**7. Optimization Criterion**

In recent years, there has been a lot of interest in finding the optimal censoring scheme in the statistical literature; for example, see Refs. [47–53]. Possible censoring schemes refer to any R1, ... , R m combinations such that n = m + ∑ m i=1 Ri and finding the optimum sampling approach means locating the progressive censoring scheme that offers the most information about the unknown parameters among all conceivable progressive censoring schemes for fixed n and m. The first difficulty is, of course, how to generate unknown parameter information measures based on specific progressive censoring data, and the second is how to compare two distinct information measures based on two different progressive censoring techniques. The next subsections go through some of the optimality criteria that were employed in this situation. In practice, we want to select the filtering scheme that delivers the most information about the unknown parameters; see Ref. [54] for further information. In our example, Table 2 presents a number of regularly used measures to help us choose the appropriate progressive censoring approach.

**Table 2.** Some practical censoring plan optimum criteria.


In terms of O1, our goal is to maximize the observed Fisher **<sup>I</sup>**3×<sup>3</sup>(.) information values. Furthermore, our goal for criterion O2 and O3 is to minimize the determinant and trace of [**<sup>I</sup>**3×<sup>3</sup>(.)]−1. Comparing multiple criteria is simple when dealing with single-parameter distributions; however, when dealing with unknown multi-parameter distributions, comparing the two Fisher information matrices becomes more difficult because the criterion O2 and O3 are not scale-invariant; see Ref. [55]. However, the optimal censoring scheme of multi-parameter distributions can be chosen using scale-invariant criteria O4. The criterion O4, which is dependent on the value of p, clearly tends to minimize the variance of logarithmic MLE of the p-th quantile, log ˆtp . As a result, the logarithmic for ˆtp of the APE distribution is supplied by

$$\log\left(\mathbf{f}\_{\mathbf{P}}\right) = \log\left\{\frac{-1}{\beta}\log\left[1 - \frac{\log\left(\mathbf{p}\left(\alpha - 1\right)\right)}{\log\alpha}\right]\right\}, \ 0 < \mathbf{p} < 1,$$

The delta approach is applied to (3) to produce the approximated variance forlog ˆtp of the APE distribution as

$$\text{Var}(\log(\mathbf{f}\_{\textsf{P}})) = \left[\nabla \log(\mathbf{f}\_{\textsf{P}})\right]^{\mathsf{T}} \mathbf{I}\_{\textsf{3}\times\textsf{3}}^{-1}(\boldsymbol{\upbeta}\_{1}, \boldsymbol{\upbeta}\_{2}, \boldsymbol{\upbeta}\_{3}) \left[\nabla \log(\mathbf{f}\_{\textsf{P}})\right].$$

where

$$\mathbb{E}\left[\nabla\log\left(\widehat{\mathbf{f}}\_{\mathsf{P}}\right)\right]^{\mathsf{T}} = \left[\frac{\partial}{\partial\boldsymbol{\beta}\_{1}}\log\left(\widehat{\mathbf{f}}\_{\mathsf{P}}\right), \frac{\partial}{\partial\boldsymbol{\beta}\_{2}}\log\left(\widehat{\mathbf{f}}\_{\mathsf{P}}\right), \frac{\partial}{\partial\boldsymbol{\beta}\_{3}}\log\left(\widehat{\mathbf{f}}\_{\mathsf{P}}\right)\right]\_{\left(\boldsymbol{\beta}\_{1}-\boldsymbol{\beta}\_{1},\boldsymbol{\beta}\_{2}=\boldsymbol{\beta}\_{2},\boldsymbol{\beta}\_{3}=\boldsymbol{\beta}\_{3}\right)}.$$

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The optimal progressive censoring, however, corresponds to a maximum value of the criterion O1 and a minimum value of the criteria Oi i, = 1, 2, 3, 4.
