# Statistical Inference for the Inverted Scale Family under General Progressive Type-II Censoring

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Exact Interval Estimation for $\mathit{\theta}$

#### 2.1. Interval Estimation with Pivotal Quantity Method

**Theorem**

**1.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Remark**

**1.**

**Example**

**1.**

**Example**

**2.**

#### 2.2. Simulation Study

- Step 1:
- Generate a random variable ${L}_{n}$ from $Beta(N-r,r+1)$.
- Step 2:
- Generate $n-r-1$ independent Uniform (0,1) random variables ${Z}_{r+1},\dots ,{Z}_{n-1}$.
- Step 3:
- Set ${L}_{r+i}={Z}^{\frac{1}{{a}_{r+i}}}$ where ${a}_{r+i}=i+{\sum}_{j=n-i+1}^{n}{R}_{j},i=1,\dots ,n-r$.
- Step 4:
- Set ${U}_{r+i}=1-{L}_{n-i+1}{L}_{n-i+2}\dots {L}_{n}$ for $i=1,\dots ,n-r$.
- Step 5:
- Set ${X}_{r+i}=-\frac{\theta}{log({U}_{r+i})}$ for $i=1,\dots ,n-r$.

#### 2.3. An Illustrated Example

$1.1$ | $1.2$ | $1.3$ | $1.4$ | $1.4$ | $1.5$ | $1.6$ | $1.6$ | $1.7$ | $1.7$ |

$1.7$ | $1.8$ | $1.8$ | $1.9$ | $2.0$ | $2.2$ | $2.3$ | $2.7$ | $3.0$ | $4.1$ |

## 3. Estimation of $\mathit{R}=\mathit{P}(\mathit{Y}<\mathit{X})$

#### 3.1. Maximum Likelihood Estimator (MLE)

#### 3.2. Approximate Maximum Likelihood Estimator (AMLE)

**Remark**

**2.**

**Proof.**

**Theorem**

**4.**

**Remark**

**3.**

**Theorem**

**5.**

**Lemma**

**2.**

**Proof.**

#### 3.3. R-Symmetric and R-Asymmetric Approximate Confidence Intervals

#### 3.3.1. R-Symmetric Interval Estimation Based on Asymptotic Distributions

**Theorem**

**6.**

**Theorem**

**7.**

**Proof.**

#### 3.3.2. R-Asymmetric Interval Estimation Based on Bootstrap Methods

- Step 1:
- Generate $({x}_{r+1}^{*},\dots ,{x}_{n}^{*})$ from $({x}_{r+1},\dots ,{x}_{n})$ and $({y}_{{r}^{\prime}+1}^{*},\dots ,{y}_{m}^{*})$ from $({y}_{{r}^{\prime}+1},\dots ,{y}_{m})$, respectively. Based on the generated samples, calculate $\widehat{{R}^{*}}$ with Equation (14);
- Step 2:
- Repeat step 1, $NBOOT$ times;
- Step 3:
- Define the CDF of $\widehat{{R}^{*}}$ as ${\mathsf{\Phi}}_{1}(z)=P(\widehat{{R}^{*}}\le z)$. Let ${\widehat{R}}_{BP}(z)={\mathsf{\Phi}}_{1}^{-1}(z)$ for a given z. Thus an approximate $100(1-\xi )\%$ confidence interval of R is $({\widehat{R}}_{BP}(\frac{\xi}{2}),{\widehat{R}}_{BP}(1-\frac{\xi}{2}))$.

