# Estimating the Entropy for Lomax Distribution Based on Generalized Progressively Hybrid Censoring

^{*}

## Abstract

**:**

## 1. Introduction

- Case I ($T<{X}_{k}<{X}_{m}$)${T}_{end}^{*}={X}_{k}$, ${R}_{1}^{*}=n-k-{\sum}_{i=1}^{k-1}{R}_{i}.$
- Case II (${X}_{k}<T<{X}_{m}$)${T}_{end}^{*}=T$, ${R}_{2}^{*}=n-D-{\sum}_{i=1}^{D}{R}_{i}.$
- Case III (${X}_{k}<{X}_{m}<T$)${T}_{end}^{*}={X}_{m}$, ${R}_{3}^{*}=n-m-{\sum}_{i=1}^{m-1}{R}_{i}={R}_{m}.$

## 2. Maximum Likelihood Estimator

## 3. Bayesian Estimation

#### 3.1. Prior and Posterior Distributions

#### 3.2. Loss Function

#### 3.3. Lindley Method

**For squared error loss function**, we take:$$g(\alpha ,\lambda )=H(f)=log(\lambda )-log(\alpha )+\frac{1}{\alpha}+1.$$Then we can compute that:$$\begin{array}{ccc}\hfill {u}_{1}& =& -\frac{1}{\alpha}-\frac{1}{{\alpha}^{2}},\hfill \\ \hfill {u}_{2}& =& \frac{1}{\lambda},\hfill \\ \hfill {u}_{12}& =& {u}_{21}=0,\hfill \\ \hfill {u}_{11}& =& \frac{1}{{\alpha}^{2}}+\frac{2}{{\alpha}^{3}},\hfill \\ \hfill {u}_{22}& =& -\frac{1}{{\lambda}^{2}}.\hfill \end{array}$$Then using Equation (18), the Bayesian estimates under squared error loss function can be obtained as:$$\begin{array}{ccc}\widehat{{H}_{S}}\hfill & =& \widehat{g}\hfill \\ & =& g(\widehat{\alpha},\widehat{\lambda})+0.5[({u}_{11}{\tau}_{11}+{u}_{22}{\tau}_{22})+{l}_{30}({u}_{1}{\tau}_{11}+{u}_{2}{\tau}_{12}){\tau}_{11}+{l}_{03}({u}_{2}{\tau}_{22}+{u}_{1}{\tau}_{21}){\tau}_{22}\hfill \\ & & +{l}_{12}(3{u}_{2}{\tau}_{22}{\tau}_{21}+{u}_{1}({\tau}_{22}{\tau}_{11}+2{\tau}_{21}^{2}))]+{p}_{1}({u}_{1}{\tau}_{11}+{u}_{2}{\tau}_{21})+{p}_{2}({u}_{2}{\tau}_{22}+{u}_{1}{\tau}_{12})\hfill \end{array}$$**Further, for the linex loss function**, we take:$$g(\alpha ,\lambda )={e}^{-hH(f)},\phantom{\rule{2.em}{0ex}}H(f)=log(\lambda )-log(\alpha )+\frac{1}{\alpha}+1.$$We can obtain that:$$\begin{array}{c}{u}_{1}=h{e}^{-hH}(\frac{1}{\alpha}+\frac{1}{{\alpha}^{2}}),\hfill \\ {u}_{2}=-\frac{h}{\lambda}{e}^{-hH},\hfill \\ {u}_{12}={u}_{21}=-\frac{{h}^{2}}{\lambda}{e}^{-hH}(\frac{1}{\alpha}+\frac{1}{{\alpha}^{2}}),\hfill \\ {u}_{11}=h{e}^{-hH}[h{(\frac{1}{\alpha}+\frac{1}{{\alpha}^{2}})}^{2}-(\frac{1}{{\alpha}^{2}}+\frac{2}{{\alpha}^{3}})],\hfill \\ {u}_{22}=\frac{h}{{\lambda}^{2}}{e}^{-hH}(1+h).