# Modeling COVID-19 Real Data Set by a New Extension of Haq Distribution

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## Abstract

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## 1. Introduction

## 2. Model Formulation

#### Reliability Functions

## 3. Statistical Properties

#### 3.1. Mode

#### 3.2. Moments

#### 3.3. Incomplete Moments

#### 3.4. Order Statistics

## 4. Estimation of PHD Parameters

## 5. Numerical Simulation

- The PHD estimators’ values show the consistency property.
- All measures in the simulation tables decrease, as the sample size increases, except for the CP.
- ${D}_{abs}$ is always smaller than ${D}_{max}$.
- The maximum spacing product is the estimation method that is most preferred according to the ranking table.

## 6. Real Data Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**These figures show the histogram of the fitted PDF (top left), CDF (top right), SF (bottom left) and P-P (bottom right) plots of the COVID-19 dataset.

**Table 1.**Numerical values of PHD simulation for $(\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =0.75)$.

n | Est. | Est.Par. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE |
---|---|---|---|---|---|---|---|---|---|---|---|---|

30 | BIAS | $\widehat{\alpha}$ | 0.0904${}^{\left\{2\right\}}$ | ${0.09794}^{\left\{6\right\}}$ | ${0.09387}^{\left\{3\right\}}$ | ${0.09027}^{\left\{1\right\}}$ | ${0.09612}^{\left\{4\right\}}$ | ${0.10526}^{\left\{10\right\}}$ | ${0.09803}^{\left\{7\right\}}$ | ${0.1005}^{\left\{8\right\}}$ | ${0.096797}^{\left\{5\right\}}$ | ${0.103329}^{\left\{9\right\}}$ |

$\widehat{\beta}$ | ${0.04455}^{\left\{2\right\}}$ | ${0.04616}^{\left\{4\right\}}$ | ${0.04654}^{\left\{5\right\}}$ | ${0.03983}^{\left\{1\right\}}$ | ${0.04538}^{\left\{3\right\}}$ | ${0.05378}^{\left\{10\right\}}$ | ${0.04703}^{\left\{7\right\}}$ | ${0.04848}^{\left\{9\right\}}$ | ${0.046722}^{\left\{6\right\}}$ | ${0.047425}^{\left\{8\right\}}$ | ||

MSE | $\widehat{\alpha}$ | ${0.01348}^{\left\{2\right\}}$ | ${0.01488}^{\left\{4\right\}}$ | ${0.01411}^{\left\{3\right\}}$ | ${0.01321}^{\left\{1\right\}}$ | ${0.01532}^{\left\{6\right\}}$ | ${0.01766}^{\left\{9\right\}}$ | ${0.01522}^{\left\{5\right\}}$ | ${0.01569}^{\left\{7\right\}}$ | ${0.01594}^{\left\{8\right\}}$ | ${0.018027}^{\left\{10\right\}}$ | |

$\widehat{\beta}$ | ${0.00345}^{\left\{4\right\}}$ | ${0.00352}^{\left\{5\right\}}$ | ${0.00376}^{\left\{8\right\}}$ | ${0.00256}^{\left\{1\right\}}$ | ${0.0034}^{\left\{2\right\}}$ | ${0.00582}^{\left\{10\right\}}$ | ${0.00342}^{\left\{3\right\}}$ | ${0.00389}^{\left\{9\right\}}$ | ${0.003526}^{\left\{6\right\}}$ | ${0.003587}^{\left\{7\right\}}$ | ||

MRE | $\widehat{\alpha}$ | ${0.1808}^{\left\{2\right\}}$ | ${0.19587}^{\left\{6\right\}}$ | ${0.18774}^{\left\{3\right\}}$ | ${0.18053}^{\left\{1\right\}}$ | ${0.19224}^{\left\{4\right\}}$ | ${0.21053}^{\left\{10\right\}}$ | ${0.19605}^{\left\{7\right\}}$ | ${0.20099}^{\left\{8\right\}}$ | ${0.193594}^{\left\{5\right\}}$ | ${0.206659}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.0594}^{\left\{2\right\}}$ | ${0.06154}^{\left\{4\right\}}$ | ${0.06205}^{\left\{5\right\}}$ | ${0.05311}^{\left\{1\right\}}$ | ${0.06051}^{\left\{3\right\}}$ | ${0.07171}^{\left\{10\right\}}$ | ${0.0627}^{\left\{7\right\}}$ | ${0.06464}^{\left\{9\right\}}$ | ${0.062296}^{\left\{6\right\}}$ | ${0.063234}^{\left\{8\right\}}$ | ||

CIL | $\widehat{\alpha}$ | ${0.43204}^{\left\{1\right\}}$ | ${0.44836}^{\left\{3\right\}}$ | ${0.46326}^{\left\{6\right\}}$ | ${0.45125}^{\left\{4\right\}}$ | ${0.46713}^{\left\{7\right\}}$ | ${0.50013}^{\left\{8\right\}}$ | ${0.45441}^{\left\{5\right\}}$ | ${0.445}^{\left\{2\right\}}$ | ${0.53819}^{\left\{10\right\}}$ | ${0.51364}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.21937}^{\left\{2\right\}}$ | ${0.22683}^{\left\{5\right\}}$ | ${0.25576}^{\left\{9\right\}}$ | ${0.1978}^{\left\{1\right\}}$ | ${0.23176}^{\left\{7\right\}}$ | ${0.312}^{\left\{10\right\}}$ | ${0.22378}^{\left\{3\right\}}$ | ${0.23056}^{\left\{6\right\}}$ | ${0.24912}^{\left\{8\right\}}$ | ${0.2241}^{\left\{4\right\}}$ | ||

CP | $\widehat{\alpha}$ | ${0.916}^{\left\{7\right\}}$ | ${0.912}^{\left\{8.5\right\}}$ | ${0.918}^{\left\{6\right\}}$ | ${0.944}^{\left\{3\right\}}$ | ${0.924}^{\left\{4\right\}}$ | ${0.92}^{\left\{5\right\}}$ | ${0.912}^{\left\{8.5\right\}}$ | ${0.896}^{\left\{10\right\}}$ | ${0.968}^{\left\{1\right\}}$ | ${0.95}^{\left\{2\right\}}$ | |

$\widehat{\beta}$ | ${0.916}^{\left\{7\right\}}$ | ${0.93}^{\left\{4\right\}}$ | ${0.904}^{\left\{10\right\}}$ | ${0.932}^{\left\{2.5\right\}}$ | ${0.932}^{\left\{2.5\right\}}$ | ${0.914}^{\left\{8.5\right\}}$ | ${0.918}^{\left\{6\right\}}$ | ${0.914}^{\left\{8.5\right\}}$ | ${0.968}^{\left\{1\right\}}$ | ${0.928}^{\left\{5\right\}}$ | ||

${D}_{abs}$ | ${0.04527}^{\left\{1\right\}}$ | ${0.04888}^{\left\{5\right\}}$ | ${0.04753}^{\left\{3\right\}}$ | ${0.04675}^{\left\{2\right\}}$ | ${0.05007}^{\left\{6\right\}}$ | ${0.05131}^{\left\{8\right\}}$ | ${0.04783}^{\left\{4\right\}}$ | ${0.0505}^{\left\{7\right\}}$ | ${0.05478}^{\left\{10\right\}}$ | ${0.05295}^{\left\{9\right\}}$ | ||

${D}_{max}$ | ${0.08054}^{\left\{2\right\}}$ | ${0.08459}^{\left\{4\right\}}$ | ${0.08534}^{\left\{5\right\}}$ | ${0.07955}^{\left\{1\right\}}$ | ${0.08651}^{\left\{6\right\}}$ | ${0.09144}^{\left\{9\right\}}$ | ${0.08444}^{\left\{3\right\}}$ | ${0.08935}^{\left\{7\right\}}$ | ${0.09536}^{\left\{10\right\}}$ | ${0.09053}^{\left\{8\right\}}$ | ||

ASAE | ${0.0363}^{\left\{2\right\}}$ | ${0.03355}^{\left\{4\right\}}$ | ${0.03262}^{\left\{5\right\}}$ | ${0.03453}^{\left\{1\right\}}$ | ${0.03267}^{\left\{6\right\}}$ | ${0.03071}^{\left\{9\right\}}$ | ${0.03245}^{\left\{3\right\}}$ | ${0.03697}^{\left\{7\right\}}$ | ${0.04829}^{\left\{10\right\}}$ | ${0.04326}^{\left\{8\right\}}$ | ||

$\sum Ranks$ | ${35}^{\left\{1\right\}}$ | ${60.5}^{\left\{4.5\right\}}$ | ${59}^{\left\{3\right\}}$ | ${36.5}^{\left\{2\right\}}$ | ${67.5}^{\left\{6\right\}}$ | ${103.5}^{\left\{8\right\}}$ | ${60.5}^{\left\{4.5\right\}}$ | ${83.5}^{\left\{7\right\}}$ | ${104}^{\left\{9\right\}}$ | ${105}^{\left\{10\right\}}$ | ||

70 | BIAS | $\widehat{\alpha}$ | ${0.05999}^{\left\{2\right\}}$ | ${0.06043}^{\left\{4\right\}}$ | ${0.06226}^{\left\{6\right\}}$ | ${0.06025}^{\left\{3\right\}}$ | ${0.06078}^{\left\{5\right\}}$ | ${0.06738}^{\left\{9\right\}}$ | ${0.05917}^{\left\{1\right\}}$ | ${0.06302}^{\left\{7\right\}}$ | ${0.072793}^{\left\{10\right\}}$ | ${0.066639}^{\left\{8\right\}}$ |