- Step 1:
- Calculate $\widehat{{\theta}_{1}}$, $\widehat{{\theta}_{2}}$ and $\widehat{R}$ with $({x}_{r+1},\dots ,{x}_{n})$ and $({y}_{{r}^{\prime}+1},\dots ,{y}_{m})$;
- Step 2:
- Generate $({x}_{r+1}^{*},\dots ,{x}_{n}^{*})$ from $({x}_{r+1},\dots ,{x}_{n})$ and $({y}_{{r}^{\prime}+1}^{*},\dots ,{y}_{m}^{*})$ from $({y}_{{r}^{\prime}+1},\dots ,{y}_{m})$, respectively;
- Step 3:
- Calculate $\widehat{{R}^{*}}$ with Equation (14) and calculate ${B}^{*}$ with ${B}^{*}=\frac{\sqrt{n-r}(\widehat{{R}^{*}}-\widehat{R})}{\sqrt{Var(\widehat{{R}^{*}})}}$ (Note that $Var(\widehat{{R}^{*}})$ can be derived from Theorem 7);
- Step 4:
- Repeat steps 2 and 3, $NBOOT$ times;
- Step 5:
- Define the CDF of ${B}^{*}$ as ${\mathsf{\Phi}}_{2}(z)=P({B}^{*}\le z)$. Let ${\widehat{R}}_{BT}(z)=\widehat{R}+{\mathsf{\Phi}}_{2}^{-1}(z)\sqrt{\frac{Var(\widehat{R})}{n-r}}$ for a given z. Thus an approximate $100(1-\xi )\%$ confidence interval of R is $({\widehat{R}}_{BT}(\frac{\xi}{2}),{\widehat{R}}_{BT}(1-\frac{\xi}{2}))$.

#### 3.4. Bayesian Estimation

- Step 1:
- Determine the initial value $({\theta}_{1}^{(0)},{\theta}_{2}^{(0)})$;
- Step 2:
- Set $k=1$;
- Step 3:
- Using the $M-H$ method, generate ${\theta}_{1}^{(k)}$ from $f({\theta}_{1}|{\theta}_{2},data)$. Here, the proposal distribution is $exp({\theta}_{1}^{(k-1)})$;
- Step 4:
- Using the $M-H$ method, generate ${\theta}_{2}^{(k)}$ from $f({\theta}_{2}|{\theta}_{1},data)$. Here, the proposal distribution is $exp({\theta}_{2}^{(k-1)})$;
- Step 5:
- Compute ${R}^{(k)}$ by Equation (14);
- Step 6:
- Set $k=k+1$;
- Step 7:
- Repeat steps 3 to 6 K times.

#### 3.5. Simulation Study

#### 3.6. An Illustrated Example

**Data Set 1.**The data below shows the breaking strength for 63 carbon fibers of 10 mm gauge length in a tension test (unit: GPA).

$1.901$ | $2.132$ | $2.203$ | $2.228$ | $2.257$ | $2.350$ | $2.361$ | $2.396$ | $2.397$ | $2.445$ |

$2.454$ | $2.474$ | $2.518$ | $2.522$ | $2.525$ | $2.532$ | $2.575$ | $2.614$ | $2.616$ | $2.618$ |

$2.624$ | $2.659$ | $2.675$ | $2.738$ | $2.740$ | $2.856$ | $2.917$ | $2.928$ | $2.937$ | $2.937$ |

$2.977$ | $2.996$ | $3.030$ | $3.125$ | $3.139$ | $3.145$ | $3.220$ | $3.223$ | $3.235$ | $3.243$ |

$3.264$ | $3.272$ | $3.294$ | $3.332$ | $3.346$ | $3.377$ | $3.408$ | $3.435$ | $3.493$ | $3.501$ |

$3.537$ | $3.554$ | $3.562$ | $3.628$ | $3.852$ | $3.871$ | $3.886$ | $3.971$ | $4.024$ | $4.027$ |

$4.225$ | $4.395$ | $5.020$ |

**Data Set 2.**The data below shows the breaking strength for 69 carbon fibers of 20 mm gauge length in a tension test (unit: GPA).

$1.312$ | $1.314$ | $1.479$ | $1.552$ | $1.700$ | $1.803$ | $1.861$ | $1.865$ | $1.944$ | $1.958$ |

$1.966$ | $1.997$ | $2.006$ | $2.021$ | $2.027$ | $2.055$ | $2.063$ | $2.098$ | $2.140$ | $2.179$ |

$2.224$ | $2.240$ | $2.253$ | $2.270$ | $2.272$ | $2.274$ | $2.301$ | $2.301$ | $2.359$ | $2.382$ |

$2.382$ | $2.426$ | $2.434$ | $2.435$ | $2.478$ | $2.490$ | $2.511$ | $2.514$ | $2.535$ | $2.554$ |

$2.566$ | $2.570$ | $2.586$ | $2.629$ | $2.633$ | $2.642$ | $2.648$ | $2.684$ | $2.697$ | $2.726$ |

$2.770$ | $2.773$ | $2.800$ | $2.809$ | $2.818$ | $2.821$ | $2.848$ | $2.880$ | $2.954$ | $3.012$ |

$3.067$ | $3.084$ | $3.090$ | $3.096$ | $3.128$ | $3.233$ | $3.433$ | $3.585$ | $3.585$ |

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**An empirical distribution function (CDF) and other CDFs based on different statistical models.