\hfill \end{array}$$Similarly, using Equation (18), we can derive the Bayesian estimates of entropy under the linex loss function as:$$\begin{array}{ccc}\widehat{{H}_{l}}\hfill & =& -\frac{1}{h}log(\widehat{g})\hfill \\ & =& -\frac{1}{h}log\{g(\widehat{\alpha},\widehat{\lambda})+0.5[({u}_{11}{\tau}_{11}+2{u}_{12}{\tau}_{12}+{u}_{22}{\tau}_{22})+{l}_{30}({u}_{1}{\tau}_{11}+{u}_{2}{\tau}_{12}){\tau}_{11}\hfill \\ & & +{l}_{03}({u}_{2}{\tau}_{22}+{u}_{1}{\tau}_{21}){\tau}_{22}+{l}_{12}(3{u}_{2}{\tau}_{22}{\tau}_{21}+{u}_{1}({\tau}_{22}{\tau}_{11}+2{\tau}_{21}^{2}))]+{p}_{1}({u}_{1}{\tau}_{11}+{u}_{2}{\tau}_{21})\hfill \\ & & +{p}_{2}({u}_{2}{\tau}_{22}+{u}_{1}{\tau}_{12}))\}.\hfill \end{array}$$**For general entropy loss function**, we take:$$g(\alpha ,\lambda )=H{(f)}^{-q},\phantom{\rule{2.em}{0ex}}H(f)=log(\lambda )-log(\alpha )+\frac{1}{\alpha}+1.$$We can derive that:$$\begin{array}{c}{u}_{1}=q{H}^{-(q+1)}(\frac{1}{\alpha}+\frac{1}{{\alpha}^{2}}),\hfill \\ {u}_{2}=-\frac{q}{\lambda}{H}^{-(q+1)},\hfill \\ {u}_{12}={u}_{21}=-\frac{q(q+1)}{\lambda}{H}^{-(q+2)}(\frac{1}{\alpha}+\frac{1}{{\alpha}^{2}}),\hfill \\ {u}_{11}=\frac{q{H}^{-(q+2)}}{{\alpha}^{2}}[(q+1){(1+\frac{1}{\alpha})}^{2}-H(1+\frac{2}{\alpha})],\hfill \\ {u}_{22}=\frac{q}{{\lambda}^{2}}{H}^{-(q+2)}(1+q+H).\hfill \end{array}$$Thus, the Bayesian estimate under general entropy loss function can be computed using:$$\begin{array}{ccc}\widehat{{H}_{e}}\hfill & =& {\widehat{g}}^{-\frac{1}{q}}=\{g(\widehat{\alpha},\widehat{\lambda})+0.5[({u}_{11}{\tau}_{11}+2{u}_{12}{\tau}_{12}+{u}_{22}{\tau}_{22})+{l}_{30}({u}_{1}{\tau}_{11}+{u}_{2}{\tau}_{12}){\tau}_{11}+\hfill \\ & & {l}_{03}({u}_{2}{\tau}_{22}+{u}_{1}{\tau}_{21}){\tau}_{22}+{l}_{12}(3{u}_{2}{\tau}_{22}{\tau}_{21}+{u}_{1}({\tau}_{22}{\tau}_{11}+2{\tau}_{21}^{2}))]+{p}_{1}({u}_{1}{\tau}_{11}+{u}_{2}{\tau}_{21})\hfill \\ & & +{p}_{2}({u}_{2}{\tau}_{22}+{u}_{1}{\tau}_{12}){)\}}^{-\frac{1}{q}}.\hfill \end{array}$$