$\widehat{\beta}$ | ${0.02696}^{\left\{1\right\}}$ | ${0.02948}^{\left\{6\right\}}$ | ${0.0286}^{\left\{4\right\}}$ | ${0.02805}^{\left\{2.5\right\}}$ | ${0.02805}^{\left\{2.5\right\}}$ | ${0.03316}^{\left\{9\right\}}$ | ${0.02947}^{\left\{5\right\}}$ | ${0.03026}^{\left\{8\right\}}$ | ${0.034518}^{\left\{10\right\}}$ | ${0.029687}^{\left\{7\right\}}$ | ||

MSE | $\widehat{\alpha}$ | ${0.00595}^{\left\{3\right\}}$ | ${0.00599}^{\left\{4.5\right\}}$ | ${0.00616}^{\left\{6\right\}}$ | ${0.00599}^{\left\{4.5\right\}}$ | ${0.00594}^{\left\{2\right\}}$ | ${0.00714}^{\left\{9\right\}}$ | ${0.00547}^{\left\{1\right\}}$ | ${0.00638}^{\left\{7\right\}}$ | ${0.008646}^{\left\{10\right\}}$ | ${0.007075}^{\left\{8\right\}}$ | |

$\widehat{\beta}$ | ${0.00121}^{\left\{1\right\}}$ | ${0.00143}^{\left\{8\right\}}$ | ${0.00131}^{\left\{4\right\}}$ | ${0.00123}^{\left\{2\right\}}$ | ${0.00126}^{\left\{3\right\}}$ | ${0.00183}^{\left\{9\right\}}$ | ${0.00133}^{\left\{5\right\}}$ | ${0.00142}^{\left\{7\right\}}$ | ${0.001928}^{\left\{10\right\}}$ | ${0.001402}^{\left\{6\right\}}$ | ||

MRE | $\widehat{\alpha}$ | ${0.11998}^{\left\{2\right\}}$ | ${0.12086}^{\left\{4\right\}}$ | ${0.12452}^{\left\{6\right\}}$ | ${0.12049}^{\left\{3\right\}}$ | ${0.12156}^{\left\{5\right\}}$ | ${0.13476}^{\left\{9\right\}}$ | ${0.11833}^{\left\{1\right\}}$ | ${0.12604}^{\left\{7\right\}}$ | ${0.145586}^{\left\{10\right\}}$ | ${0.133278}^{\left\{8\right\}}$ | |

$\widehat{\beta}$ | ${0.03594}^{\left\{1\right\}}$ | ${0.03931}^{\left\{6\right\}}$ | ${0.03813}^{\left\{4\right\}}$ | ${0.0374}^{\left\{2.5\right\}}$ | ${0.0374}^{\left\{2.5\right\}}$ | ${0.04421}^{\left\{9\right\}}$ | ${0.03929}^{\left\{5\right\}}$ | ${0.04034}^{\left\{8\right\}}$ | ${0.046024}^{\left\{10\right\}}$ | ${0.039583}^{\left\{7\right\}}$ | ||

CIL | $\widehat{\alpha}$ | ${0.28523}^{\left\{1\right\}}$ | ${0.28615}^{\left\{2\right\}}$ | ${0.29743}^{\left\{6\right\}}$ | ${0.28883}^{\left\{3\right\}}$ | ${0.29927}^{\left\{7\right\}}$ | ${0.31397}^{\left\{8\right\}}$ | ${0.29146}^{\left\{5\right\}}$ | ${0.29093}^{\left\{4\right\}}$ | ${0.35924}^{\left\{10\right\}}$ | ${0.33303}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.13502}^{\left\{2\right\}}$ | ${0.13957}^{\left\{4\right\}}$ | ${0.14404}^{\left\{7\right\}}$ | ${0.12851}^{\left\{1\right\}}$ | ${0.13901}^{\left\{3\right\}}$ | ${0.16132}^{\left\{9\right\}}$ | ${0.13997}^{\left\{5\right\}}$ | ${0.14207}^{\left\{6\right\}}$ | ${0.16335}^{\left\{10\right\}}$ | ${0.14891}^{\left\{8\right\}}$ | ||

CP | $\widehat{\alpha}$ | ${0.936}^{\left\{5.5\right\}}$ | ${0.934}^{\left\{7\right\}}$ | ${0.922}^{\left\{8.5\right\}}$ | ${0.94}^{\left\{4\right\}}$ | ${0.942}^{\left\{3\right\}}$ | ${0.922}^{\left\{8.5\right\}}$ | ${0.936}^{\left\{5.5\right\}}$ | ${0.916}^{\left\{10\right\}}$ | ${0.948}^{\left\{2\right\}}$ | ${0.95}^{\left\{1\right\}}$ | |

$\widehat{\beta}$ | ${0.946}^{\left\{1.5\right\}}$ | ${0.944}^{\left\{4\right\}}$ | ${0.93}^{\left\{8.5\right\}}$ | ${0.93}^{\left\{8.5\right\}}$ | ${0.944}^{\left\{4\right\}}$ | ${0.916}^{\left\{10\right\}}$ | ${0.944}^{\left\{4\right\}}$ | ${0.936}^{\left\{7\right\}}$ | ${0.942}^{\left\{6\right\}}$ | ${0.946}^{\left\{1.5\right\}}$ | ||

${D}_{abs}$ | ${0.0298}^{\left\{1\right\}}$ | ${0.03019}^{\left\{2\right\}}$ | ${0.03192}^{\left\{6\right\}}$ | ${0.03114}^{\left\{4\right\}}$ | ${0.03134}^{\left\{5\right\}}$ | ${0.03234}^{\left\{7\right\}}$ | ${0.03059}^{\left\{3\right\}}$ | ${0.03332}^{\left\{8\right\}}$ | ${0.03822}^{\left\{10\right\}}$ | ${0.03541}^{\left\{9\right\}}$ | ||

${D}_{max}$ | ${0.05214}^{\left\{1\right\}}$ | ${0.05377}^{\left\{2\right\}}$ | ${0.05574}^{\left\{6\right\}}$ | ${0.05412}^{\left\{3\right\}}$ | ${0.05531}^{\left\{5\right\}}$ | ${0.05852}^{\left\{7\right\}}$ | ${0.05491}^{\left\{4\right\}}$ | ${0.05858}^{\left\{8\right\}}$ | ${0.06707}^{\left\{10\right\}}$ | ${0.06061}^{\left\{9\right\}}$ | ||

ASAE | ${0.0205}^{\left\{1\right\}}$ | ${0.01974}^{\left\{2\right\}}$ | ${0.01935}^{\left\{6\right\}}$ | ${0.02038}^{\left\{3\right\}}$ | ${0.01973}^{\left\{5\right\}}$ | ${0.01849}^{\left\{7\right\}}$ | ${0.01959}^{\left\{4\right\}}$ | ${0.02064}^{\left\{8\right\}}$ | ${0.02762}^{\left\{10\right\}}$ | ${0.02555}^{\left\{9\right\}}$ | ||

$\sum Ranks$ | ${37}^{\left\{1\right\}}$ | ${58.5}^{\left\{4\right\}}$ | ${62}^{\left\{6\right\}}$ | ${44}^{\left\{2\right\}}$ | ${59}^{\left\{5\right\}}$ | ${89.5}^{\left\{8\right\}}$ | ${50.5}^{\left\{3\right\}}$ | ${83}^{\left\{7\right\}}$ | ${124}^{\left\{10\right\}}$ | ${107.5}^{\left\{9\right\}}$ | ||

150 | BIAS | $\widehat{\alpha}$ | ${0.03962}^{\left\{3\right\}}$ | ${0.03909}^{\left\{2\right\}}$ | ${0.04489}^{\left\{8\right\}}$ | ${0.03629}^{\left\{1\right\}}$ | ${0.04484}^{\left\{7\right\}}$ | ${0.04415}^{\left\{6\right\}}$ | ${0.04031}^{\left\{4\right\}}$ | ${0.04509}^{\left\{9\right\}}$ | ${0.051067}^{\left\{10\right\}}$ | ${0.041882}^{\left\{5\right\}}$ |

$\widehat{\beta}$ | ${0.01844}^{\left\{2\right\}}$ | ${0.01896}^{\left\{3\right\}}$ | ${0.02104}^{\left\{7\right\}}$ | ${0.01688}^{\left\{1\right\}}$ | ${0.02086}^{\left\{6\right\}}$ | ${0.02121}^{\left\{8\right\}}$ | ${0.01941}^{\left\{4\right\}}$ | ${0.02142}^{\left\{9\right\}}$ | ${0.023948}^{\left\{10\right\}}$ | ${0.01955}^{\left\{5\right\}}$ | ||

MSE | $\widehat{\alpha}$ | ${0.00251}^{\left\{2\right\}}$ | ${0.00254}^{\left\{3\right\}}$ | ${0.00326}^{\left\{8\right\}}$ | ${0.00225}^{\left\{1\right\}}$ | ${0.00327}^{\left\{9\right\}}$ | ${0.00313}^{\left\{6\right\}}$ | ${0.00264}^{\left\{4\right\}}$ | ${0.00315}^{\left\{7\right\}}$ | ${0.004204}^{\left\{10\right\}}$ | ${0.003023}^{\left\{5\right\}}$ | |