**Table 1.**Average confidence intervals (CI), interval lengths (IL), and interval coverage percentages (CP).

$1-\mathit{\xi}=0.90$ | $1-\mathit{\xi}=0.95$ | |||||
---|---|---|---|---|---|---|

$(N,r,n)$ | CI | IL | CP | CI | IL | CP |

$(20,1,10)$ | $(2.97,13.18)$ | $10.21$ | $0.904$ | $(2.63,15.62)$ | $12.99$ | $0.953$ |

$(20,3,12)$ | $(2.87,13.85)$ | $10.98$ | $0.900$ | $(2.52,16.39)$ | $13.87$ | $0.951$ |

$(20,1,16)$ | $(2.93,10.84)$ | $7.91$ | $0.901$ | $(2.60,12.33)$ | $9.73$ | $0.951$ |

$(20,3,18)$ | $(2.76,11.97)$ | $9.21$ | $0.902$ | $(2.42,13.77)$ | $11.35$ | $0.948$ |

$(30,1,10)$ | $(3.37,8.40)$ | $5.03$ | $0.897$ | $(3.08,9.12)$ | $6.04$ | $0.952$ |

$(30,3,12)$ | $(3.27,9.00)$ | $5.73$ | $0.896$ | $(2.99,9.91)$ | $6.92$ | $0.952$ |

$(30,1,16)$ | $(3.28,9.13)$ | $5.85$ | $0.897$ | $(2.99,10.21)$ | $7.22$ | $0.947$ |

$(30,3,18)$ | $(3.13,10.46)$ | $7.33$ | $0.895$ | $(2.80,11.78)$ | $8.98$ | $0.951$ |

$(40,1,10)$ | $(3.53,7.83)$ | $4.30$ | $0.900$ | $(3.39,8.48)$ | $5.09$ | $0.949$ |

$(40,3,12)$ | $(3.46,8.39)$ | $4.93$ | $0.903$ | $(3.19,9.17)$ | $5.98$ | $0.950$ |

$(40,1,16)$ | $(3.54,7.71)$ | $4.17$ | $0.900$ | $(3.29,8.32)$ | $5.03$ | $0.950$ |

$(40,3,18)$ | $(3.41,8.57)$ | $5.16$ | $0.897$ | $(3.14,9.37)$ | $6.23$ | $0.950$ |

Model | $\mathit{\alpha}$ | $\mathit{\theta}$ | −ln | $\mathit{K}-\mathit{S}$ | p-Value | $\mathbf{AIC}$ | $\mathbf{BIC}$ |
---|---|---|---|---|---|---|---|

IWD | $3.777054$ | $1.58526$ | $15.19896$ | $0.1306$ | $0.9187$ | $34.39793$ | $36.17867$ |

GIED | $19.519704$ | $6.189745$ | $16.22355$ | $0.14001$ | $0.8721$ | $36.4471$ | $38.22784$ |

WD | $2.785545$ | $8.779702$ | $19.06114$ | $0.17819$ | $0.6171$ | $42.12228$ | $43.90302$ |

$\mathit{i},\mathit{j}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${X}_{i}$ | − | − | $1.3$ | $1.4$ | $1.6$ | $1.7$ | $1.7$ | $1.8$ | $1.8$ | $1.9$ | $2.0$ | $2.2$ | $2.3$ | $2.7$ | $3.0$ | $4.1$ |

