# A Probability Mass Function for Various Shapes of the Failure Rates, Asymmetric and Dispersed Data with Applications to Coronavirus and Kidney Dysmorphogenesis

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## Abstract

**:**

## 1. Introduction

## 2. Synthesis of the DHLo-II Model

## 3. Statistical Properties

#### 3.1. Moments and Generating Functions

#### 3.2. Conditional Moments

#### 3.3. Stress-Strength Analysis

#### 3.4. Residual Entropy and Cumulative Residual Entropy

#### 3.5. Order Statistics

## 4. Maximum Likelihood Estimation (MLE)

## 5. Simulation

**R**software. The assessment is based on a simulation study: generate 10,000 samples of size $n=$$10,12,14,\dots ,60$ from DHLo-II${}_{\alpha =0.5}^{\beta =0.5}$ and DHLo-II${}_{\alpha =0.8}^{\beta =0.3}$, respectively; compute the MLEs for the 10,000 samples, say ${\widehat{\varrho}}_{l}$ for $l=1,2,\dots ,\mathrm{10,000}$; and compute the biases and mean-squared errors (MSEs), where bias$\left(\varrho \right)=\frac{1}{10,000}{\sum}_{l=1}^{10,000}\left(\widehat{{\varrho}_{j}}-\varrho \right)\phantom{\rule{4pt}{0ex}}$and MSE$\left(\varrho \right)=\frac{1}{10,000}{\sum}_{l=1}^{10,000}{\left(\widehat{{\varrho}_{j}}-\varrho \right)}^{2}$. The empirical results are given in Figure 2 and Figure 3, respectively.

## 6. Applications

#### 6.1. Data set I: COVID-19 in Armenia

#### 6.2. Data Set II: Kidney Dysmorphogenesis

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**$\mathrm{The}\phantom{\rule{4.pt}{0ex}}\mathrm{bias}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{MSE}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}$DHLo-II${}_{\alpha =0.5}^{\beta =0.5}$.

**Figure 3.**$\mathrm{The}\phantom{\rule{4.pt}{0ex}}\mathrm{bias}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{MSE}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}$DHLo-II${}_{\alpha =0.8}^{\beta =0.3}$.

$\mathit{\alpha}$ | |||||||
---|---|---|---|---|---|---|---|

Measure | $\mathbf{0.001}$ | $\mathbf{0.01}$ | $\mathbf{0.1}$ | $\mathbf{0.3}$ | $\mathbf{0.5}$ | $\mathbf{0.7}$ | $\mathbf{0.9}$ |

Mean | $0.00200$ | $0.01109$ | $0.11201$ | $0.42927$ | $1.00050$ | $2.33363$ | $9.00010$ |

Variance | $0.00200$ | $0.01117$ | $0.12415$ | $0.61234$ | $1.99950$ | $7.77667$ | $89.9982$ |

$\mathbf{IOD}$ | $1.00000$ | $1.00748$ | $1.10844$ | $1.42647$ | $1.99850$ | $3.33243$ | $9.99969$ |

Skewness | $22.3608$ | $9.60244$ | $3.45601$ | $2.37058$ | $2.12123$ | $2.03213$ | $2.00282$ |

Kurtosis | $503.015$ | $96.5834$ | $16.9092$ | $10.6215$ | $9.50161$ | $9.13014$ | $9.01133$ |

$\mathit{\alpha}$ | |||||||
---|---|---|---|---|---|---|---|

Measure | $\mathbf{0.001}$ | $\mathbf{0.01}$ | $\mathbf{0.1}$ | $\mathbf{0.3}$ | $\mathbf{0.5}$ | $\mathbf{0.7}$ | $\mathbf{0.9}$ |

Mean | $0.01099$ | $0.02000$ | $0.12012$ | $0.43559$ | $1.00502$ | $2.33635$ | $9.00100$ |

Variance | $0.01107$ | $0.02001$ | $0.13058$ | $0.61338$ | $1.99510$ | $7.76679$ | $89.9828$ |

$\mathbf{IOD}$ | $1.00775$ | $1.00039$ | $1.08711$ | $1.40815$ | $1.98512$ | $3.32432$ | $9.99697$ |

Skewness | $9.64876$ | $7.07593$ | $3.26866$ | $2.34450$ | $2.12024$ | $2.03437$ | $2.00328$ |