$\widehat{\beta}$ | ${0.00056}^{\left\{2\right\}}$ | $6\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$${}^{\left\{3.5\right\}}$ | ${0.00073}^{\left\{8.5\right\}}$ | $5\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$${}^{\left\{1\right\}}$ | ${0.00072}^{\left\{7\right\}}$ | ${0.00073}^{\left\{8.5\right\}}$ | $6\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$${}^{\left\{3.5\right\}}$ | ${0.00071}^{\left\{6\right\}}$ | ${0.000998}^{\left\{10\right\}}$ | ${0.000641}^{\left\{5\right\}}$ | ||

MRE | $\widehat{\alpha}$ | ${0.07924}^{\left\{3\right\}}$ | ${0.07818}^{\left\{2\right\}}$ | ${0.08977}^{\left\{8\right\}}$ | ${0.07258}^{\left\{1\right\}}$ | ${0.08969}^{\left\{7\right\}}$ | ${0.0883}^{\left\{6\right\}}$ | ${0.08063}^{\left\{4\right\}}$ | ${0.09018}^{\left\{9\right\}}$ | ${0.102135}^{\left\{10\right\}}$ | ${0.083763}^{\left\{5\right\}}$ | |

$\widehat{\beta}$ | ${0.02458}^{\left\{2\right\}}$ | ${0.02527}^{\left\{3\right\}}$ | ${0.02806}^{\left\{7\right\}}$ | ${0.0225}^{\left\{1\right\}}$ | ${0.02781}^{\left\{6\right\}}$ | ${0.02828}^{\left\{8\right\}}$ | ${0.02588}^{\left\{4\right\}}$ | ${0.02856}^{\left\{9\right\}}$ | ${0.031931}^{\left\{10\right\}}$ | ${0.026067}^{\left\{5\right\}}$ | ||

CIL | $\widehat{\alpha}$ | ${0.18966}^{\left\{1\right\}}$ | ${0.1963}^{\left\{4\right\}}$ | ${0.20051}^{\left\{6\right\}}$ | ${0.19119}^{\left\{2\right\}}$ | ${0.20139}^{\left\{7\right\}}$ | ${0.214}^{\left\{8\right\}}$ | ${0.19942}^{\left\{5\right\}}$ | ${0.19609}^{\left\{3\right\}}$ | ${0.24756}^{\left\{10\right\}}$ | ${0.22449}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.08878}^{\left\{2\right\}}$ | ${0.09307}^{\left\{3\right\}}$ | ${0.09571}^{\left\{7\right\}}$ | ${0.08675}^{\left\{1\right\}}$ | ${0.09445}^{\left\{4\right\}}$ | ${0.10319}^{\left\{9\right\}}$ | ${0.0946}^{\left\{5\right\}}$ | ${0.09473}^{\left\{6\right\}}$ | ${0.11191}^{\left\{10\right\}}$ | ${0.10183}^{\left\{8\right\}}$ | ||

CP | $\widehat{\alpha}$ | ${0.944}^{\left\{4.5\right\}}$ | ${0.932}^{\left\{6\right\}}$ | ${0.906}^{\left\{9\right\}}$ | ${0.944}^{\left\{4.5\right\}}$ | ${0.912}^{\left\{8\right\}}$ | ${0.948}^{\left\{3\right\}}$ | ${0.92}^{\left\{7\right\}}$ | ${0.902}^{\left\{10\right\}}$ | ${0.952}^{\left\{2\right\}}$ | ${0.958}^{\left\{1\right\}}$ | |

$\widehat{\beta}$ | ${0.946}^{\left\{2.5\right\}}$ | ${0.93}^{\left\{6\right\}}$ | ${0.914}^{\left\{8\right\}}$ | ${0.916}^{\left\{7\right\}}$ | ${0.912}^{\left\{9\right\}}$ | ${0.936}^{\left\{4.5\right\}}$ | ${0.936}^{\left\{4.5\right\}}$ | ${0.908}^{\left\{10\right\}}$ | ${0.948}^{\left\{1\right\}}$ | ${0.946}^{\left\{2.5\right\}}$ | ||

${D}_{abs}$ | ${0.0194}^{\left\{2\right\}}$ | ${0.01977}^{\left\{3\right\}}$ | ${0.02239}^{\left\{9\right\}}$ | ${0.01914}^{\left\{1\right\}}$ | ${0.02214}^{\left\{8\right\}}$ | ${0.02176}^{\left\{5\right\}}$ | ${0.02097}^{\left\{4\right\}}$ | ${0.02204}^{\left\{7\right\}}$ | ${0.02547}^{\left\{10\right\}}$ | ${0.02202}^{\left\{6\right\}}$ | ||

${D}_{max}$ | ${0.03424}^{\left\{2\right\}}$ | ${0.0357}^{\left\{3\right\}}$ | ${0.03935}^{\left\{7\right\}}$ | ${0.03337}^{\left\{1\right\}}$ | ${0.0395}^{\left\{8\right\}}$ | ${0.03886}^{\left\{6\right\}}$ | ${0.03733}^{\left\{4\right\}}$ | ${0.03988}^{\left\{9\right\}}$ | ${0.04516}^{\left\{10\right\}}$ | ${0.03827}^{\left\{5\right\}}$ | ||

ASAE | ${0.013}^{\left\{2\right\}}$ | ${0.01202}^{\left\{3\right\}}$ | ${0.01224}^{\left\{7\right\}}$ | ${0.01317}^{\left\{1\right\}}$ | ${0.01199}^{\left\{8\right\}}$ | ${0.01133}^{\left\{6\right\}}$ | ${0.01213}^{\left\{4\right\}}$ | ${0.01312}^{\left\{9\right\}}$ | ${0.01725}^{\left\{10\right\}}$ | ${0.01573}^{\left\{5\right\}}$ | ||

$\sum Ranks$ | ${41}^{\left\{2\right\}}$ | ${48.5}^{\left\{3\right\}}$ | ${82.5}^{\left\{7.5\right\}}$ | ${36.5}^{\left\{1\right\}}$ | ${73}^{\left\{5\right\}}$ | ${83}^{\left\{9\right\}}$ | ${62}^{\left\{4\right\}}$ | ${80}^{\left\{6\right\}}$ | ${126}^{\left\{10\right\}}$ | ${82.5}^{\left\{7.5\right\}}$ | ||

350 | BIAS | $\widehat{\alpha}$ | ${0.02361}^{\left\{2\right\}}$ | ${0.02285}^{\left\{1\right\}}$ | ${0.02821}^{\left\{6\right\}}$ | ${0.02387}^{\left\{3\right\}}$ | ${0.02846}^{\left\{7\right\}}$ | ${0.03116}^{\left\{9\right\}}$ | ${0.02657}^{\left\{5\right\}}$ | ${0.02944}^{\left\{8\right\}}$ | ${0.031269}^{\left\{10\right\}}$ | ${0.025512}^{\left\{4\right\}}$ |

$\widehat{\beta}$ | ${0.0112}^{\left\{2\right\}}$ | ${0.01058}^{\left\{1\right\}}$ | ${0.01277}^{\left\{6\right\}}$ | ${0.01148}^{\left\{3\right\}}$ | ${0.01323}^{\left\{7\right\}}$ | ${0.01535}^{\left\{10\right\}}$ | ${0.01254}^{\left\{5\right\}}$ | ${0.01399}^{\left\{8\right\}}$ | ${0.014824}^{\left\{9\right\}}$ | ${0.011611}^{\left\{4\right\}}$ | ||

MSE | $\widehat{\alpha}$ | ${0.00105}^{\left\{3\right\}}$ | ${0.00099}^{\left\{1\right\}}$ | ${0.00127}^{\left\{6\right\}}$ | ${0.00102}^{\left\{2\right\}}$ | ${0.00129}^{\left\{7\right\}}$ | ${0.00151}^{\left\{9\right\}}$ | ${0.00113}^{\left\{4\right\}}$ | ${0.00132}^{\left\{8\right\}}$ | ${0.001808}^{\left\{10\right\}}$ | ${0.001196}^{\left\{5\right\}}$ | |

$\widehat{\beta}$ | ${0.00024}^{\left\{3.5\right\}}$ | ${0.00021}^{\left\{1\right\}}$ | ${0.00027}^{\left\{6\right\}}$ | ${0.00023}^{\left\{2\right\}}$ | ${0.00028}^{\left\{7\right\}}$ | ${0.00038}^{\left\{9\right\}}$ | ${0.00024}^{\left\{3.5\right\}}$ | $3\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$${}^{\left\{8\right\}}$ | ${0.00046}^{\left\{10\right\}}$ | ${0.000249}^{\left\{5\right\}}$ | ||

MRE | $\widehat{\alpha}$ | ${0.04722}^{\left\{2\right\}}$ | ${0.0457}^{\left\{1\right\}}$ | ${0.05642}^{\left\{6\right\}}$ | ${0.04774}^{\left\{3\right\}}$ | ${0.05692}^{\left\{7\right\}}$ | ${0.06231}^{\left\{9\right\}}$ | ${0.05315}^{\left\{5\right\}}$ | ${0.05888}^{\left\{8\right\}}$ | ${0.062539}^{\left\{10\right\}}$ | ${0.051025}^{\left\{4\right\}}$ | |