${R}_{i}$ | − | − | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

$\mathit{C}.\mathit{S}.$ | $(\mathit{r},\phantom{\rule{0.277778em}{0ex}}\mathit{n},\phantom{\rule{0.277778em}{0ex}}\mathit{N},\phantom{\rule{0.277778em}{0ex}}{\mathit{R}}_{\mathit{r}+1},\phantom{\rule{0.277778em}{0ex}}{\mathit{R}}_{\mathit{r}+2},\phantom{\rule{0.277778em}{0ex}}\dots \phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}{\mathit{R}}_{\mathit{n}})$ |
---|---|

${R}_{1}$ | $(2,\phantom{\rule{0.277778em}{0ex}}10,\phantom{\rule{0.277778em}{0ex}}29,\phantom{\rule{0.277778em}{0ex}}3,\phantom{\rule{0.277778em}{0ex}}1,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}3,\phantom{\rule{0.277778em}{0ex}}1,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}3,\phantom{\rule{0.277778em}{0ex}}4)$ |

${R}_{2}$ | $(2,\phantom{\rule{0.277778em}{0ex}}10,\phantom{\rule{0.277778em}{0ex}}26,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}2)$ |

${R}_{3}$ | $(2,\phantom{\rule{0.277778em}{0ex}}10,\phantom{\rule{0.277778em}{0ex}}26,\phantom{\rule{0.277778em}{0ex}}4,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}4,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}1,\phantom{\rule{0.277778em}{0ex}}1,\phantom{\rule{0.277778em}{0ex}}1,\phantom{\rule{0.277778em}{0ex}}1)$ |

$\mathit{MLE}$ | $\mathit{AMLE}$ | $\mathit{Bayes}$ | |||||
---|---|---|---|---|---|---|---|

$({\mathbf{\theta}}_{\mathbf{1}},{\mathbf{\theta}}_{\mathbf{2}})$ | $\mathit{C}.\mathit{S}.$ | $\mathit{R}$ | $\mathit{Bias}(\mathit{Mse})$ | $\mathit{R}$ | $\mathit{Bias}(\mathit{Mse})$ | $\mathit{R}$ | $\mathit{Bias}(\mathit{Mse})$ |

$(1,1)$ | $({R}_{1},{R}_{1})$ | $0.50085$ | $0.00085(0.00513)$ | $0.50291$ | $0.00291(0.00481)$ | $0.50016$ | $0.00016(0.00018)$ |

$({R}_{1},{R}_{2})$ | $0.50468$ | $0.01468(0.00479)$ | $0.50699$ | $0.00699(0.00576)$ | $0.50049$ | $0.00049(0.00017)$ | |

$({R}_{1},{R}_{3})$ | $0.51370$ | $0.01370(0.00555)$ | $0.52850$ | $0.02850(0.00649)$ | $0.50050$ | $0.00050(0.00018)$ | |

$({R}_{2},{R}_{2})$ | $0.50230$ | $0.00230(0.00515)$ | $0.50231$ | $0.00231(0.00523)$ | $0.50009$ | $0.00009(0.00016)$ | |

$({R}_{2},{R}_{3})$ | $0.50833$ | $0.00833(0.00550)$ | $0.51731$ | $0.01731(0.00634)$ | $0.50078$ | $0.00078(0.00016)$ | |

$({R}_{3},{R}_{3})$ | $0.50295$ | $0.00295(0.00592)$ | $0.50262$ | $0.00262(0.00541)$ | $0.50066$ | $0.00066(0.00017)$ | |

$(1,2)$ | $({R}_{1},{R}_{1})$ | $0.33615$ | $0.00315(0.00425)$ | $0.34790$ | $0.01490(0.00603)$ | $0.33341$ | $0.00041(0.00007)$ |

$({R}_{1},{R}_{2})$ | $0.34654$ | $0.01354(0.00402)$ | $0.34425$ | $0.01125(0.00434)$ | $0.35512$ | $0.02212(0.00327)$ | |

$({R}_{1},{R}_{3})$ | $0.34605$ | $0.01305(0.00473)$ | $0.34831$ | $0.01531(0.00473)$ | $0.35051$ | $0.01751(0.00289)$ | |

$({R}_{2},{R}_{2})$ | $0.33812$ | $0.00512(0.00436)$ | $0.33780$ | $0.00480(0.00494)$ | $0.35882$ | $0.02582(0.00370)$ | |