#### 3.4. Tierney and Kadane Method

**For squared error loss function**, we take:$$\begin{array}{c}g(\alpha ,\lambda )=H(f)=log(\lambda )-log(\alpha )+\frac{1}{\alpha}+1,\hfill \\ {\delta}_{S}^{*}=\delta (\alpha ,\lambda )+\frac{logH}{n}.\hfill \end{array}$$Let ${\delta}_{S}^{*}$ take derivatives with respect to $\alpha $ and $\lambda $, we obtain:$$\frac{\partial {\delta}_{S}^{*}}{\partial \alpha}=\frac{\partial \delta}{\partial \alpha}-\frac{1}{nH}(\frac{1}{\alpha}+\frac{1}{{\alpha}^{2}})=0,$$$$\frac{\partial {\delta}_{S}^{*}}{\partial \lambda}=\frac{\partial \delta}{\partial \lambda}+\frac{1}{nH\lambda}=0.$$According to the equations above, we can obtain $(\widehat{{\alpha}_{T}^{*}},\widehat{{\lambda}_{T}^{*}})$. In order to compute $|{\sum}^{*}|={[\frac{{\partial}^{2}{\delta}_{S}^{*}}{\partial {\alpha}^{2}}\frac{{\partial}^{2}{\delta}_{S}^{*}}{\partial {\lambda}^{2}}-{(\frac{{\partial}^{2}{\delta}_{S}^{*}}{\partial \alpha \partial \lambda})}^{2}]}^{-1}$ at $(\widehat{{\alpha}_{T}^{*}},\widehat{{\lambda}_{T}^{*}})$, we also need to compute:$$\begin{array}{c}\frac{{\partial}^{2}{\delta}_{S}^{*}}{\partial \alpha \partial \lambda}=\frac{{\partial}^{2}\delta}{\partial \alpha \partial \lambda}+[\frac{1}{n{H}^{2}\lambda}(\frac{1}{\alpha}+\frac{1}{{\alpha}^{2}})],\hfill \\ \frac{{\partial}^{2}{\delta}_{S}^{*}}{\partial {\alpha}^{2}}=\frac{{\partial}^{2}\delta}{\partial {\alpha}^{2}}+\frac{1}{n{H}^{2}{\alpha}^{2}}[H(1+\frac{2}{\alpha})-{(1+\frac{1}{\alpha})}^{2}],\hfill \\ \frac{{\partial}^{2}{\delta}_{S}^{*}}{\partial {\lambda}^{2}}=\frac{{\partial}^{2}\delta}{\partial {\lambda}^{2}}-\frac{H+1}{n{H}^{2}{\lambda}^{2}}.\hfill \end{array}$$So the Bayesian estimate under squared error loss function is:$$\widehat{{H}_{S}}=\widehat{g}=\sqrt{\frac{|{\sum}^{*}|}{|\sum |}}{e}^{n[{\delta}^{*}(\widehat{{\alpha}_{T}^{*}},\widehat{{\lambda}_{T}^{*}})-\delta (\widehat{{\alpha}_{T}},\widehat{{\lambda}_{T}})]}$$**Further, for linex loss function**, we take:$$\begin{array}{c}g(\alpha ,\lambda )={e}^{-hH(f)},\phantom{\rule{2.em}{0ex}}H(f)=log(\lambda )-log(\alpha )+\frac{1}{\alpha}+1,\hfill \\ {\delta}_{l}^{*}=\delta (\alpha ,\lambda )-\frac{hH}{n}.\hfill \end{array}$$Thus, we have:$$\frac{\partial {\delta}_{l}^{*}}{\partial \alpha}=\frac{\partial \delta}{\partial \alpha}+\frac{h}{n}(\frac{1}{\alpha}+\frac{1}{{\alpha}^{2}})=0,$$$$\frac{\partial {\delta}_{l}^{*}}{\partial \lambda}=\frac{\partial \delta}{\partial \lambda}-\frac{h}{n\lambda}=0.$$Using Equations (23) and (24), we can obtain $(\widehat{{\alpha}_{T}^{*}},\widehat{{\lambda}_{T}^{*}})$.We can also compute that:$$\frac{{\partial}^{2}{\delta}_{l}^{*}}{\partial \alpha \partial \lambda}=\frac{{\partial}^{2}\delta}{\partial \alpha \partial \lambda},$$$$\frac{{\partial}^{2}{\delta}_{l}^{*}}{\partial {\alpha}^{2}}=\frac{{\partial}^{2}\delta}{\partial {\alpha}^{2}}-\frac{h}{n{\alpha}^{2}}(1+\frac{2}{\alpha}),$$$$\frac{{\partial}^{2}{\delta}_{l}^{*}}{\partial {\lambda}^{2}}=\frac{{\partial}^{2}\delta}{\partial {\lambda}^{2}}+\frac{h}{n{\lambda}^{2}}.