$\widehat{\beta}$ | ${0.01493}^{\left\{2\right\}}$ | ${0.01411}^{\left\{1\right\}}$ | ${0.01703}^{\left\{6\right\}}$ | ${0.01531}^{\left\{3\right\}}$ | ${0.01763}^{\left\{7\right\}}$ | ${0.02047}^{\left\{10\right\}}$ | ${0.01672}^{\left\{5\right\}}$ | ${0.01866}^{\left\{8\right\}}$ | ${0.019766}^{\left\{9\right\}}$ | ${0.015482}^{\left\{4\right\}}$ | ||

CIL | $\widehat{\alpha}$ | ${0.12282}^{\left\{2\right\}}$ | ${0.12529}^{\left\{3\right\}}$ | ${0.132}^{\left\{7\right\}}$ | ${0.12174}^{\left\{1\right\}}$ | ${0.13137}^{\left\{6\right\}}$ | ${0.1397}^{\left\{8\right\}}$ | ${0.12918}^{\left\{5\right\}}$ | ${0.12803}^{\left\{4\right\}}$ | ${0.16036}^{\left\{10\right\}}$ | ${0.1441}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.056949}^{\left\{2\right\}}$ | ${0.05891}^{\left\{3\right\}}$ | ${0.06214}^{\left\{7\right\}}$ | ${0.05592}^{\left\{1\right\}}$ | ${0.06169}^{\left\{5\right\}}$ | ${0.06754}^{\left\{9\right\}}$ | ${0.06109}^{\left\{4\right\}}$ | ${0.06174}^{\left\{6\right\}}$ | ${0.0738}^{\left\{10\right\}}$ | ${0.06546}^{\left\{8\right\}}$ | ||

CP | $\widehat{\alpha}$ | ${0.936}^{\left\{4\right\}}$ | ${0.936}^{\left\{4\right\}}$ | ${0.928}^{\left\{6.5\right\}}$ | ${0.928}^{\left\{6.5\right\}}$ | ${0.916}^{\left\{9.5\right\}}$ | ${0.916}^{\left\{9.5\right\}}$ | ${0.936}^{\left\{4\right\}}$ | ${0.92}^{\left\{8\right\}}$ | ${0.95}^{\left\{2\right\}}$ | ${0.954}^{\left\{1\right\}}$ | |

$\widehat{\beta}$ | ${0.932}^{\left\{5\right\}}$ | ${0.936}^{\left\{4\right\}}$ | ${0.926}^{\left\{7.5\right\}}$ | ${0.93}^{\left\{6\right\}}$ | ${0.918}^{\left\{9\right\}}$ | ${0.902}^{\left\{10\right\}}$ | ${0.94}^{\left\{2.5\right\}}$ | ${0.926}^{\left\{7.5\right\}}$ | ${0.948}^{\left\{1\right\}}$ | ${0.94}^{\left\{2.5\right\}}$ | ||

${D}_{abs}$ | ${0.01163}^{\left\{1\right\}}$ | ${0.01199}^{\left\{3\right\}}$ | ${0.01448}^{\left\{6\right\}}$ | ${0.01194}^{\left\{2\right\}}$ | ${0.01463}^{\left\{7\right\}}$ | ${0.01482}^{\left\{9\right\}}$ | ${0.01426}^{\left\{5\right\}}$ | ${0.01477}^{\left\{8\right\}}$ | ${0.01626}^{\left\{10\right\}}$ | ${0.01355}^{\left\{4\right\}}$ | ||

${D}_{max}$ | ${0.02087}^{\left\{1\right\}}$ | ${0.02101}^{\left\{2\right\}}$ | ${0.0254}^{\left\{6\right\}}$ | ${0.02126}^{\left\{3\right\}}$ | ${0.02588}^{\left\{7\right\}}$ | ${0.02692}^{\left\{9\right\}}$ | ${0.02465}^{\left\{5\right\}}$ | ${0.02612}^{\left\{8\right\}}$ | ${0.02906}^{\left\{10\right\}}$ | ${0.02393}^{\left\{4\right\}}$ | ||

ASAE | ${0.00803}^{\left\{1\right\}}$ | ${0.00778}^{\left\{2\right\}}$ | ${0.00706}^{\left\{6\right\}}$ | ${0.00798}^{\left\{3\right\}}$ | ${0.00703}^{\left\{7\right\}}$ | ${0.00677}^{\left\{9\right\}}$ | ${0.0071}^{\left\{5\right\}}$ | ${0.00766}^{\left\{8\right\}}$ | ${0.00988}^{\left\{10\right\}}$ | ${0.0097}^{\left\{4\right\}}$ | ||

$\sum Ranks$ | ${41.5}^{\left\{3\right\}}$ | ${37}^{\left\{1\right\}}$ | ${73}^{\left\{6\right\}}$ | ${39.5}^{\left\{2\right\}}$ | ${72.5}^{\left\{5\right\}}$ | ${93.5}^{\left\{9\right\}}$ | ${66}^{\left\{4\right\}}$ | ${87.5}^{\left\{8\right\}}$ | ${126}^{\left\{10\right\}}$ | ${78.5}^{\left\{7\right\}}$ |

**Table 2.**Numerical values of PHD simulation for $(\alpha =0.75,\phantom{\rule{3.33333pt}{0ex}}\beta =2.5)$.

n | Est. | Est.Par. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE |
---|---|---|---|---|---|---|---|---|---|---|---|---|

30 | BIAS | $\widehat{\alpha}$ | ${0.10605}^{\left\{3\right\}}$ | ${0.10826}^{\left\{4\right\}}$ | ${0.1134}^{\left\{7\right\}}$ | ${0.10396}^{\left\{1\right\}}$ | ${0.10866}^{\left\{6\right\}}$ | ${0.11602}^{\left\{8\right\}}$ | ${0.1047}^{\left\{2\right\}}$ | ${0.10848}^{\left\{5\right\}}$ | ${0.132517}^{\left\{10\right\}}$ | ${0.117941}^{\left\{9\right\}}$ |

$\widehat{\beta}$ | ${0.26678}^{\left\{1\right\}}$ | ${0.28325}^{\left\{3\right\}}$ | ${0.33322}^{\left\{10\right\}}$ | ${0.27098}^{\left\{2\right\}}$ | ${0.31744}^{\left\{9\right\}}$ | ${0.31233}^{\left\{8\right\}}$ | ${0.29313}^{\left\{6\right\}}$ | ${0.31055}^{\left\{7\right\}}$ | ${0.284707}^{\left\{4\right\}}$ | ${0.290149}^{\left\{5\right\}}$ | ||

MSE | $\widehat{\alpha}$ | ${0.01765}^{\left\{3\right\}}$ | ${0.01801}^{\left\{4\right\}}$ | ${0.02073}^{\left\{7\right\}}$ | ${0.01683}^{\left\{1\right\}}$ | ${0.01928}^{\left\{6\right\}}$ | ${0.02113}^{\left\{8\right\}}$ | ${0.0171}^{\left\{2\right\}}$ | ${0.01894}^{\left\{5\right\}}$ | ${0.029514}^{\left\{10\right\}}$ | ${0.021965}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.1246}^{\left\{2\right\}}$ | ${0.13164}^{\left\{4\right\}}$ | ${0.18484}^{\left\{10\right\}}$ | ${0.11079}^{\left\{1\right\}}$ | ${0.15962}^{\left\{7\right\}}$ | ${0.16438}^{\left\{8\right\}}$ | ${0.13755}^{\left\{6\right\}}$ | ${0.16739}^{\left\{9\right\}}$ | ${0.135753}^{\left\{5\right\}}$ | ${0.128227}^{\left\{3\right\}}$ | ||

MRE | $\widehat{\alpha}$ | ${0.1414}^{\left\{3\right\}}$ | ${0.14435}^{\left\{4\right\}}$ | ${0.1512}^{\left\{7\right\}}$ | ${0.13862}^{\left\{1\right\}}$ | ${0.14488}^{\left\{6\right\}}$ | ${0.1547}^{\left\{8\right\}}$ | ${0.1396}^{\left\{2\right\}}$ | ${0.14464}^{\left\{5\right\}}$ | ${0.17669}^{\left\{10\right\}}$ | ${0.157254}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.10671}^{\left\{1\right\}}$ | ${0.1133}^{\left\{3\right\}}$ | ${0.13329}^{\left\{10\right\}}$ | ${0.10839}^{\left\{2\right\}}$ | ${0.12698}^{\left\{9\right\}}$ | ${0.12493}^{\left\{8\right\}}$ | ${0.11725}^{\left\{6\right\}}$ | ${0.12422}^{\left\{7\right\}}$ | ${0.113883}^{\left\{4\right\}}$ | ${0.11606}^{\left\{5\right\}}$ | ||