$({R}_{2},{R}_{3})$ | $0.33332$ | $0.00032(0.00464)$ | $0.34141$ | $0.00841(0.00485)$ | $0.35614$ | $0.02314(0.00352)$ | |

$({R}_{3},{R}_{3})$ | $0.33670$ | $0.00370(0.00459)$ | $0.33432$ | $0.00132(0.00454)$ | $0.36062$ | $0.02762(0.00377)$ | |

$(1,3)$ | $({R}_{1},{R}_{1})$ | $0.25153$ | $0.00153(0.00292)$ | $0.25651$ | $0.00651(0.00308)$ | $0.32660$ | $0.07660(0.01191)$ |

$({R}_{1},{R}_{2})$ | $0.25812$ | $0.00812(0.00290)$ | $0.25621$ | $0.00621(0.00291)$ | $0.32476$ | $0.07476(0.01080)$ | |

$({R}_{1},{R}_{3})$ | $0.25847$ | $0.00847(0.00371)$ | $0.27132$ | $0.02132(0.00353)$ | $0.32228$ | $0.07228(0.01192)$ | |

$({R}_{2},{R}_{2})$ | $0.25594$ | $0.00594(0.00354)$ | $0.25403$ | $0.00403(0.00362)$ | $0.32491$ | $0.07491(0.01116)$ | |

$({R}_{2},{R}_{3})$ | $0.25292$ | $0.00292(0.00337)$ | $0.25022$ | $0.00022(0.00341)$ | $0.32323$ | $0.07323(0.01163)$ | |

$({R}_{3},{R}_{3})$ | $0.25245$ | $0.00245(0.00301)$ | $0.25808$ | $0.00808(0.00346)$ | $0.32516$ | $0.07516(0.01053)$ | |

$(2,2)$ | $({R}_{1},{R}_{1})$ | $0.50245$ | $0.00245(0.00495)$ | $0.50019$ | $0.00019(0.00520)$ | $0.50005$ | $0.00005(0.00007)$ |

$({R}_{1},{R}_{2})$ | $0.50341$ | $0.00341(0.00550)$ | $0.51469$ | $0.01469(0.00575)$ | $0.50044$ | $0.00044(0.00007)$ | |

$({R}_{1},{R}_{3})$ | $0.50896$ | $0.00896(0.00552)$ | $0.52607$ | $0.02607(0.00611)$ | $0.50027$ | $0.00027(0.00006)$ | |

$({R}_{2},{R}_{2})$ | $0.50033$ | $0.00033(0.00565)$ | $0.50055$ | $0.00055(0.00542)$ | $0.50013$ | $0.00013(0.00007)$ | |

$({R}_{2},{R}_{3})$ | $0.50055$ | $0.00055(0.00618)$ | $0.51098$ | $0.01098(0.00578)$ | $0.50010$ | $0.00010(0.00008)$ | |

$({R}_{3},{R}_{3})$ | $0.50365$ | $0.00365(0.00554)$ | $0.50225$ | $0.00255(0.00596)$ | $0.50079$ | $0.00079(0.00007)$ | |

$(2,3)$ | $({R}_{1},{R}_{1})$ | $0.40467$ | $0.00467(0.00464)$ | $0.40345$ | $0.00345(0.00465)$ | $0.49350$ | $0.09350(0.00879)$ |

$({R}_{1},{R}_{2})$ | $0.41019$ | $0.01019(0.00485)$ | $0.40838$ | $0.00838(0.00458)$ | $0.49376$ | $0.09376(0.00885)$ | |

$({R}_{1},{R}_{3})$ | $0.40929$ | $0.00929(0.00445)$ | $0.42179$ | $0.02179(0.00531)$ | $0.49361$ | $0.09361(0.00882)$ | |

$({R}_{2},{R}_{2})$ | $0.40152$ | $0.00152(0.00449)$ | $0.40775$ | $0.00775(0.00604)$ | $0.49382$ | $0.09382(0.00885)$ | |

$({R}_{2},{R}_{3})$ | $0.40044$ | $0.00044(0.00530)$ | $0.41298$ | $0.01298(0.00560)$ | $0.49362$ | $0.09362(0.00882)$ | |

