14.5504$ | $13.8789$ | $9.9187$ | $9.0893$ | $12.3449$ |

$\mathbf{4}$ | 8 | $10.0864$ | $10.2257$ | $9.5696$ | $6.0537$ | $5.5023$ | $7.6045$ |

$\mathbf{5}$ | 7 | $6.9287$ | $6.9849$ | $6.5406$ | $4.0199$ | $3.6500$ | $4.9354$ |

$\mathbf{6}$ | 6 | $4.6771$ | $4.6791$ | $4.4475$ | $2.8368$ | $2.5837$ | $3.3619$ |

$\mathbf{7}$ | 5 | $3.1145$ | $3.0936$ | $3.0147$ | $2.0949$ | $1.9183$ | $2.3864$ |

$\mathbf{8}$ | 5 | $2.0515$ | $2.0277$ | $2.0393$ | $1.6026$ | $1.4770$ | $1.7534$ |

$\mathbf{9}$ | 1 | $3.7429$ | $3.7495$ | $4.2335$ | $10.5061$ | $10.6798$ | $8.3763$ |

Total | $\mathbf{122}$ | $\mathbf{122}$ | $\mathbf{122}$ | $\mathbf{122}$ | $\mathbf{122}$ | $\mathbf{122}$ | $\mathbf{122}$ |

$-l$ | $\mathbf{249}.\mathbf{0626}$ | $249.2000$ | $249.5807$ | $262.3222$ | $263.5383$ | $256.7394$ | |

AIC | $\mathbf{502}.\mathbf{1251}$ | $502.3999$ | $503.1614$ | $528.6444$ | $531.0766$ | $517.4788$ | |

CAIC | $\mathbf{502}.\mathbf{2260}$ | $502.5008$ | $503.2622$ | $528.7453$ | $531.1774$ | $517.5796$ | |

${\chi}^{2}$ | $\mathbf{2}.\mathbf{2108}$ | $2.3634$ | $2.3941$ | $12.3012$ | $14.1272$ | $5.5044$ | |

p-value | $\mathbf{0}.\mathbf{8193}$ | $0.7969$ | $0.7924$ | $0.0152$ | $0.0069$ | $0.2393$ |

Type ↓ Measures → | Mean | Variance | IxD | Skewness | Kurtosis |
---|---|---|---|---|---|

Theoretical | $2.37742$ | $6.41498$ | $2.69829$ | $1.66410$ | $6.95252$ |

Empirical | $2.37705$ | $5.757485$ | $2.42211$ | $0.98615$ | $3.07782$ |

Parameter → | p | $\mathit{\theta}$ | ||||
---|---|---|---|---|---|---|

Model ↓ | MLE | Std-er | C. I | MLE | Std-er | C. I |

DBiExII | $0.2576$ | $0.0377$ | $[0.1837,0.3315]$ | $0.6439$ | $0.1472$ | $[0.3556,0.9324]$ |

DEx | $0.3933$ | $0.0211$ | $[0.3518,0.4347]$ | − | − | − |

GDEx | $0.3179$ | $0.0407$ | $[0.2375,0.3971]$ | $1.5917$ | $0.3804$ | $[0.8462,2.3373]$ |

DGExII | $0.3379$ | $0.0366$ | $[0.2661,0.4096]$ | $1.3317$ | $0.2231$ | $[0.8943,1.7690]$ |

DR | $0.6216$ | $0.0175$ | $[0.5873,0.6559]$ | − | − | − |

DIR | $0.5747$ | $0.0274$ | $[0.5209,0.6285]$ | − | − | − |

DBe | $0.4809$ | $0.0149$ | $[0.4516,0.5102]$ | − | − | − |

DBH | $0.6555$ | $0.0356$ | $[0.5877,0.7232]$ | − | − | − |

DPa | $0.1913$ | $0.0182$ | $[0.1556,0.2271]$ | − | − | − |

DIW | $0.5744$ | $0.0276$ | $[0.5202,0.6285]$ | $2.0178$ | $0.1672$ | $[1.6902,2.3455]$ |

DB-XII | $0.2858$ | $0.0267$ | $[0.2335,0.3381]$ | $2.0128$ | $0.1892$ | $[1.6419,2.3836]$ |

DLogL | $0.8846$ | $0.0466$ | $[0.7933,0.9759]$ | $2.2880$ | $0.1786$ | $[1.9379,2.6381]$ |

No. | Observed | Expected Frequency | ||||||
---|---|---|---|---|---|---|---|---|

X | Frequency | DBiExII | DEx | DR | DIR | DBe | DBH | DPa |

$\mathbf{0}$ | 188 | $186.7622$ | $196.5842$ | $122.5868$ | $186.2094$ | $171.2839$ | $217.8157$ | $221.0309$ |

$\mathbf{1}$ | 83 | $88.0289$ | $77.3084$ | $153.0275$ | $95.8946$ | $108.7492$ | $59.7847$ | $50.3079$ |

$\mathbf{2}$ | 36 | $32.9622$ | $30.4022$ | $43.8937$ | $22.5578$ | $32.8374$ | $23.5898$ | $19.9369$ |

$\mathbf{3}$ | 14 | $11.1415$ | $11.9559$ | $4.3308$ | $8.3142$ | $8.4488$ | $10.8491$ | $10.0986$ |

$\mathbf{4}$ | 2 | $3.5529$ | $4.7018$ | $0.1589$ | $3.9247$ | $2.0489$ | $5.4276$ | $5.8896$ |

$\mathbf{5}$ | 1 | $1.5523$ | $3.0475$ | $0.0023$ | $7.0993$ | $0.6318$ | $6.5331$ | $16.7361$ |