CIL | $\widehat{\alpha}$ | ${0.51392}^{\left\{3\right\}}$ | ${0.5133}^{\left\{2\right\}}$ | ${0.54792}^{\left\{8\right\}}$ | ${0.49439}^{\left\{1\right\}}$ | ${0.53215}^{\left\{5\right\}}$ | ${0.54138}^{\left\{7\right\}}$ | ${0.51598}^{\left\{4\right\}}$ | ${0.53864}^{\left\{6\right\}}$ | ${0.63506}^{\left\{10\right\}}$ | ${0.58381}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${1.34573}^{\left\{4\right\}}$ | ${1.30256}^{\left\{3\right\}}$ | ${1.67021}^{\left\{10\right\}}$ | ${1.08778}^{\left\{1\right\}}$ | ${1.50623}^{\left\{7\right\}}$ | ${1.52421}^{\left\{8\right\}}$ | ${1.39938}^{\left\{5\right\}}$ | ${1.56061}^{\left\{9\right\}}$ | ${1.43464}^{\left\{6\right\}}$ | ${1.2256}^{\left\{2\right\}}$ | ||

CP | $\widehat{\alpha}$ | ${0.934}^{\left\{4.5\right\}}$ | ${0.93}^{\left\{7\right\}}$ | ${0.914}^{\left\{10\right\}}$ | ${0.938}^{\left\{3\right\}}$ | ${0.932}^{\left\{6\right\}}$ | ${0.918}^{\left\{9\right\}}$ | ${0.946}^{\left\{2\right\}}$ | ${0.922}^{\left\{8\right\}}$ | ${0.934}^{\left\{4.5\right\}}$ | ${0.952}^{\left\{1\right\}}$ | |

$\widehat{\beta}$ | ${0.866}^{\left\{8\right\}}$ | ${0.93}^{\left\{3\right\}}$ | ${0.9}^{\left\{7\right\}}$ | ${0.846}^{\left\{10\right\}}$ | ${0.918}^{\left\{5\right\}}$ | ${0.928}^{\left\{4\right\}}$ | ${0.938}^{\left\{2\right\}}$ | ${0.914}^{\left\{6\right\}}$ | ${0.958}^{\left\{1\right\}}$ | ${0.85}^{\left\{9\right\}}$ | ||

${D}_{abs}$ | ${0.0445}^{\left\{1\right\}}$ | ${0.04683}^{\left\{2\right\}}$ | ${0.05}^{\left\{8\right\}}$ | ${0.04764}^{\left\{4\right\}}$ | ${0.04912}^{\left\{6\right\}}$ | ${0.04955}^{\left\{7\right\}}$ | ${0.04707}^{\left\{3\right\}}$ | ${0.04909}^{\left\{5\right\}}$ | ${0.05837}^{\left\{10\right\}}$ | ${0.05328}^{\left\{9\right\}}$ | ||

${D}_{max}$ | ${0.07143}^{\left\{1\right\}}$ | ${0.07507}^{\left\{3\right\}}$ | ${0.08297}^{\left\{8\right\}}$ | ${0.07491}^{\left\{2\right\}}$ | ${0.07885}^{\left\{5\right\}}$ | ${0.07994}^{\left\{6\right\}}$ | ${0.076}^{\left\{4\right\}}$ | ${0.08071}^{\left\{7\right\}}$ | ${0.0909}^{\left\{10\right\}}$ | ${0.08411}^{\left\{9\right\}}$ | ||

ASAE | ${0.03572}^{\left\{1\right\}}$ | ${0.03225}^{\left\{3\right\}}$ | ${0.03368}^{\left\{8\right\}}$ | ${0.03406}^{\left\{2\right\}}$ | ${0.03232}^{\left\{5\right\}}$ | ${0.03317}^{\left\{6\right\}}$ | ${0.03141}^{\left\{4\right\}}$ | ${0.03556}^{\left\{7\right\}}$ | ${0.04476}^{\left\{10\right\}}$ | ${0.04204}^{\left\{9\right\}}$ | ||

$\sum Ranks$ | ${39.5}^{\left\{2\right\}}$ | ${46}^{\left\{3\right\}}$ | ${95}^{\left\{9\right\}}$ | ${31}^{\left\{1\right\}}$ | ${80}^{\left\{5.5\right\}}$ | ${89}^{\left\{7\right\}}$ | ${59}^{\left\{4\right\}}$ | ${80}^{\left\{5.5\right\}}$ | ${105.5}^{\left\{10\right\}}$ | ${90}^{\left\{8\right\}}$ | ||

70 | BIAS | $\widehat{\alpha}$ | ${0.06856}^{\left\{2\right\}}$ | ${0.07195}^{\left\{6\right\}}$ | ${0.07673}^{\left\{8\right\}}$ | ${0.06718}^{\left\{1\right\}}$ | ${0.06916}^{\left\{3\right\}}$ | ${0.07376}^{\left\{7\right\}}$ | ${0.07073}^{\left\{5\right\}}$ | ${0.06962}^{\left\{4\right\}}$ | ${0.087905}^{\left\{10\right\}}$ | ${0.08025}^{\left\{9\right\}}$ |

$\widehat{\beta}$ | ${0.15983}^{\left\{1\right\}}$ | ${0.17303}^{\left\{3\right\}}$ | ${0.20522}^{\left\{8\right\}}$ | ${0.17467}^{\left\{4\right\}}$ | ${0.19925}^{\left\{7\right\}}$ | ${0.17199}^{\left\{2\right\}}$ | ${0.17556}^{\left\{5\right\}}$ | ${0.20621}^{\left\{9\right\}}$ | ${0.208565}^{\left\{10\right\}}$ | ${0.19013}^{\left\{6\right\}}$ | ||

MSE | $\widehat{\alpha}$ | ${0.00717}^{\left\{2\right\}}$ | ${0.00804}^{\left\{6\right\}}$ | ${0.00909}^{\left\{8\right\}}$ | ${0.00714}^{\left\{1\right\}}$ | ${0.0076}^{\left\{4\right\}}$ | ${0.00835}^{\left\{7\right\}}$ | ${0.00778}^{\left\{5\right\}}$ | ${0.00733}^{\left\{3\right\}}$ | ${0.012075}^{\left\{10\right\}}$ | ${0.010354}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.04209}^{\left\{1\right\}}$ | ${0.04718}^{\left\{3\right\}}$ | ${0.07014}^{\left\{9\right\}}$ | ${0.04581}^{\left\{2\right\}}$ | ${0.06284}^{\left\{7\right\}}$ | ${0.04942}^{\left\{4\right\}}$ | ${0.05057}^{\left\{5\right\}}$ | ${0.06907}^{\left\{8\right\}}$ | ${0.072159}^{\left\{10\right\}}$ | ${0.053924}^{\left\{6\right\}}$ | ||

MRE | $\widehat{\alpha}$ | ${0.09142}^{\left\{2\right\}}$ | ${0.09593}^{\left\{6\right\}}$ | ${0.10231}^{\left\{8\right\}}$ | ${0.08957}^{\left\{1\right\}}$ | ${0.09222}^{\left\{3\right\}}$ | ${0.09834}^{\left\{7\right\}}$ | ${0.09431}^{\left\{5\right\}}$ | ${0.09282}^{\left\{4\right\}}$ | ${0.117207}^{\left\{10\right\}}$ | ${0.107}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.06393}^{\left\{1\right\}}$ | ${0.06921}^{\left\{3\right\}}$ | ${0.08209}^{\left\{8\right\}}$ | ${0.06987}^{\left\{4\right\}}$ | ${0.0797}^{\left\{7\right\}}$ | ${0.0688}^{\left\{2\right\}}$ | ${0.07023}^{\left\{5\right\}}$ | ${0.08248}^{\left\{9\right\}}$ | ${0.083426}^{\left\{10\right\}}$ | ${0.076052}^{\left\{6\right\}}$ | ||

CIL | $\widehat{\alpha}$ | ${0.32703}^{\left\{2\right\}}$ | ${0.33103}^{\left\{3\right\}}$ | ${0.34819}^{\left\{7\right\}}$ | ${0.32019}^{\left\{1\right\}}$ | ${0.34562}^{\left\{6\right\}}$ | ${0.35217}^{\left\{8\right\}}$ | ${0.33541}^{\left\{5\right\}}$ | ${0.33313}^{\left\{4\right\}}$ | ${0.40838}^{\left\{10\right\}}$ | ${0.38085}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.81055}^{\left\{2\right\}}$ | ${0.82548}^{\left\{4\right\}}$ | ${1.01044}^{\left\{10\right\}}$ | ${0.72005}^{\left\{1\right\}}$ | ${0.95589}^{\left\{9\right\}}$ | ${0.92663}^{\left\{6\right\}}$ | ${0.88022}^{\left\{5\right\}}$ | ${0.9459}^{\left\{7\right\}}$ | ${0.94995}^{\left\{8\right\}}$ | ${0.82149}^{\left\{3\right\}}$ | ||

CP | $\widehat{\alpha}$ | ${0.92}^{\left\{9\right\}}$ | ${0.916}^{\left\{10\right\}}$ | ${0.922}^{\left\{8\right\}}$ | ${0.936}^{\left\{2\right\}}$ | ${0.932}^{\left\{6\right\}}$ | ${0.934}^{\left\{4\right\}}$ | ${0.938}^{\left\{1\right\}}$ | ${0.934}^{\left\{4\right\}}$ | ${0.928}^{\left\{7\right\}}$ | ${0.934}^{\left\{4\right\}}$ | |

$\widehat{\beta}$ | ${0.916}^{\left\{7.5\right\}}$ | ${0.946}^{\left\{3\right\}}$ | ${0.938}^{\left\{4\right\}}$ | ${0.852}^{\left\{10\right\}}$ | ${0.932}^{\left\{5\right\}}$ | ${0.952}^{\left\{1\right\}}$ | ${0.948}^{\left\{2\right\}}$ | ${0.916}^{\left\{7.5\right\}}$ | ${0.918}^{\left\{6\right\}}$ | ${0.89}^{\left\{9\right\}}$ | ||