$({R}_{3},{R}_{3})$ | $0.40132$ | $0.00132(0.00526)$ | $0.40099$ | $0.00099(0.00526)$ | $0.49382$ | $0.09382(0.00886)$ | |

$(3,3)$ | $({R}_{1},{R}_{1})$ | $0.50172$ | $0.00172(0.00522)$ | $0.50209$ | $0.00209(0.00495)$ | $0.50009$ | $0.00009(0.00005)$ |

$({R}_{1},{R}_{2})$ | $0.50577$ | $0.00577(0.00519)$ | $0.51180$ | $0.01180(0.00581)$ | $0.50010$ | $0.00010(0.00005)$ | |

$({R}_{1},{R}_{3})$ | $0.50220$ | $0.00220(0.00598)$ | $0.52334$ | $0.02334(0.00622)$ | $0.50086$ | $0.00086(0.00005)$ | |

$({R}_{2},{R}_{2})$ | $0.50062$ | $0.00062(0.00550)$ | $0.50583$ | $0.00583(0.00590)$ | $0.50003$ | $0.00003(0.00004)$ | |

$({R}_{2},{R}_{3})$ | $0.50077$ | $0.00077(0.00545)$ | $0.50499$ | $0.00499(0.00601)$ | $0.50043$ | $0.00043(0.00005)$ | |

$({R}_{3},{R}_{3})$ | $0.50363$ | $0.00363(0.00602)$ | $0.50470$ | $0.00470(0.00629)$ | $0.50020$ | $0.00020(0.00004)$ |

$({\mathit{\theta}}_{1},{\mathit{\theta}}_{2})$ | $\mathit{C}.\mathit{S}.$ | $\mathit{MLE}$ | $\mathit{AMLE}$ | $\mathit{Bayes}$ | $\mathit{Boot}-\mathit{p}$ | $\mathit{Boot}-\mathit{t}$ |
---|---|---|---|---|---|---|