Total | $\mathbf{324}$ | $\mathbf{324}$ | $\mathbf{324}$ | $\mathbf{324}$ | $\mathbf{324}$ | $\mathbf{324}$ | $\mathbf{324}$ | $\mathbf{324}$ |

$-l$ | $\mathbf{355}.\mathbf{9001}$ | $357.8779$ | $404.4854$ | $366.2275$ | $360.5431$ | $369.7014$ | $387.8939$ | |

AIC | $\mathbf{715}.\mathbf{8001}$ | $717.7558$ | $810.9708$ | $734.4551$ | $723.0862$ | $741.4027$ | $777.7877$ | |

CAIC | $\mathbf{715}.\mathbf{8375}$ | $717.7682$ | $810.9832$ | $734.4675$ | $723.0986$ | $741.4152$ | $777.8001$ | |

${\mathbf{\xd8}}^{2}$ | $\mathbf{2.1769}$ | $5.0847$ | $67.3905$ | $16.1573$ | $11.1293$ | $27.3908$ | $57.9946$ | |

pvalue | $\mathbf{0}.\mathbf{3367}$ | $0.1657$ | <0.0001 | $0.0011$ | $0.0038$ | <0.0001 | <0.0001 |

No. | Observed | Expected Frequency | |||||
---|---|---|---|---|---|---|---|

X | Frequency | DBiExII | GDEx | DGExII | DIW | DsB-XII | DLogL |

$\mathbf{0}$ | 188 | $186.7622$ | $186.2301$ | $187.1074$ | $186.0956$ | $188.0009$ | $184.5744$ |

$\mathbf{1}$ | 83 | $88.0289$ | $88.7678$ | $88.5925$ | $96.4464$ | $93.2129$ | $96.0252$ |

$\mathbf{2}$ | 36 | $32.9622$ | $32.8333$ | $31.7641$ | $22.4638$ | $24.9509$ | $24.7251$ |

$\mathbf{3}$ | 14 | $11.1415$ | $10.9727$ | $10.9246$ | $8.2235$ | $8.7052$ | $8.7293$ |

$\mathbf{4}$ | 2 | $3.5529$ | $3.5407$ | $3.7126$ | $3.8629$ | $3.7882$ | $3.9026$ |

$\mathbf{5}$ | 1 | $1.5523$ | $1.6554$ | $1.8988$ | $6.9078$ | $5.3419$ | $6.0434$ |

Total | $\mathbf{324}$ | $\mathbf{324}$ | $\mathbf{324}$ | $\mathbf{324}$ | $\mathbf{324}$ | $\mathbf{324}$ | $\mathbf{324}$ |

$-l$ | $\mathbf{355}.\mathbf{9001}$ | $356.0700$ | $356.4021$ | $366.2218$ | $362.6942$ | $363.8550$ | |

AIC | $\mathbf{715}.\mathbf{8001}$ | $716.1400$ | $716.8042$ | $736.4436$ | $729.3883$ | $731.7100$ | |

CAIC | $\mathbf{715}.\mathbf{8375}$ | $716.1774$ | $716.8416$ | $736.4810$ | $729.4257$ | $731.7474$ | |

${\mathbf{\xd8}}^{2}$ | $\mathbf{2.1769}$ | $2.4604$ | $3.0032$ | $19.7147$ | $10.5252$ | $12.0788$ | |

pvalue | $\mathbf{0}.\mathbf{3367}$ | $0.2922$ | $0.2228$ | <0.0001 | $0.0052$ | $0.0024$ |

Type ↓ Measures → | Mean | Variance | IxD | Skewness | Kurtosis |
---|---|---|---|---|---|

Theoretical | $0.64798$ | $0.88666$ | $1.36834$ | $1.83365$ | $7.44446$ |

Empirical | $0.64814$ | $0.84795$ | $1.30828$ | $1.52091$ | $5.23804$ |

Data Set | p | $\mathit{\theta}$ | ||
---|---|---|---|---|

Estimate (PStd-er) | HPD (Width) | Estimate (PStd-er) | HPD (Width) | |

Data set I | 0.5970 (0.0179) | [0.5626, 0.6326] (0.0700) | 0.6423 (0.0425) | [0.5639, 0.7278] (0.1638) |

Data set II | 0.2578 (0.0159) | [0.2251, 0.2876] (0.0624) | 0.6443 (0.0412) | [0.5662, 0.7278] (0.1616) |

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## Share and Cite

**MDPI and ACS Style**

Eliwa, M.S.; Tyagi, A.; Almohaimeed, B.; El-Morshedy, M.
Modelling Coronavirus and Larvae Pyrausta Data: A Discrete Binomial Exponential II Distribution with Properties, Classical and Bayesian Estimation. *Axioms* **2022**, *11*, 646.
https://doi.org/10.3390/axioms11110646

**AMA Style**

Eliwa MS, Tyagi A, Almohaimeed B, El-Morshedy M.
Modelling Coronavirus and Larvae Pyrausta Data: A Discrete Binomial Exponential II Distribution with Properties, Classical and Bayesian Estimation. *Axioms*. 2022; 11(11):646.
https://doi.org/10.3390/axioms11110646

**Chicago/Turabian Style**

Eliwa, Mohamed S., Abhishek Tyagi, Bader Almohaimeed, and Mahmoud El-Morshedy.
2022. "Modelling Coronavirus and Larvae Pyrausta Data: A Discrete Binomial Exponential II Distribution with Properties, Classical and Bayesian Estimation" *Axioms* 11, no. 11: 646.
https://doi.org/10.3390/axioms11110646