${D}_{abs}$ | ${0.02991}^{\left\{1\right\}}$ | ${0.03146}^{\left\{5\right\}}$ | ${0.03383}^{\left\{8\right\}}$ | ${0.03008}^{\left\{2\right\}}$ | ${0.03194}^{\left\{6\right\}}$ | ${0.03093}^{\left\{3\right\}}$ | ${0.03115}^{\left\{4\right\}}$ | ${0.0328}^{\left\{7\right\}}$ | ${0.03833}^{\left\{10\right\}}$ | ${0.03665}^{\left\{9\right\}}$ | ||

${D}_{max}$ | ${0.04776}^{\left\{1\right\}}$ | ${0.05035}^{\left\{5\right\}}$ | ${0.05516}^{\left\{8\right\}}$ | ${0.04781}^{\left\{2\right\}}$ | ${0.05197}^{\left\{6\right\}}$ | ${0.04931}^{\left\{3\right\}}$ | ${0.05002}^{\left\{4\right\}}$ | ${0.05382}^{\left\{7\right\}}$ | ${0.06068}^{\left\{10\right\}}$ | ${0.0581}^{\left\{9\right\}}$ | ||

ASAE | ${0.02068}^{\left\{1\right\}}$ | ${0.01939}^{\left\{5\right\}}$ | ${0.01989}^{\left\{8\right\}}$ | ${0.02028}^{\left\{2\right\}}$ | ${0.01993}^{\left\{6\right\}}$ | ${0.01991}^{\left\{3\right\}}$ | ${0.01928}^{\left\{4\right\}}$ | ${0.02108}^{\left\{7\right\}}$ | ${0.02772}^{\left\{10\right\}}$ | ${0.02626}^{\left\{9\right\}}$ | ||

$\sum Ranks$ | ${27.5}^{\left\{1\right\}}$ | ${55}^{\left\{3\right\}}$ | ${95}^{\left\{9\right\}}$ | ${35}^{\left\{2\right\}}$ | ${74}^{\left\{6\right\}}$ | ${70}^{\left\{5\right\}}$ | ${68}^{\left\{4\right\}}$ | ${80.5}^{\left\{7\right\}}$ | ${117}^{\left\{10\right\}}$ | ${93}^{\left\{8\right\}}$ | ||

150 | BIAS | $\widehat{\alpha}$ | ${0.04839}^{\left\{3\right\}}$ | ${0.04882}^{\left\{5\right\}}$ | ${0.05187}^{\left\{8\right\}}$ | ${0.04642}^{\left\{2\right\}}$ | ${0.04981}^{\left\{7\right\}}$ | ${0.04878}^{\left\{4\right\}}$ | ${0.04894}^{\left\{6\right\}}$ | ${0.04527}^{\left\{1\right\}}$ | ${0.058573}^{\left\{10\right\}}$ | ${0.054359}^{\left\{9\right\}}$ |

$\widehat{\beta}$ | ${0.10898}^{\left\{1\right\}}$ | ${0.12269}^{\left\{4\right\}}$ | ${0.13676}^{\left\{9\right\}}$ | ${0.10966}^{\left\{2\right\}}$ | ${0.12902}^{\left\{7\right\}}$ | ${0.12878}^{\left\{6\right\}}$ | ${0.12181}^{\left\{3\right\}}$ | ${0.13365}^{\left\{8\right\}}$ | ${0.139931}^{\left\{10\right\}}$ | ${0.125376}^{\left\{5\right\}}$ | ||

MSE | $\widehat{\alpha}$ | ${0.00366}^{\left\{5\right\}}$ | ${0.00364}^{\left\{3\right\}}$ | ${0.00419}^{\left\{8\right\}}$ | ${0.00334}^{\left\{1\right\}}$ | ${0.00411}^{\left\{7\right\}}$ | ${0.00371}^{\left\{6\right\}}$ | ${0.00365}^{\left\{4\right\}}$ | ${0.00338}^{\left\{2\right\}}$ | ${0.005487}^{\left\{10\right\}}$ | ${0.004537}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.01914}^{\left\{1\right\}}$ | ${0.0232}^{\left\{3\right\}}$ | ${0.03059}^{\left\{9\right\}}$ | ${0.01936}^{\left\{2\right\}}$ | ${0.02756}^{\left\{8\right\}}$ | ${0.02652}^{\left\{6\right\}}$ | ${0.02459}^{\left\{5\right\}}$ | ${0.02753}^{\left\{7\right\}}$ | ${0.031513}^{\left\{10\right\}}$ | ${0.024105}^{\left\{4\right\}}$ | ||

MRE | $\widehat{\alpha}$ | ${0.06453}^{\left\{3\right\}}$ | ${0.0651}^{\left\{5\right\}}$ | ${0.06917}^{\left\{8\right\}}$ | ${0.06189}^{\left\{2\right\}}$ | ${0.06641}^{\left\{7\right\}}$ | ${0.06504}^{\left\{4\right\}}$ | ${0.06525}^{\left\{6\right\}}$ | ${0.06037}^{\left\{1\right\}}$ | ${0.078097}^{\left\{10\right\}}$ | ${0.072479}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.04359}^{\left\{1\right\}}$ | ${0.04908}^{\left\{4\right\}}$ | ${0.05471}^{\left\{9\right\}}$ | ${0.04387}^{\left\{2\right\}}$ | ${0.05161}^{\left\{7\right\}}$ | ${0.05151}^{\left\{6\right\}}$ | ${0.04873}^{\left\{3\right\}}$ | ${0.05346}^{\left\{8\right\}}$ | ${0.055972}^{\left\{10\right\}}$ | ${0.05015}^{\left\{5\right\}}$ | ||

CIL | $\widehat{\alpha}$ | ${0.2235}^{\left\{2\right\}}$ | ${0.22638}^{\left\{4\right\}}$ | ${0.23513}^{\left\{6\right\}}$ | ${0.22103}^{\left\{1\right\}}$ | ${0.23556}^{\left\{7\right\}}$ | ${0.24114}^{\left\{8\right\}}$ | ${0.22563}^{\left\{3\right\}}$ | ${0.22809}^{\left\{5\right\}}$ | ${0.27973}^{\left\{10\right\}}$ | ${0.25909}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.53384}^{\left\{2\right\}}$ | ${0.55709}^{\left\{3\right\}}$ | ${0.66036}^{\left\{10\right\}}$ | ${0.49685}^{\left\{1\right\}}$ | ${0.65033}^{\left\{9\right\}}$ | ${0.62286}^{\left\{6\right\}}$ | ${0.58012}^{\left\{5\right\}}$ | ${0.6381}^{\left\{7\right\}}$ | ${0.64766}^{\left\{8\right\}}$ | ${0.56991}^{\left\{4\right\}}$ | ||

CP | $\widehat{\alpha}$ | ${0.93}^{\left\{7\right\}}$ | ${0.942}^{\left\{5\right\}}$ | ${0.928}^{\left\{8.5\right\}}$ | ${0.952}^{\left\{1\right\}}$ | ${0.908}^{\left\{10\right\}}$ | ${0.946}^{\left\{3\right\}}$ | ${0.928}^{\left\{8.5\right\}}$ | ${0.946}^{\left\{3\right\}}$ | ${0.932}^{\left\{6\right\}}$ | ${0.946}^{\left\{3\right\}}$ | |

$\widehat{\beta}$ | ${0.916}^{\left\{7\right\}}$ | ${0.932}^{\left\{3\right\}}$ | ${0.934}^{\left\{2\right\}}$ | ${0.872}^{\left\{10\right\}}$ | ${0.93}^{\left\{4\right\}}$ | ${0.918}^{\left\{5.5\right\}}$ | ${0.918}^{\left\{5.5\right\}}$ | ${0.948}^{\left\{1\right\}}$ | ${0.91}^{\left\{8\right\}}$ | ${0.902}^{\left\{9\right\}}$ | ||

${D}_{abs}$ | ${0.02089}^{\left\{3\right\}}$ | ${0.02129}^{\left\{5\right\}}$ | ${0.02239}^{\left\{8\right\}}$ | ${0.02017}^{\left\{1\right\}}$ | ${0.02187}^{\left\{7\right\}}$ | ${0.02072}^{\left\{2\right\}}$ | ${0.02133}^{\left\{6\right\}}$ | ${0.02112}^{\left\{4\right\}}$ | ${0.02574}^{\left\{10\right\}}$ | ${0.02367}^{\left\{9\right\}}$ | ||

${D}_{max}$ | ${0.03339}^{\left\{2\right\}}$ | ${0.03448}^{\left\{5\right\}}$ | ${0.03643}^{\left\{8\right\}}$ | ${0.03207}^{\left\{1\right\}}$ | ${0.03539}^{\left\{7\right\}}$ | ${0.03371}^{\left\{3\right\}}$ | ${0.03428}^{\left\{4\right\}}$ | ${0.03491}^{\left\{6\right\}}$ | ${0.04114}^{\left\{10\right\}}$ | ${0.03749}^{\left\{9\right\}}$ | ||