$$Thus, we can compute $|{\sum}^{*}|={[\frac{{\partial}^{2}{\delta}_{S}^{*}}{\partial {\alpha}^{2}}\frac{{\partial}^{2}{\delta}_{S}^{*}}{\partial {\lambda}^{2}}-{(\frac{{\partial}^{2}{\delta}_{S}^{*}}{\partial \alpha \partial \lambda})}^{2}]}^{-1}$ at $(\widehat{{\alpha}_{T}^{*}},\widehat{{\lambda}_{T}^{*}})$. The Bayesian estimate under linex loss function can be derived as:$$\widehat{{H}_{l}}=-\frac{1}{h}log(\widehat{g})=-\frac{1}{h}log[\sqrt{\frac{|{\sum}^{*}|}{|\sum |}}{e}^{n[{\delta}^{*}(\widehat{{\alpha}_{T}^{*}},\widehat{{\lambda}_{T}^{*}})-\delta (\widehat{{\alpha}_{T}},\widehat{{\lambda}_{T}})]}].$$**As for general entropy loss function**, we take:$$\begin{array}{c}g(\alpha ,\lambda )=H{(f)}^{-q},\phantom{\rule{2.em}{0ex}}H(f)=log(\lambda )-log(\alpha )+\frac{1}{\alpha}+1,\hfill \\ {\delta}_{e}^{*}=\delta (\alpha ,\lambda )-\frac{q}{n}log\phantom{\rule{3.33333pt}{0ex}}H.\hfill \end{array}$$We can compute that:$$\frac{\partial {\delta}_{e}^{*}}{\partial \alpha}=\frac{\partial \delta}{\partial \alpha}+\frac{q}{nH}(\frac{1}{\alpha}+\frac{1}{{\alpha}^{2}})=0,$$$$\frac{\partial {\delta}_{e}^{*}}{\partial \lambda}=\frac{\partial \delta}{\partial \lambda}-\frac{q}{nH\lambda}=0.$$Using Equations (25) and (26), we can compute $(\widehat{{\alpha}_{T}^{*}},\widehat{{\lambda}_{T}^{*}})$.We can also compute that:$$\frac{{\partial}^{2}{\delta}_{e}^{*}}{\partial \alpha \partial \lambda}=\frac{{\partial}^{2}\delta}{\partial \alpha \partial \lambda}-\frac{q}{n{H}^{2}\lambda}(\frac{1}{\alpha}+\frac{1}{{\alpha}^{2}}),$$$$\frac{{\partial}^{2}{\delta}_{e}^{*}}{\partial {\alpha}^{2}}=\frac{{\partial}^{2}\delta}{\partial {\alpha}^{2}}+\frac{q}{n{H}^{2}{\alpha}^{2}}[{(1+\frac{1}{\alpha})}^{2}-H(1+\frac{2}{\alpha})],$$$$\frac{{\partial}^{2}{\delta}_{e}^{*}}{\partial {\lambda}^{2}}=\frac{{\partial}^{2}\delta}{\partial {\lambda}^{2}}+\frac{q}{n{H}^{2}{\lambda}^{2}}(1+H).$$Then, we can obtain $|{\sum}^{*}|={[\frac{{\partial}^{2}{\delta}_{e}^{*}}{\partial {\alpha}^{2}}\frac{{\partial}^{2}{\delta}_{e}^{*}}{\partial {\lambda}^{2}}-{(\frac{{\partial}^{2}{\delta}_{e}^{*}}{\partial \alpha \partial \lambda})}^{2}]}^{-1}$ at $(\widehat{{\alpha}_{T}^{*}},\widehat{{\lambda}_{T}^{*}})$.Obviously, the Bayesian estimate under the general entropy loss function can be derived as:$$\widehat{{H}_{e}}={\widehat{g}}^{-\frac{1}{q}}={[\sqrt{\frac{|{\sum}^{*}|}{|\sum |}}{e}^{n[{\delta}^{*}(\widehat{{\alpha}_{T}^{*}},\widehat{{\lambda}_{T}^{*}})-\delta (\widehat{{\alpha}_{T}},\widehat{{\lambda}_{T}})]}]}^{-\frac{1}{q}}.$$