Kurtosis | $97.5325$ | $53.1472$ | $15.3539$ | $10.5121$ | $9.51522$ | $9.14427$ | $9.01338$ |

$\mathit{\alpha}$ | |||||||
---|---|---|---|---|---|---|---|

Measure | $\mathbf{0.001}$ | $\mathbf{0.01}$ | $\mathbf{0.1}$ | $\mathbf{0.3}$ | $\mathbf{0.5}$ | $\mathbf{0.7}$ | $\mathbf{0.9}$ |

Mean | $0.10283$ | $0.11111$ | $0.20384$ | $0.50229$ | $1.05385$ | $2.36639$ | $9.01128$ |

Variance | $0.11674$ | $0.12345$ | $0.21126$ | $0.63993$ | $1.96223$ | $7.66870$ | $89.8131$ |

$\mathbf{IOD}$ | $1.13534$ | $1.11111$ | $1.03643$ | $1.27401$ | $1.86195$ | $3.24066$ | $9.96674$ |

Skewness | $3.69573$ | $3.47850$ | $2.36398$ | $2.07340$ | $2.08971$ | $2.05342$ | $2.00819$ |

Kurtosis | $19.0106$ | $17.1000$ | $9.21101$ | $9.18706$ | $9.53455$ | $9.27863$ | $9.19153$ |

$\mathit{\alpha}$ | |||||||
---|---|---|---|---|---|---|---|

Measure | $\mathbf{0.001}$ | $\mathbf{0.01}$ | $\mathbf{0.1}$ | $\mathbf{0.3}$ | $\mathbf{0.5}$ | $\mathbf{0.7}$ | $\mathbf{0.9}$ |

Mean | $0.35189$ | $0.35891$ | $0.43830$ | $0.70225$ | $1.21122$ | $2.47100$ | $9.05015$ |

Variance | $0.57726$ | $0.57948$ | $0.61832$ | $0.90749$ | $2.03517$ | $7.47027$ | $89.2348$ |

$\mathbf{IOD}$ | $1.64044$ | $1.61455$ | $1.41070$ | $1.29225$ | $1.68026$ | $3.02317$ | $9.86003$ |

Skewness | $2.76593$ | $2.72830$ | $2.32845$ | $1.70329$ | $1.84145$ | $2.05212$ | $2.02320$ |

Kurtosis | $12.7583$ | $12.5653$ | $10.3780$ | $6.97051$ | $8.37013$ | $9.43783$ | $9.11141$ |

$\mathit{\alpha}$ | |||||||
---|---|---|---|---|---|---|---|

Measure | $\mathbf{0.001}$ | $\mathbf{0.01}$ | $\mathbf{0.1}$ | $\mathbf{0.3}$ | $\mathbf{0.5}$ | $\mathbf{0.7}$ | $\mathbf{0.9}$ |

Mean | $0.76516$ | $0.77124$ | $0.84016$ | $1.07150$ | $1.52899$ | $2.70637$ | $9.15007$ |

Variance | $1.99106$ | $1.98797$ | $1.96588$ | $2.05052$ | $2.81392$ | $7.57726$ | $88.0408$ |

$\mathbf{IOD}$ | $2.60213$ | $2.57760$ | $2.33988$ | $1.91367$ | $1.84036$ | $2.79978$ | $9.62186$ |

Skewness | $2.48246$ | $2.47637$ | $2.39012$ | $1.99815$ | $1.58944$ | $1.87366$ | $2.04677$ |

Kurtosis | $10.9062$ | $10.8970$ | $10.6659$ | $8.86802$ | $6.69554$ | $8.67895$ | $8.23654$ |

Parameter → | $\mathit{\alpha}$ | $\mathit{\beta}$ | ||||
---|---|---|---|---|---|---|