ASAE | ${0.01303}^{\left\{2\right\}}$ | ${0.01217}^{\left\{5\right\}}$ | ${0.01243}^{\left\{8\right\}}$ | ${0.01297}^{\left\{1\right\}}$ | ${0.01244}^{\left\{7\right\}}$ | ${0.01267}^{\left\{3\right\}}$ | ${0.01217}^{\left\{4\right\}}$ | ${0.01365}^{\left\{6\right\}}$ | ${0.01813}^{\left\{10\right\}}$ | ${0.01639}^{\left\{9\right\}}$ | ||

$\sum Ranks$ | ${38}^{\left\{2\right\}}$ | ${57}^{\left\{4\right\}}$ | ${97.5}^{\left\{9\right\}}$ | ${32}^{\left\{1\right\}}$ | ${85}^{\left\{7\right\}}$ | ${69.5}^{\left\{5\right\}}$ | ${54}^{\left\{3\right\}}$ | ${75}^{\left\{6\right\}}$ | ${116}^{\left\{10\right\}}$ | ${91}^{\left\{8\right\}}$ | ||

350 | BIAS | $\widehat{\alpha}$ | ${0.03208}^{\left\{6\right\}}$ | ${0.03185}^{\left\{4\right\}}$ | ${0.03297}^{\left\{7\right\}}$ | ${0.03009}^{\left\{1\right\}}$ | ${0.03199}^{\left\{5\right\}}$ | ${0.03517}^{\left\{9\right\}}$ | ${0.03154}^{\left\{3\right\}}$ | ${0.03153}^{\left\{2\right\}}$ | ${0.037721}^{\left\{10\right\}}$ | ${0.034645}^{\left\{8\right\}}$ |

$\widehat{\beta}$ | ${0.07032}^{\left\{2\right\}}$ | ${0.07817}^{\left\{3\right\}}$ | ${0.09107}^{\left\{10\right\}}$ | ${0.07003}^{\left\{1\right\}}$ | ${0.09063}^{\left\{9\right\}}$ | ${0.08678}^{\left\{6\right\}}$ | ${0.07856}^{\left\{4\right\}}$ | ${0.08811}^{\left\{7\right\}}$ | ${0.088915}^{\left\{8\right\}}$ | ${0.081702}^{\left\{5\right\}}$ | ||

MSE | $\widehat{\alpha}$ | ${0.00163}^{\left\{6\right\}}$ | ${0.0016}^{\left\{5\right\}}$ | ${0.00167}^{\left\{7\right\}}$ | ${0.00143}^{\left\{1\right\}}$ | ${0.00158}^{\left\{4\right\}}$ | ${0.00196}^{\left\{9\right\}}$ | ${0.00153}^{\left\{3\right\}}$ | ${0.0015}^{\left\{2\right\}}$ | ${0.002239}^{\left\{10\right\}}$ | ${0.001921}^{\left\{8\right\}}$ | |

$\widehat{\beta}$ | ${0.00786}^{\left\{2\right\}}$ | ${0.00955}^{\left\{3\right\}}$ | ${0.01296}^{\left\{10\right\}}$ | ${0.00777}^{\left\{1\right\}}$ | ${0.01281}^{\left\{8\right\}}$ | ${0.0117}^{\left\{6\right\}}$ | ${0.0096}^{\left\{4\right\}}$ | ${0.0118}^{\left\{7\right\}}$ | ${0.012956}^{\left\{9\right\}}$ | ${0.010593}^{\left\{5\right\}}$ | ||

MRE | $\widehat{\alpha}$ | ${0.04277}^{\left\{6\right\}}$ | ${0.04247}^{\left\{4\right\}}$ | ${0.04396}^{\left\{7\right\}}$ | ${0.04012}^{\left\{1\right\}}$ | ${0.04266}^{\left\{5\right\}}$ | ${0.04689}^{\left\{9\right\}}$ | ${0.04205}^{\left\{3\right\}}$ | ${0.04204}^{\left\{2\right\}}$ | ${0.050295}^{\left\{10\right\}}$ | ${0.046194}^{\left\{8\right\}}$ | |

$\widehat{\beta}$ | ${0.02813}^{\left\{2\right\}}$ | ${0.03127}^{\left\{3\right\}}$ | ${0.03643}^{\left\{10\right\}}$ | ${0.02801}^{\left\{1\right\}}$ | ${0.03625}^{\left\{9\right\}}$ | ${0.03471}^{\left\{6\right\}}$ | ${0.03143}^{\left\{4\right\}}$ | ${0.03525}^{\left\{7\right\}}$ | ${0.035566}^{\left\{8\right\}}$ | ${0.032681}^{\left\{5\right\}}$ | ||

CIL | $\widehat{\alpha}$ | ${0.14599}^{\left\{2\right\}}$ | ${0.14821}^{\left\{3\right\}}$ | ${0.15274}^{\left\{7\right\}}$ | ${0.14314}^{\left\{1\right\}}$ | ${0.15253}^{\left\{6\right\}}$ | ${0.15763}^{\left\{8\right\}}$ | ${0.14921}^{\left\{5\right\}}$ | ${0.14894}^{\left\{4\right\}}$ | ${0.18237}^{\left\{10\right\}}$ | ${0.16883}^{\left\{9\right\}}$ | |

$\widehat{\beta}$ | ${0.34319}^{\left\{2\right\}}$ | ${0.36566}^{\left\{3\right\}}$ | ${0.42925}^{\left\{10\right\}}$ | ${0.32466}^{\left\{1\right\}}$ | ${0.42448}^{\left\{8\right\}}$ | ${0.40707}^{\left\{6\right\}}$ | ${0.37558}^{\left\{4\right\}}$ | ${0.41882}^{\left\{7\right\}}$ | ${0.42838}^{\left\{9\right\}}$ | ${0.37714}^{\left\{5\right\}}$ | ||

CP | $\widehat{\alpha}$ | ${0.922}^{\left\{8.5\right\}}$ | ${0.918}^{\left\{10\right\}}$ | ${0.938}^{\left\{5\right\}}$ | ${0.928}^{\left\{7\right\}}$ | ${0.946}^{\left\{2.5\right\}}$ | ${0.922}^{\left\{8.5\right\}}$ | ${0.946}^{\left\{2.5\right\}}$ | ${0.95}^{\left\{1\right\}}$ | ${0.932}^{\left\{6\right\}}$ | ${0.942}^{\left\{4\right\}}$ | |

$\widehat{\beta}$ | ${0.928}^{\left\{6\right\}}$ | ${0.942}^{\left\{2\right\}}$ | ${0.924}^{\left\{7\right\}}$ | ${0.906}^{\left\{10\right\}}$ | ${0.934}^{\left\{3.5\right\}}$ | ${0.93}^{\left\{5\right\}}$ | ${0.934}^{\left\{3.5\right\}}$ | ${0.956}^{\left\{1\right\}}$ | ${0.91}^{\left\{9\right\}}$ | ${0.916}^{\left\{8\right\}}$ | ||

${D}_{abs}$ | ${0.01384}^{\left\{3\right\}}$ | ${0.01411}^{\left\{4\right\}}$ | ${0.01464}^{\left\{7\right\}}$ | ${0.01318}^{\left\{1\right\}}$ | ${0.01437}^{\left\{5\right\}}$ | ${0.01508}^{\left\{8\right\}}$ | ${0.01378}^{\left\{2\right\}}$ | ${0.01462}^{\left\{6\right\}}$ | ${0.01642}^{\left\{10\right\}}$ | ${0.01519}^{\left\{9\right\}}$ | ||

${D}_{max}$ | ${0.02195}^{\left\{2\right\}}$ | ${0.02266}^{\left\{4\right\}}$ | ${0.02391}^{\left\{7\right\}}$ | ${0.02094}^{\left\{1\right\}}$ | ${0.02344}^{\left\{5\right\}}$ | ${0.02418}^{\left\{8\right\}}$ | ${0.02213}^{\left\{3\right\}}$ | ${0.02389}^{\left\{6\right\}}$ | ${0.02633}^{\left\{10\right\}}$ | ${0.02425}^{\left\{9\right\}}$ | ||

ASAE | ${0.00807}^{\left\{2\right\}}$ | ${0.00772}^{\left\{4\right\}}$ | ${0.00779}^{\left\{7\right\}}$ | ${0.00806}^{\left\{1\right\}}$ | ${0.00775}^{\left\{5\right\}}$ | ${0.00786}^{\left\{8\right\}}$ | ${0.00757}^{\left\{3\right\}}$ | ${0.0081}^{\left\{6\right\}}$ | ${0.01083}^{\left\{10\right\}}$ | ${0.01006}^{\left\{9\right\}}$ | ||

$\sum Ranks$ | ${47.5}^{\left\{2\right\}}$ | ${48}^{\left\{3\right\}}$ | ${96}^{\left\{9\right\}}$ | ${21}^{\left\{1\right\}}$ | ${83}^{\left\{6\right\}}$ | ${88.5}^{\left\{7\right\}}$ | ${52}^{\left\{4\right\}}$ | ${78}^{\left\{5\right\}}$ | ${111}^{\left\{10\right\}}$ | ${90}^{\left\{8\right\}}$ |

**Table 3.**Numerical values of PHD simulation for $(\alpha =2.5,\phantom{\rule{3.33333pt}{0ex}}\beta =1.5)$.

n | Est. | Est.Par. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE |
---|---|---|---|---|---|---|---|---|---|---|---|---|

