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This paper provides an estimation method for an unknown parameter by extending weighted least-squared and pivot-based methods to the Gompertz distribution with the shape and scale parameters under the progressive Type-II censoring scheme, which induces a consistent estimator and an unbiased estimator of the scale parameter. In addition, a way to deal with a nuisance parameter is provided in the pivot-based approach. For evaluation and comparison, the Monte Carlo simulations are conducted, and real data are analyzed.

The probability density function (PDF) and cumulative distribution function (CDF) of the random variable

In addition to this distribution, inferences based on the pivotal quantity have been studied for many distributions because the pivotal-based approach provides exact CIs even for small samples, as well as more efficient estimators than the maximum likelihood estimators (MLEs) in terms of bias. Wu [

Recently, a new estimation method based on a weighted regression framework has been proposed under some censoring schemes. Lu and Tao [

This paper focuses on point estimation using the weighted regression framework and pivot-based methods based on the progressive Type-II censored data from the Gompertz distribution with the PDF (

This section gives a brief description for the progressive Type-II censoring scheme that is the generalization of the Type-II censoring scheme and that is one of the most popular censoring schemes and provides different approaches on estimation for unknown parameters of the Gompertz distribution with the PDF (

Exp (1): the standard exponential distribution

By Balakrishnan and Aggarwala [

Suppose that

The MLEs

Let

Then,

By minimizing the following quantity for

However, the approach gives same weight on each point, and it is not proper because the variances of

By minimizing the following quantity with the weighted square term for

For known

Let

Then, the estimator

Here, both

Therefore, the fraction term in (

For unknown

The pivotal quantity provided in Wu et al. [

Note that the estimator

A quantity

By Wang et al. [

By Lemma 1 in the work of Seo and Kang [

This section assesses and compares the estimation methods provided in Section

For evaluation and comparison, the mean squared errors (MSEs) and biases of the provided estimators are reported in Table

Scheme I:

Scheme II:

Scheme III:

Scheme IV:

MSEs(biases) for

Scheme | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

0.1 | 0.5 | 20 | 20 | 0.052 (0.098) | 0.032 (0.029) | 0.061 (−0.035) | 0.033 (−0.020) | 0.023 (−0.003) | 0.060 (0.059) | |

18 | I | 0.104 (0.138) | 0.056 (0.062) | 0.129 (−0.033) | 0.040 (−0.025) | 0.024 (−0.020) | 0.079 (0.064) | |||

II | 0.061 (0.113) | 0.037 (0.039) | 0.076 (−0.033) | 0.034 (−0.027) | 0.025 (−0.010) | 0.070 (0.060) | ||||

III | 0.082 (0.125) | 0.047 (0.052) | 0.100 (−0.030) | 0.038 (−0.026) | 0.024 (−0.016) | 0.074 (0.061) | ||||

IV | 0.066 (0.119) | 0.040 (0.043) | 0.084 (−0.035) | 0.033 (−0.026) | 0.024 (−0.011) | 0.069 (0.060) | ||||

14 | I | 0.380 (0.274) | 0.179 (0.145) | 0.529 (−0.070) | 0.062 (−0.036) | 0.031 (−0.040) | 0.150 (0.096) | |||

II | 0.095 (0.150) | 0.054 (0.054) | 0.109 (−0.027) | 0.046 (−0.034) | 0.032 (−0.015) | 0.099 (0.068) | ||||

III | 0.223 (0.214) | 0.113 (0.103) | 0.268 (−0.036) | 0.057 (−0.034) | 0.031 (−0.032) | 0.120 (0.081) | ||||

IV | 0.123 (0.173) | 0.074 (0.070) | 0.163 (−0.042) | 0.042 (−0.030) | 0.030 (−0.019) | 0.095 (0.070) | ||||

30 | 30 | 0.029 (0.068) | 0.023 (0.027) | 0.031 (−0.016) | 0.021 (−0.020) | 0.018 (−0.008) | 0.029 (0.029) | |||