## 4. Simulation Results

- generate ${Z}_{1},{Z}_{2},\cdots ,{Z}_{m}$, where ${Z}_{i}$$(i=1,2,\cdots ,m)$ is the random variable from standard exponential distribution.
- Let ${Y}_{1}=\frac{{Z}_{1}}{n}$, ${Y}_{i}={Y}_{i-1}+\frac{{Z}_{i}}{n-(i-1)-{\sum}_{k=1}^{i-1}{R}_{k}}$, then $Y=({Y}_{1},{Y}_{2},\cdots ,{Y}_{m})$ is the Type II progressively censored sample from standard exponential distribution.
- Further, let ${X}_{i}={F}^{-1}(1-{e}^{-{Y}_{i}})$, where ${F}^{-1}$ is the inverse of the cumulative distribution function. Then $X=({X}_{1},{X}_{2},\cdots ,{X}_{m})$ is the Type II progressively censored sample for Lomax distribution.
- For pre-fixed T and k, if $T<{X}_{k}<{X}_{m}$, then the generalized progressively hybrid censored sample X is $X=({X}_{1},{X}_{2},\cdots ,{X}_{k})$; if ${X}_{k}<T<{X}_{m}$, then the corresponding generalized progressively hybrid censored sample X is $X=({X}_{1},{X}_{2},\cdots ,{X}_{D})$; if ${X}_{k}<{X}_{m}<T$, then the corresponding generalized progressively hybrid censored sample X is $X=({X}_{1},{X}_{2},\cdots ,{X}_{m})$.

## 5. Data Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Likelihood and Log-Likelihood Functions

## Appendix B. Lindley Method

## References

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**Table 1.**The relative MSEs and biases of entropy estimates with MLE and the Bayes estimates with non-informative priors ($\widehat{H}$ for the MLEs, ${\widehat{H}}_{s}$ for the Bayesian estimates under the squared error loss function, ${\widehat{H}}_{l}$ for the Bayesian estimates under the linex loss function, ${\widehat{H}}_{e}$ for the Bayesian estimates under general entropy loss function, Lindley for the Lindley method and TK for the Tierney and Kadane method).

n | m | k | $\mathit{Sch}$ | $\widehat{\mathit{H}}$ | ${\widehat{\mathit{H}}}_{\mathit{s}}$ | ${\widehat{\mathit{H}}}_{\mathit{l}}$ | ${\widehat{\mathit{H}}}_{\mathit{e}}$ | $\mathit{Method}$ | ||
---|---|---|---|---|---|---|---|---|---|---|

h = 0.2 | h = 0.5 | q = 0.2 | q = −0.2 | |||||||

60 | 52 | 40 | I | −0.0157 | 0.0518 | 0.0171 | −0.0402 | −0.0712 | −0.0079 | $Lindley$ |

(0.3544) | (0.4323) | (0.4141) | (0.4054) | (0.5412) | (0.4432) | |||||

0.0282 | −0.0058 | −0.0566 | −0.0474 | −0.022 | $TK$ | |||||

(0.3818) | (0.3706) | (0.3562) | (0.3770) | (0.3766) | ||||||

II | −0.066 | 0.0331 | 0.0001 | −0.0539 | −0.0619 | −0.0303 | $Lindley$ | |||

(0.374) | (0.4233) | (0.4065) | (0.395) | (0.4669) | (0.4521) | |||||

−0.0062 | −0.0394 | −0.0912 | −0.0818 | −0.0577 | $TK$ | |||||

(0.35) | (0.341) | (0.3329) | (0.3516) | (0.3489) | ||||||

48 | I | −0.0265 | 0.1006 | 0.077 | 0.0116 | 0.0433 | 0.0668 | $Lindley$ | ||

(0.2412) | (0.4526) | (0.4618) | (0.3594) | (0.5667) | (0.5138) | |||||

0.0861 | 0.0633 | 0.0238 | 0.0303 | 0.047 | $TK$ | |||||

(0.2981) | (0.2873) | (0.2744) | (0.2871) | (0.2917) | ||||||

II | −0.0224 | 0.0569 | 0.0312 | −0.0079 | 0.0013 | 0.02 | $Lindley$ | |||

(0.2501) | (0.2706) | (0.2632) | (0.2553) | (0.2662) | (0.2669) | |||||

0.0457 | 0.0221 | −0.018 | −0.0135 | 0.0067 | $TK$ | |||||

(0.2745) | (0.267) | (0.2574) | (0.2744) | (0.2706) | ||||||

100 | 80 | 64 | I | −0.0346 | −0.0692 | −0.0978 | −0.1384 | −0.1935 | −0.1383 | $Lindley$ |