30 | BIAS | $\widehat{\alpha}$ | ${0.38182}^{\left\{6\right\}}$ | ${0.36672}^{\left\{4\right\}}$ | ${0.43965}^{\left\{9\right\}}$ | ${0.32326}^{\left\{1\right\}}$ | ${0.41934}^{\left\{8\right\}}$ | ${0.35354}^{\left\{3\right\}}$ | ${0.38721}^{\left\{7\right\}}$ | ${0.48828}^{\left\{10\right\}}$ | ${0.375462}^{\left\{5\right\}}$ | ${0.333218}^{\left\{2\right\}}$ |

$\widehat{\beta}$ | ${0.16983}^{\left\{3\right\}}$ | ${0.16462}^{\left\{1\right\}}$ | ${0.20601}^{\left\{8\right\}}$ | ${0.167}^{\left\{2\right\}}$ | ${0.21083}^{\left\{9\right\}}$ | ${0.17504}^{\left\{4\right\}}$ | ${0.18252}^{\left\{5\right\}}$ | ${0.21681}^{\left\{10\right\}}$ | ${0.204852}^{\left\{7\right\}}$ | ${0.186542}^{\left\{6\right\}}$ | ||

MSE | $\widehat{\alpha}$ | ${0.25276}^{\left\{5\right\}}$ | ${0.22803}^{\left\{4\right\}}$ | ${0.42909}^{\left\{9\right\}}$ | ${0.16548}^{\left\{1\right\}}$ | ${0.37347}^{\left\{8\right\}}$ | ${0.21613}^{\left\{3\right\}}$ | ${0.28947}^{\left\{7\right\}}$ | ${0.5349}^{\left\{10\right\}}$ | ${0.279213}^{\left\{6\right\}}$ | ${0.18436}^{\left\{2\right\}}$ | |

$\widehat{\beta}$ | ${0.04991}^{\left\{3\right\}}$ | ${0.04396}^{\left\{2\right\}}$ | ${0.07554}^{\left\{9\right\}}$ | ${0.0431}^{\left\{1\right\}}$ | ${0.07331}^{\left\{8\right\}}$ | ${0.05198}^{\left\{5\right\}}$ | ${0.05306}^{\left\{6\right\}}$ | ${0.07953}^{\left\{10\right\}}$ | ${0.071142}^{\left\{7\right\}}$ | ${0.051791}^{\left\{4\right\}}$ | ||

MRE | $\widehat{\alpha}$ | ${0.15273}^{\left\{6\right\}}$ | ${0.14669}^{\left\{4\right\}}$ | ${0.17586}^{\left\{9\right\}}$ | ${0.1293}^{\left\{1\right\}}$ | ${0.16774}^{\left\{8\right\}}$ | ${0.14142}^{\left\{3\right\}}$ | ${0.15488}^{\left\{7\right\}}$ | ${0.19531}^{\left\{10\right\}}$ | ${0.150185}^{\left\{5\right\}}$ | ${0.133287}^{\left\{2\right\}}$ | |

$\widehat{\beta}$ | ${0.11322}^{\left\{3\right\}}$ | ${0.10975}^{\left\{1\right\}}$ | ${0.13734}^{\left\{8\right\}}$ | ${0.11133}^{\left\{2\right\}}$ | ${0.14055}^{\left\{9\right\}}$ | ${0.11669}^{\left\{4\right\}}$ | ${0.12168}^{\left\{5\right\}}$ | ${0.14454}^{\left\{10\right\}}$ | ${0.136568}^{\left\{7\right\}}$ | ${0.124361}^{\left\{6\right\}}$ | ||

CIL | $\widehat{\alpha}$ | ${1.97994}^{\left\{6\right\}}$ | ${1.97965}^{\left\{5\right\}}$ | ${2.6424}^{\left\{9\right\}}$ | ${1.47309}^{\left\{1\right\}}$ | ${2.19813}^{\left\{8\right\}}$ | ${1.97044}^{\left\{4\right\}}$ | ${2.01843}^{\left\{7\right\}}$ | ${3.1013}^{\left\{10\right\}}$ | ${1.92404}^{\left\{3\right\}}$ | ${1.76956}^{\left\{2\right\}}$ | |

$\widehat{\beta}$ | ${0.84902}^{\left\{4\right\}}$ | ${0.84759}^{\left\{3\right\}}$ | ${1.05147}^{\left\{9\right\}}$ | ${0.70878}^{\left\{1\right\}}$ | ${0.95931}^{\left\{7\right\}}$ | ${0.93416}^{\left\{6\right\}}$ | ${0.89461}^{\left\{5\right\}}$ | ${1.066}^{\left\{10\right\}}$ | ${0.97114}^{\left\{8\right\}}$ | ${0.81297}^{\left\{2\right\}}$ | ||

CP | $\widehat{\alpha}$ | ${0.878}^{\left\{10\right\}}$ | ${0.936}^{\left\{3\right\}}$ | ${0.906}^{\left\{9\right\}}$ | ${0.916}^{\left\{7.5\right\}}$ | ${0.928}^{\left\{5\right\}}$ | ${0.934}^{\left\{4\right\}}$ | ${0.916}^{\left\{7.5\right\}}$ | ${0.926}^{\left\{6\right\}}$ | ${0.964}^{\left\{1\right\}}$ | ${0.956}^{\left\{2\right\}}$ | |

$\widehat{\beta}$ | ${0.898}^{\left\{8\right\}}$ | ${0.948}^{\left\{1\right\}}$ | ${0.918}^{\left\{7\right\}}$ | ${0.86}^{\left\{10\right\}}$ | ${0.93}^{\left\{5.5\right\}}$ | ${0.94}^{\left\{3\right\}}$ | ${0.938}^{\left\{4\right\}}$ | ${0.93}^{\left\{5.5\right\}}$ | ${0.946}^{\left\{2\right\}}$ | ${0.89}^{\left\{9\right\}}$ | ||

${D}_{abs}$ | ${0.04584}^{\left\{2\right\}}$ | ${0.04477}^{\left\{1\right\}}$ | ${0.0483}^{\left\{5\right\}}$ | ${0.04624}^{\left\{3\right\}}$ | ${0.05096}^{\left\{8\right\}}$ | ${0.04792}^{\left\{4\right\}}$ | ${0.0489}^{\left\{6\right\}}$ | ${0.05038}^{\left\{7\right\}}$ | ${0.05526}^{\left\{10\right\}}$ | ${0.05142}^{\left\{9\right\}}$ | ||

${D}_{max}$ | ${0.074}^{\left\{3\right\}}$ | ${0.07262}^{\left\{1\right\}}$ | ${0.07999}^{\left\{6\right\}}$ | ${0.07279}^{\left\{2\right\}}$ | ${0.0831}^{\left\{8\right\}}$ | ${0.07712}^{\left\{4\right\}}$ | ${0.07876}^{\left\{5\right\}}$ | ${0.08368}^{\left\{9\right\}}$ | ${0.08802}^{\left\{10\right\}}$ | ${0.08099}^{\left\{7\right\}}$ | ||

ASAE | ${0.035}^{\left\{3\right\}}$ | ${0.03171}^{\left\{1\right\}}$ | ${0.0334}^{\left\{6\right\}}$ | ${0.0331}^{\left\{2\right\}}$ | ${0.0339}^{\left\{8\right\}}$ | ${0.03138}^{\left\{4\right\}}$ | ${0.03154}^{\left\{5\right\}}$ | ${0.03815}^{\left\{9\right\}}$ | ${0.04377}^{\left\{10\right\}}$ | ${0.04078}^{\left\{7\right\}}$ | ||

$\sum Ranks$ | ${52}^{\left\{3\right\}}$ | ${47}^{\left\{2\right\}}$ | ${92}^{\left\{7\right\}}$ | ${23.5}^{\left\{1\right\}}$ | ${98.5}^{\left\{9\right\}}$ | ${56}^{\left\{4\right\}}$ | ${72.5}^{\left\{6\right\}}$ | ${114.5}^{\left\{10\right\}}$ | ${97}^{\left\{8\right\}}$ | ${62}^{\left\{5\right\}}$ | ||

70 | BIAS | $\widehat{\alpha}$ | ${0.23569}^{\left\{4\right\}}$ | ${0.24021}^{\left\{5\right\}}$ | ${0.25999}^{\left\{8\right\}}$ | ${0.22506}^{\left\{2\right\}}$ | ${0.26562}^{\left\{9\right\}}$ | ${0.21461}^{\left\{1\right\}}$ | ${0.23276}^{\left\{3\right\}}$ | ${0.30826}^{\left\{10\right\}}$ | ${0.24301}^{\left\{6\right\}}$ | ${0.251691}^{\left\{7\right\}}$ |

$\widehat{\beta}$ | ${0.11055}^{\left\{1\right\}}$ | ${0.11273}^{\left\{3\right\}}$ | ${0.13443}^{\left\{7\right\}}$ | ${0.11348}^{\left\{5\right\}}$ | ${0.13466}^{\left\{8\right\}}$ | ${0.11305}^{\left\{4\right\}}$ | ${0.11182}^{\left\{2\right\}}$ | ${0.14247}^{\left\{10\right\}}$ | ${0.136529}^{\left\{9\right\}}$ | ${0.122137}^{\{6}$ |