Model↓ | MLE | Se | CI | MLE | Se | CI |

DHLo-II | $0.620$ | $0.048$ | $[0.527,0.714]$ | $0.815$ | $0.014$ | $[0.788,0.842]$ |

DIW | $0.201$ | $0.026$ | $[0.149,0.252]$ | $0.958$ | $0.060$ | $[0.839,1.076]$ |

DGL | $0.784$ | $0.037$ | $[0.712,0.855]$ | $0.228$ | $0.089$ | $[0.053,0.404]$ |

DB-II | $0.643$ | $0.034$ | $[0.576,0.711]$ | $1.811$ | $0.210$ | $[1.399,2.223]$ |

DLL | $2.871$ | $0.2426$ | $[2.395,3.346]$ | $1.388$ | $0.086$ | $[1.219,1.557]$ |

DIR | $0.112$ | $0.019$ | $[0.075,0.149]$ | − | − | − |

DBH | $0.976$ | $0.01136$ | $[0.953,0.998]$ | − | − | − |

DL | $0.692$ | $0.012$ | $[0.668,0.716]$ | − | − | − |

DP | $0.493$ | $0.0229$ | $[0.448,0.538]$ | − | − | − |

Expected Frequency | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

X | Of | DHLo-II | DIW | DGL | DB-II | DLL | DIR | DBH | DL | DP |

$\mathbf{0}$ | 56 | $48.53$ | $46.60$ | $43.85$ | $61.12$ | $43.59$ | $25.96$ | $118.83$ | $28.24$ | $89.88$ |

$\mathbf{1}$ | 31 | $37.23$ | $54.93$ | $35.74$ | $51.49$ | $43.91$ | $108.22$ | $39.57$ | $32.85$ | $35.42$ |

$\mathbf{2}$ | 22 | $28.83$ | $30.93$ | $29.08$ | $28.24$ | $32.05$ | $47.70$ | $19.75$ | $31.94$ | $19.64$ |

$\mathbf{3}$ | 25 | $22.55$ | $19.14$ | $23.63$ | $17.09$ | $22.69$ | $20.44$ | $11.83$ | $28.47$ | $12.70$ |

$\mathbf{4}$ | 11 | $17.83$ | $12.94$ | $19.18$ | $11.39$ | $16.34$ | $10.22$ | $7.865$ | $24.12$ | $8.99$ |

$\mathbf{5}$ | 14 | $14.23$ | $9.31$ | $15.55$ | $8.15$ | $12.08$ | $5.76$ | $5.59$ | $19.74$ | $6.75$ |

$\mathbf{6}$ | 14 | $11.46$ | $7.02$ | $12.59$ | $6.12$ | $9.15$ | $3.55$ | $4.18$ | $15.77$ | $5.28$ |

$\mathbf{7}$ | 10 | $9.29$ | $5.49$ | $10.18$ | $4.78$ | $7.12$ | $2.34$ | $3.24$ | $12.38$ | $4.26$ |

$\mathbf{8}$ | 11 | $7.57$ | $4.41$ | $8.23$ | $3.84$ | $5.65$ | $1.62$ | $2.58$ | $9.58$ | $3.52$ |

$\mathbf{9}$ | 3 | $6.19$ | $3.62$ | $6.64$ | $3.16$ | $4.55$ | $1.17$ | $2.10$ | $7.33$ | $2.97$ |

$\mathbf{10}$ | 10 | $5.08$ | $3.02$ | $5.36$ | $2.65$ | $3.74$ | $0.86$ | $1.74$ | $5.56$ | $2.54$ |

$\mathbf{11}$ | 7 | $4.17$ | $2.57$ | $4.32$ | $2.26$ | $3.13$ | $0.66$ | $1.47$ | $4.18$ | $2.20$ |

$\mathbf{12}$ | 4 | $3.43$ | $2.20$ | $3.48$ | $1.94$ | $2.62$ | $0.51$ | $1.25$ | $3.13$ | $1.93$ |

$\mathbf{13}$ | 5 | $2.82$ | $1.91$ | $2.80$ | $1.69$ | $2.23$ | $0.41$ | $1.07$ | $2.32$ | $1.71$ |

$\mathbf{14}$ | 2 | $2.32$ | $1.68$ | $2.25$ | $1.49$ | $1.92$ | $0.33$ | $0.93$ | $1.72$ | $1.52$ |

$\mathbf{15}$ | 2 | $1.90$ | $1.49$ | $1.81$ | $1.33$ | $1.68$ | $0.27$ | $0.82$ | $1.27$ | $1.37$ |

≥$\mathbf{16}$ | 6 | $8.57$ | $24.74$ | $7.31$ | $25.26$ | $19.55$ | $1.98$ | $9.19$ | $3.40$ | $31.32$ |