(0.206) | (0.2741) | (0.2948) | (0.3182) | (0.4859) | (0.3755) | |||||

0.0319 | 0.0116 | −0.0189 | -0.0129 | 0.0033 | $TK$ | |||||

(0.2074) | (0.2025) | (0.1967) | (0.2053) | (0.2056) | ||||||

II | −0.0601 | −0.025 | −0.046 | −0.077 | −0.081 | −0.0567 | $Lindley$ | |||

(0.2282) | (0.2405) | (0.238) | (0.235) | (0.2712) | (0.2443) | |||||

0.0099 | −0.0105 | −0.042 | −0.0361 | −0.0213 | $TK$ | |||||

(0.2159) | (0.2115) | (0.2068) | (0.2141) | (0.2147) | ||||||

72 | I | −0.031 | 0.0379 | 0.0154 | −0.0125 | −0.0047 | 0.0066 | $Lindley$ | ||

(0.1713) | (0.2766) | (0.2481) | (0.2291) | (0.2766) | (0.2538) | |||||

0.022 | 0.0062 | −0.0198 | −0.0148 | −0.0026 | $TK$ | |||||

(0.1773) | (0.174) | (0.1705) | (0.1763) | (0.1765) | ||||||

II | −0.0138 | 0.03 | 0.0131 | −0.0128 | −0.0092 | 0.0059 | $Lindley$ | |||

(0.1814) | (0.1914) | (0.1864) | (0.1809) | (0.1944) | (0.1877) | |||||

0.0324 | 0.0161 | −0.0105 | −0.0052 | 0.0067 | $TK$ | |||||

(0.1805) | (0.1767) | (0.1724) | (0.1779) | (0.1796) |

**Table 2.**The relative MSEs and biases of entropy estimates with MLE and the Bayes estimates with informative priors ($\widehat{H}$ for the MLEs, ${\widehat{H}}_{s}$ for the Bayesian estimates under the squared error loss function, ${\widehat{H}}_{l}$ for the Bayesian estimates under the linex loss function, ${\widehat{H}}_{e}$ for the Bayesian estimates under general entropy loss function, Lindley for the Lindley method and TK for the Tierney and Kadane method).

n | m | k | $\mathit{Sch}$ | $\widehat{\mathit{H}}$ | ${\widehat{\mathit{H}}}_{\mathit{s}}$ | ${\widehat{\mathit{H}}}_{\mathit{l}}$ | ${\widehat{\mathit{H}}}_{\mathit{e}}$ | $\mathit{Method}$ | ||
---|---|---|---|---|---|---|---|---|---|---|

h = 0.2 | h = 0.5 | q = 0.2 | q = $-0.2$ | |||||||

60 | 52 | 40 | I | −0.0157 | 0.152 | 0.1181 | 0.0733 | 0.0718 | 0.101 | $Lindley$ |

(0.3544) | (0.3253) | (0.3242) | (0.3577) | (0.3696) | (0.3403) | |||||

0.078 | 0.0481 | 0.0005 | 0.0095 | 0.0323 | $TK$ | |||||

(0.3025) | (0.2911) | (0.2774) | (0.2936) | (0.2961) | ||||||

II | −0.066 | 0.0687 | 0.0379 | −0.0173 | −0.0374 | 0.0012 | $Lindley$ | |||

(0.374) | (0.289) | (0.2823) | (0.3057) | (0.4011) | (0.3508) | |||||

0.0592 | 0.0294 | −0.0177 | −0.0106 | 0.0139 | $TK$ | |||||

(0.2919) | (0.2818) | (0.2703) | (0.284) | (0.2873) | ||||||

48 | I | −0.0265 | 0.064 | 0.0391 | 0.0011 | 0.01 | 0.0282 | $Lindley$ | ||

(0.2412) | (0.227) | (0.2199) | (0.2122) | (0.2226) | (0.2233) | |||||

0.0322 | 0.0091 | −0.0284 | −0.0219 | −0.004 | $TK$ | |||||

(0.2242) | (0.2184) | (0.2125) | (0.2225) | (0.2233) | ||||||

II | −0.0224 | 0.0692 | 0.0437 | 0.0048 | 0.0116 | 0.03 | $Lindley$ | |||

(0.2501) | (0.2307) | (0.2223) | (0.2129) | (0.2312) | (0.2331) | |||||

0.0338 | 0.0109 | −0.0268 | −0.0213 | −0.0032 | $TK$ | |||||

(0.2408) | (0.2352) | (0.2289) | (0.2399) | (0.2394) | ||||||

100 | 80 | 64 | I | −0.0346 | 0.0229 | −0.0009 | −0.0309 | −0.0393 | −0.016 | $Lindley$ |