26 | I | 0.073 (0.100) | 0.045 (0.056) | 0.081 (−0.011) | 0.029 (−0.018) | 0.019 (−0.019) | 0.041 (0.036) | |||

II | 0.035 (0.077) | 0.025 (0.029) | 0.037 (−0.017) | 0.024 (−0.020) | 0.020 (−0.007) | 0.035 (0.034) | ||||

III | 0.054 (0.089) | 0.035 (0.043) | 0.058 (−0.011) | 0.027 (−0.018) | 0.020 (−0.015) | 0.039 (0.034) | ||||

IV | 0.039 (0.083) | 0.028 (0.033) | 0.042 (−0.019) | 0.023 (−0.019) | 0.019 (−0.009) | 0.034 (0.033) | ||||

18 | I | 0.394 (0.246) | 0.181 (0.158) | 0.555 (−0.077) | 0.048 (−0.025) | 0.024 (−0.040) | 0.103 (0.078) | |||

II | 0.060 (0.111) | 0.037 (0.039) | 0.073 (−0.027) | 0.034 (−0.025) | 0.024 (−0.009) | 0.066 (0.055) | ||||

III | 0.202 (0.181) | 0.100 (0.103) | 0.266 (−0.041) | 0.044 (−0.024) | 0.023 (−0.030) | 0.088 (0.067) | ||||

IV | 0.083 (0.134) | 0.054 (0.057) | 0.119 (−0.040) | 0.029 (−0.020) | 0.022 (−0.014) | 0.062 (0.054) | ||||

50 | 50 | 0.013 (0.039) | 0.012 (0.008) | 0.017 (−0.016) | 0.012 (−0.012) | 0.012 (0.001) | 0.017 (0.020) | |||

42 | I | 0.044 (0.066) | 0.031 (0.042) | 0.053 (−0.002) | 0.018 (−0.015) | 0.013 (−0.017) | 0.023 (0.017) | |||

II | 0.018 (0.049) | 0.016 (0.017) | 0.021 (−0.012) | 0.015 (−0.016) | 0.014 (−0.005) | 0.019 (0.018) | ||||

III | 0.031 (0.058) | 0.024 (0.030) | 0.036 (−0.003) | 0.017 (−0.015) | 0.013 (−0.012) | 0.022 (0.016) | ||||

IV | 0.024 (0.058) | 0.023 (0.029) | 0.030 (−0.015) | 0.012 (−0.011) | 0.012 (−0.009) | 0.016 (0.014) | ||||

26 | I | 0.355 (0.201) | 0.180 (0.167) | 0.435 (−0.034) | 0.035 (−0.019) | 0.019 (−0.038) | 0.051 (0.043) | |||

II | 0.035 (0.077) | 0.025 (0.029) | 0.037 (−0.015) | 0.024 (−0.019) | 0.020 (−0.007) | 0.035 (0.033) | ||||

III | 0.164 (0.140) | 0.090 (0.102) | 0.194 (−0.014) | 0.032 (−0.017) | 0.018 (−0.030) | 0.045 (0.038) | ||||

IV | 0.049 (0.093) | 0.039 (0.046) | 0.059 (−0.022) | 0.020 (−0.013) | 0.017 (−0.012) | 0.029 (0.028) |

From Table

Chen [

Progressive Type-II censored tumor-free time data and corresponding censoring scheme.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.60 | 0.63 | 0.66 | 0.66 | 0.68 | 0.70 | 0.70 | 0.77 | 0.77 | 0.84 | 0.91 | 0.91 | 0.94 | 0.98 | 1.01 | 1.08 | 1.09 | |

2 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 1 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 5 |

Box-scatter plots with (a) the MLEs

Estimates of

5.549 | 5.362 | 5.448 | 0.019 | 0.021 | 0.021 |

This paper provides approaches based on the weighted regression framework and pivotal quantity to estimate unknown parameters of the Gompertz distribution with the PDF (

The censored data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work was supported by the National Research Foundation of Korea (NRF) Grant Funded by the Korea Government (Ministry of Education) (no. 2019R1I1A3A01062838).