$(1,1)$ | $({R}_{1},{R}_{1})$ | $0.268(0.943)$ | $0.272(0.925)$ | $0.250(0.954)$ | $0.268(0.922)$ | $0.299(0.948)$ |

$({R}_{1},{R}_{2})$ | $0.272(0.899)$ | $0.273(0.911)$ | $0.250(0.938)$ | $0.275(0.924)$ | $0.302(0.943)$ | |

$({R}_{1},{R}_{3})$ | $0.278(0.914)$ | $0.276(0.875)$ | $0.250(0.930)$ | $0.276(0.931)$ | $0.305(0.954)$ | |

$({R}_{2},{R}_{2})$ | $0.282(0.912)$ | $0.284(0.924)$ | $0.250(0.947)$ | $0.282(0.914)$ | $0.305(0.926)$ | |

$({R}_{2},{R}_{3})$ | $0.286(0.919)$ | $0.280(0.907)$ | $0.247(0.946)$ | $0.287(0.930)$ | $0.309(0.935)$ | |

$({R}_{3},{R}_{3})$ | $0.284(0.935)$ | $0.284(0.925)$ | $0.247(0.955)$ | $0.311(0.954)$ | $0.311(0.946)$ | |

$(1,2)$ | $({R}_{1},{R}_{1})$ | $0.237(0.914)$ | $0.239(0.929)$ | $0.249(0.935)$ | $0.241(0.929)$ | $0.266(0.915)$ |

$({R}_{1},{R}_{2})$ | $0.248(0.927)$ | $0.245(0.911)$ | $0.249(0.953)$ | $0.253(0.937)$ | $0.270(0.929)$ | |

$({R}_{1},{R}_{3})$ | $0.255(0.912)$ | $0.247(0.909)$ | $0.249(0.943)$ | $0.258(0.928)$ | $0.272(0.941)$ | |

$({R}_{2},{R}_{2})$ | $0.250(0.905)$ | $0.251(0.914)$ | $0.250(0.931)$ | $0.255(0.943)$ | $0.276(0.920)$ | |

$({R}_{2},{R}_{3})$ | $0.256(0.922)$ | $0.253(0.910)$ | $0.249(0.947)$ | $0.261(0.944)$ | $0.277(0.930)$ | |

$({R}_{3},{R}_{3})$ | $0.257(0.933)$ | $0.257(0.906)$ | $0.250(0.949)$ | $0.261(0.946)$ | $0.279(0.931)$ | |

$(1,3)$ | $({R}_{1},{R}_{1})$ | $0.205(0.924)$ | $0.205(0.934)$ | $0.249(0.930)$ | $0.210(0.939)$ | $0.228(0.928)$ |

$({R}_{1},{R}_{2})$ | $0.215(0.910)$ | $0.208(0.913)$ | $0.250(0.951)$ | $0.219(0.936)$ | $0.231(0.919)$ | |

$({R}_{1},{R}_{3})$ | $0.221(0.924)$ | $0.211(0.896)$ | $0.249(0.947)$ | $0.226(0.917)$ | $0.232(0.902)$ | |

$({R}_{2},{R}_{2})$ | $0.214(0.914)$ | $0.214(0.919)$ | $0.249(0.940)$ | $0.221(0.938)$ | $0.236(0.926)$ | |

$({R}_{2},{R}_{3})$ | $0.221(0.909)$ | $0.217(0.905)$ | $0.249(0.943)$ | $0.226(0.937)$ | $0.236(0.910)$ | |

$({R}_{3},{R}_{3})$ | $0.217(0.931)$ | $0.217(0.909)$ | $0.250(0.957)$ | $0.223(0.957)$ | $0.236(0.927)$ | |

$(2,2)$ | $({R}_{1},{R}_{1})$ | $0.269(0.920)$ | $0.265(0.925)$ | $0.250(0.937)$ | $0.268(0.949)$ | $0.298(0.924)$ |

$({R}_{1},{R}_{2})$ | $0.274(0.912)$ | $0.275(0.921)$ | $0.250(0.936)$ | $0.276(0.932)$ | $0.303(0.924)$ | |

$({R}_{1},{R}_{3})$ | $0.274(0.900)$ | $0.276(0.911)$ | $0.249(0.946)$ | $0.277(0.926)$ | $0.306(0.859)$ | |

$({R}_{2},{R}_{2})$ | $0.281(0.911)$ | $0.279(0.911)$ | $0.249(0.962)$ | $0.282(0.942)$ | $0.307(0.920)$ | |

$({R}_{2},{R}_{3})$ | $0.281(0.907)$ | $0.281(0.908)$ | $0.250(0.955)$ | $0.283(0.933)$ | $0.308(0.934)$ | |

$({R}_{3},{R}_{3})$ | $0.286(0.930)$ | $0.284(0.914)$ | $0.251(0.945)$ | $0.287(0.949)$ | $0.309(0.931)$ | |

$(2,3)$ | $({R}_{1},{R}_{1})$ | $0.257(0.909)$ | $0.256(0.900)$ | $0.250(0.943)$ | $0.260(0.925)$ | $0.285(0.911)$ |

$({R}_{1},{R}_{2})$ | $0.267(0.926)$ | $0.262(0.911)$ | $0.249(0.956)$ | $0.269(0.952)$ | $0.293(0.934)$ | |

$({R}_{1},{R}_{3})$ | $0.270(0.912)$ | $0.264(0.896)$ | $0.250(0.934)$ | $0.272(0.929)$ | $0.293(0.922)$ | |

$({R}_{2},{R}_{2})$ | $0.267(0.907)$ | $0.270(0.916)$ | $0.250(0.947)$ | $0.272(0.939)$ | $0.296(0.918)$ | |