(0.206) | (0.1948) | (0.2015) | (0.1963) | (0.2426) | (0.2195) | |||||

0.0563 | 0.0371 | 0.0073 | 0.0133 | 0.0278 | $TK$ | |||||

(0.1748) | (0.1702) | (0.1644) | (0.1709) | (0.1716) | ||||||

II | −0.0601 | 0.0254 | 0.0052 | −0.0256 | −0.0186 | −0.0037 | $Lindley$ | |||

(0.2282) | (0.2015) | (0.1965) | (0.1912) | (0.1986) | (0.1989) | |||||

0.0488 | 0.0295 | −0.0007 | 0.0048 | 0.0197 | $TK$ | |||||

(0.1989) | (0.1942) | (0.188) | (0.1954) | (0.1966) | ||||||

72 | I | −0.031 | 0.0289 | 0.012 | −0.0137 | −0.0077 | 0.0046 | $Lindley$ | ||

(0.1713) | (0.16) | (0.1568) | (0.1534) | (0.1585) | (0.1586) | |||||

0.0426 | 0.0269 | 0.0013 | 0.006 | 0.0183 | $TK$ | |||||

(0.1649) | (0.1614) | (0.1573) | (0.1623) | (0.1631) | ||||||

II | −0.0138 | 0.0484 | 0.0316 | 0.0058 | 0.012 | 0.0244 | $Lindley$ | |||

(0.1814) | (0.172) | (0.1659) | (0.1599) | (0.1659) | (0.1671) | |||||

0.0395 | 0.0231 | −0.0026 | 0.0018 | 0.0149 | $TK$ | |||||

(0.1573) | (0.154) | (0.1502) | (0.1556) | (0.1552) |

No. | Distribution | $\mathit{MLEs}$ | $\mathit{AIC}$ | $\mathit{BIC}$ | K-S | p Value |
---|---|---|---|---|---|---|

1. | Lomax distribution | $(\widehat{\alpha},\widehat{\lambda})=(1.3055,150.6285)$ | 681.4757 | 685.3782 | 0.0952 | 0.7337 |

2. | Gamma distribution | $(\widehat{\alpha},\widehat{\lambda})=(0.5606,0.0021)$ | 682.1883 | 686.0908 | 0.0893 | 0.8009 |

3. | Generalized inverted exponential distribution | $(\widehat{\alpha},\widehat{\lambda})=(0.1425,0.0819)$ | 788.2046 | 792.1071 | 0.4184 | $2.4\times {10}^{-8}$ |

$\widehat{\mathit{H}}$ | ${\widehat{\mathit{H}}}_{\mathit{s}}$ | ${\widehat{\mathit{H}}}_{\mathit{l}}$ | ${\widehat{\mathit{H}}}_{\mathit{e}}$ | $\mathit{Method}$ | |||
---|---|---|---|---|---|---|---|

h = 0.2 | h = 0.5 | q = 0.2 | q = $-0.2$ | ||||

I | 7.514680 | 7.391941 | 7.199010 | 6.966619 | 7.236140 | 7.286647 | Lindley |

8.047029 | 7.699564 | 7.452938 | 7.731689 | 8.073602 | TK | ||

II | 6.595185 | 6.167338 | 6.102223 | 6.029770 | 6.108076 | 6.126530 | Lindley |

6.975125 | 6.821545 | 6.717372 | 6.856352 | 6.883334 | TK | ||

III | 6.956195 | 6.586184 | 6.462234 | 6.324426 | 6.478268 | 6.512196 | Lindley |

7.462613 | 7.187117 | 7.023052 | 7.259597 | 7.390569 | TK |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, S.; Gui, W.
Estimating the Entropy for Lomax Distribution Based on Generalized Progressively Hybrid Censoring. *Symmetry* **2019**, *11*, 1219.
https://doi.org/10.3390/sym11101219

**AMA Style**

Liu S, Gui W.
Estimating the Entropy for Lomax Distribution Based on Generalized Progressively Hybrid Censoring. *Symmetry*. 2019; 11(10):1219.
https://doi.org/10.3390/sym11101219

**Chicago/Turabian Style**

Liu, Shuhan, and Wenhao Gui.
2019. "Estimating the Entropy for Lomax Distribution Based on Generalized Progressively Hybrid Censoring" *Symmetry* 11, no. 10: 1219.
https://doi.org/10.3390/sym11101219