Total | $\mathbf{232}$ | $\mathbf{232}$ | $\mathbf{232}$ | $\mathbf{232}$ | $\mathbf{232}$ | $\mathbf{232}$ | $\mathbf{232}$ | $\mathbf{232}$ | $\mathbf{232}$ | $\mathbf{232}$ |

$-L$ | $590.63$ | $625.48$ | $590.86$ | $629.89$ | $609.58$ | $719.92$ | $657.92$ | $604.57$ | $644.98$ | |

Aic | $1185.26$ | $1254.96$ | $1185.72$ | $1263.77$ | $1223.16$ | $1441.84$ | $1317.85$ | $1211.13$ | $1291.96$ | |

Caic | $1185.31$ | $1255.01$ | $1185.77$ | $1263.83$ | $1223.22$ | $1441.86$ | $1317.87$ | $1211.15$ | $1291.98$ | |

Bic | $1192.15$ | $1261.85$ | $1192.61$ | $1270.67$ | $1230.06$ | $1445.29$ | $1321.29$ | $1214.58$ | $1295.41$ | |

Hqic | $1188.04$ | $1257.74$ | $1188.49$ | $1266.55$ | $1225.94$ | $1443.23$ | $1319.24$ | $1212.52$ | $1293.35$ | |

Chi${}^{2}$ | $18.22$ | $75.53$ | $19.03$ | $82.63$ | $39.97$ | $397.61$ | $185.04$ | $53.71$ | $113.54$ | |

Df | 11 | 9 | 11 | 8 | 10 | 6 | 8 | 11 | 9 | |

Pv | $0.08$ | ≤$0.001$ | $0.06$ | $\le 0.001$ | ≤$0.001$ | ≤$0.001$ | ≤$0.001$ | ≤$0.001$ | $\le 0.001$ |

Parameter ⟶ | $\mathit{\alpha}$ | $\mathit{\beta}$ | ||||
---|---|---|---|---|---|---|

Model↓ | MLE | Se | CI | MLE | Se | CI |

DHLo-II | $0.052$ | $0.056$ | $[0,0.162]$ | $0.659$ | $0.035$ | $[0.589,0.728]$ |

DIW | $0.581$ | $0.048$ | $[0.488,0.675]$ | $1.049$ | $0.146$ | $[0.763,1.335]$ |

DGL | $0.582$ | $0.045$ | $[0.493,0.671]$ | $0.351$ | $0.0654$ | $[0.223,0.479]$ |

DB-II | $0.278$ | $0.045$ | $[0.189,0.366]$ | $1.053$ | $0.167$ | $[0.725,1.381]$ |

DLL | $0.780$ | $0.136$ | $[0.514,1.046]$ | $1.208$ | $0.159$ | $[0.895,1.520]$ |

DIR | $0.554$ | $0.049$ | $[0.458,0.649]$ | − | − | − |

DBH | $0.874$ | $0.041$ | $[0.794,0.954]$ | − | − | − |

DL | $0.436$ | $0.026$ | $[0.385,0.488]$ | − | − | − |

DP | $0.268$ | $0.034$ | $[0.201,0.336]$ | − | − | − |

Expected Frequency | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{X}$ | Of | DHLo-II | DIW | DGL | DB-II | DLL | DIR | DBH | DL | DP |

$\mathbf{0}$ | 64 | $62.88$ | $63.91$ | $46.01$ | $64.74$ | $63.19$ | $60.89$ | $61.94$ | $40.29$ | $65.84$ |

$\mathbf{1}$ | 14 | $13.64$ | $20.69$ | $26.76$ | $19.18$ | $20.10$ | $33.99$ | $20.06$ | $29.83$ | $18.27$ |

$\mathbf{2}$ | 10 | $9.03$ | $8.05$ | $15.57$ | $8.48$ | $8.64$ | $8.12$ | $9.65$ | $18.36$ | $8.16$ |

$\mathbf{3}$ | 6 | $7.03$ | $4.23$ | $9.06$ | $4.63$ | $4.66$ | $3.00$ | $5.52$ | $10.34$ | $4.51$ |

$\mathbf{4}$ | 4 | $5.29$ | $2.59$ | $5.27$ | $2.86$ | $2.86$ | $1.42$ | $3.49$ | $5.52$ | $2.82$ |