$({R}_{2},{R}_{3})$ | $0.274(0.921)$ | $0.271(0.910)$ | $0.250(0.940)$ | $0.275(0.943)$ | $0.297(0.929)$ | |

$({R}_{3},{R}_{3})$ | $0.272(0.912)$ | $0.271(0.907)$ | $0.249(0.952)$ | $0.275(0.944)$ | $0.296(0.933)$ | |

$(3,3)$ | $({R}_{1},{R}_{1})$ | $0.267(0.908)$ | $0.267(0.923)$ | $0.249(0.959)$ | $0.269(0.939)$ | $0.296(0.923)$ |

$({R}_{1},{R}_{2})$ | $0.274(0.914)$ | $0.275(0.895)$ | $0.249(0.947)$ | $0.275(0.935)$ | $0.302(0.939)$ | |

$({R}_{1},{R}_{3})$ | $0.276(0.888)$ | $0.276(0.899)$ | $0.250(0.934)$ | $0.276(0.913)$ | $0.305(0.921)$ | |

$({R}_{2},{R}_{2})$ | $0.281(0.915)$ | $0.282(0.927)$ | $0.250(0.929)$ | $0.283(0.934)$ | $0.308(0.925)$ | |

$({R}_{2},{R}_{3})$ | $0.283(0.917)$ | $0.280(0.908)$ | $0.250(0.925)$ | $0.284(0.950)$ | $0.308(0.909)$ | |

$({R}_{3},{R}_{3})$ | $0.285(0.907)$ | $0.283(0.911)$ | $0.251(0.945)$ | $0.285(0.939)$ | $0.307(0.911)$ |

Data Set | Model | $\mathit{\alpha}$ | $\mathit{\theta}$ | −ln | $\mathit{K}-\mathit{S}$ | p-Value | $\mathbf{AIC}$ | $\mathbf{BIC}$ |
---|---|---|---|---|---|---|---|---|

1 | GIED | $175.2879$ | $16.8110$ | $57.7251$ | $0.8601$ | $0.7399$ | $119.4504$ | $123.7367$ |

WD | $5.0494$ | $424.5597$ | $61.9570$ | $0.0876$ | $0.7192$ | $127.914$ | $132.2002$ | |

IWD | $5.433788$ | $2.721432$ | $58.9022$ | $0.1001$ | $0.5528$ | $121.8043$ | $126.0906$ | |

2 | GIED | $205.87851$ | $13.88254$ | $49.16796$ | $0.041419$ | $0.9998$ | $102.3359$ | $106.8041$ |

WD | $5.504847$ | $214.130524$ | $49.59614$ | $0.056132$ | $0.9816$ | $103.1923$ | $107.6605$ | |

IWD | $4.126731$ | $2.143723$ | $63.62361$ | $0.13363$ | $0.1700$ | $131.2472$ | $135.7154$ |

$\mathit{i},\mathit{j}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${X}_{i}$ | - | - | - | $2.228$ | $2.397$ | $2.522$ | $2.616$ | $2.738$ | $2.937$ | $3.125$ | $3.235$ | $3.332$ | $3.493$ | $3.628$ | $4.024$ |

${R}_{i}$ | - | - | - | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |

${Y}_{j}$ | - | - | - | $1.552$ | $1.944$ | $2.021$ | $2.140$ | $2.270$ | $3.359$ | $2.435$ | $2.535$ | $2.629$ | $2.697$ | $2.807$ | $2.954$ |

${R}_{j}^{\prime}$ | - | - | - | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 10 |

Estimation | MLE | AMLE | Bayes | Bootstrap-p | Bootstrap-t |
---|---|---|---|---|---|

Point Estimation | $0.782$ | $0.794$ | $0.782$ | − | − |

Interval Estimation | $(0.680,0.885)$ | $(0.690,0.898)$ | $(0.651,0.871)$ | $(0.665,0.878)$ | $(0.694,0.913)$ |

© 2020 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**

Gao, J.; Bai, K.; Gui, W.
Statistical Inference for the Inverted Scale Family under General Progressive Type-II Censoring. *Symmetry* **2020**, *12*, 731.
https://doi.org/10.3390/sym12050731

**AMA Style**

Gao J, Bai K, Gui W.
Statistical Inference for the Inverted Scale Family under General Progressive Type-II Censoring. *Symmetry*. 2020; 12(5):731.
https://doi.org/10.3390/sym12050731

**Chicago/Turabian Style**

Gao, Jing, Kehan Bai, and Wenhao Gui.
2020. "Statistical Inference for the Inverted Scale Family under General Progressive Type-II Censoring" *Symmetry* 12, no. 5: 731.
https://doi.org/10.3390/sym12050731