$\mathbf{5}$ | 2 | $3.83$ | $1.75$ | $3.07$ | $1.92$ | $1.92$ | $0.78$ | $2.35$ | $2.85$ | $1.91$ |

$\mathbf{6}$ | 2 | $2.69$ | $1.26$ | $1.78$ | $1.37$ | $1.37$ | $0.47$ | $1.65$ | $1.44$ | $1.37$ |

$\mathbf{7}$ | 2 | $1.85$ | $0.95$ | $1.04$ | $1.01$ | $1.02$ | $0.31$ | $1.19$ | $0.71$ | $1.02$ |

$\mathbf{8}$ | 1 | $1.25$ | $0.74$ | $0.60$ | $0.78$ | $0.79$ | $0.21$ | $0.89$ | $0.35$ | $0.79$ |

$\mathbf{9}$ | 1 | $0.84$ | $0.59$ | $0.35$ | $0.61$ | $0.62$ | $0.15$ | $0.67$ | $0.17$ | $0.63$ |

$\mathbf{10}$ | 1 | $0.56$ | $0.49$ | $0.20$ | $0.49$ | $0.50$ | $0.11$ | $0.52$ | $0.08$ | $0.51$ |

$\mathbf{11}$ | 2 | $1.11$ | $4.75$ | $0.29$ | $3.93$ | $4.33$ | $0.55$ | $2.07$ | $0.06$ | $4.17$ |

Total | $\mathbf{110}$ | $\mathbf{110}$ | $\mathbf{110}$ | $\mathbf{110}$ | $\mathbf{110}$ | $\mathbf{110}$ | $\mathbf{110}$ | $\mathbf{110}$ | $\mathbf{110}$ | $\mathbf{110}$ |

$-L$ | $167.52$ | $172.94$ | $178.77$ | $171.14$ | $171.72$ | $186.55$ | $169.89$ | $189.11$ | $171.19$ | |

Aic | $339.03$ | $349.87$ | $361.53$ | $346.28$ | $347.43$ | $375.09$ | $341.78$ | $380.22$ | $344.38$ | |

Caic | $339.15$ | $349.98$ | $361.65$ | $346.39$ | $347.55$ | $375.13$ | $341.82$ | $380.26$ | $344.42$ | |

Bic | $344.44$ | $355.28$ | $366.94$ | $351.68$ | $352.84$ | $377.80$ | $344.48$ | $382.92$ | $347.08$ | |

Hqic | $341.22$ | $352.06$ | $363.72$ | $348.47$ | $349.62$ | $376.19$ | $342.88$ | $381.32$ | $345.48$ | |

Chi${}^{2}$ | $1.97$ | $6.45$ | $19.09$ | $2.59$ | $4.03$ | $40.46$ | $2.61$ | $34.64$ | $3.43$ | |

Df | 4 | 3 | 3 | 2 | 3 | 2 | 4 | 4 | 4 | |

Pv | $0.74$ | $0.09$ | ≤$0.001$ | $0.27$ | $0.26$ | ≤$0.001$ | $0.63$ | ≤$0.001$ | $0.49$ |

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**MDPI and ACS Style**

El-Morshedy, M.; Alizadeh, M.; Al-Bossly, A.; Eliwa, M.S. A Probability Mass Function for Various Shapes of the Failure Rates, Asymmetric and Dispersed Data with Applications to Coronavirus and Kidney Dysmorphogenesis. *Symmetry* **2021**, *13*, 1790.
https://doi.org/10.3390/sym13101790

**AMA Style**

El-Morshedy M, Alizadeh M, Al-Bossly A, Eliwa MS. A Probability Mass Function for Various Shapes of the Failure Rates, Asymmetric and Dispersed Data with Applications to Coronavirus and Kidney Dysmorphogenesis. *Symmetry*. 2021; 13(10):1790.
https://doi.org/10.3390/sym13101790

**Chicago/Turabian Style**

El-Morshedy, Mahmoud, Morad Alizadeh, Afrah Al-Bossly, and Mohamed S. Eliwa. 2021. "A Probability Mass Function for Various Shapes of the Failure Rates, Asymmetric and Dispersed Data with Applications to Coronavirus and Kidney Dysmorphogenesis" *Symmetry* 13, no. 10: 1790.
https://doi.org/10.3390/sym13101790