Entropy is one of the most often used metrics for assessing the degree of uncertainty regarding a random variable. It was first applied in physics, particularly in the context of the second law of thermodynamics. Measuring entropy is crucial in many fields of science, including statistics, physics, chemistry, economics, insurance, financial analysis, and biological phenomena. Less information in a sample is referred to as possessing higher entropy. Entropy was defined by Shannon [
1] as a measure of information that provides a quantifiable measure of uncertainty using the methods of probability and statistics. This idea was strengthened by additional entropy measurements from various real-world applications; for a comprehensive survey, see Amigó et al. [
2]. Three of the most used entropy metrics are the subject of this paper, namely, Rényi entropy (RE) by Rényi [
3], the
q-entropy (QE) by Tsallis [
4], and Shannon entropy (SE) by [
1].
Suppose that
X is a random variable with probability density function (PDF)
, where
is the vector of the unknown parameters. Following that, the RE, QE, and SE of
X are defined, respectively, as:
and
In reality, both the entropy and
are unknown. Due to this, the estimation of the parameters and entropy has received the most attention several research papers, including, but not limited to, Wong and Chen [
5], Baratpour et al. [
6], Morabbi and Razmkhah [
7], Abo-Eleneen [
8], Cramer and Bagh [
9], Cho et al. [
10], Hassan and Zaky [
11], Bantan et al. [
12], and Okasha and Nassar [
13]. In the next two subsections, some detailed descriptions of preliminary concepts used in this study are presented.
1.1. Inverse Weibull Distribution and Its Entropy Indices
The first and foremost step that researchers must consider before thinking about dealing with the unknown entropy and
is to assume a suitable probability (lifetime) distribution and accordingly define the underlying PDF
and the corresponding cumulative distribution function (CDF). This study considers the inverse Weibull (IW) distribution, which is a handy probability distribution to model various types of data, including reliability and actuarial sciences data, because its hazard rate can be decreasing or unimodal depending on the value of the shape parameter. The IW distribution was considered by Keller [
14] to model failures of mechanical components subject to degradation. Afterward, many researchers studied the IW distribution, including, but not limited to, Calabria and Pulcini [
15], Jiang et al. [
16], Mahmoud et al. [
17], Sultan [
18], Kundu and Howlader [
19], Hassan et al. [
20], Kumar and Kumar [
21], and Al-Duais [
22]. The random variable
Y follows the two-parameter IW distribution, denoted by IW
, if the corresponding PDF and CDF, are given by:
and
respectively, where
is the shape parameter and
is the rate parameter. From (
1) and (
4), the RE of the random variable
Y can be expressed as follows:
with
,
, and
. Similarly, from (
2), (
3), and (
4) the QE and SE of the random variable
Y can be written, respectively, as:
with
,
and
and
where
is the Euler constant.
1.2. Progressive Censoring Scheme and Some of Its Modifications
Typically, researchers conduct life-testing experiments on a random sample of
n objects of interest to obtain data from which they can estimate
and accordingly estimate the entropy. However, waiting for the whole sample to fail is costly and time-consuming. Therefore, researchers obtain an incomplete dataset by a censoring scheme. Censoring is a widespread technique in life-testing experiments. In simple terms, data censoring means reducing cost and saving time, but at the same time, losing some information that might be important. The most common censoring schemes are the conventional Type-I and Type-II censoring schemes. While Type-I censoring ends the experiment at a predetermined time, Type-II censoring ends the experiment whenever a predetermined number of failures are attained. Because technology is developing rapidly, researchers frequently want to reduce expenses and test time. As a result, the progressive Type-II censoring (PT-IIC) scheme can be used as a generalized censoring technique. The PT-IIC sample can be described as follows: suppose a life-testing experiment involving
n units is conducted according to a prefixed progressive censoring scheme
. Here, the number of observed failure times, say
m, where
is predetermined. When the first failure occurs,
items are removed from the remaining
experimental units. Then, at the time of the second failure,
units are removed from the
remaining units. The process is repeated until the experiment reaches the
mth failure, and at this point, all the remaining units are removed; afterward, the experiment is terminated. It is important to emphasize that the progressive censoring plan
must satisfy
. Estimating the parameters of some lifetime distributions based on the PT-IIC scheme has been the subject of several studies during the past two decades, see for example Rastogi et al. [
23], Ahmed [
24], and Dey et al. [
25]. For more comprehensive details, see, for example, Balakrishnan and Aggarwala [
26], Balakrishnan and Cramer [
27], Dey et al. [
28], and Kumar et al. [
29].
Practically, the conventional PT-IIC scheme might take a while to reach the required failure times for the tested units; therefore, Ng et al. [
30] proposed the adaptive progressive type-II censoring (APT-IIC) scheme as an alternative. For additional details about the latter censoring scheme, see, for example, Ye et al. [
31], Sobhi and Soliman [
32], and EL-Sagheer et al. [
33]. Since the APT-TIIC scheme might not solve the problem of consuming experimental time, especially when the experimental units are highly reliable, Yan et al. [
34] recently proposed the improved adaptive progressive Type-II censoring (IAPT-IIC) scheme. The process of the IAP-TIIC scheme can be defined as follows: assume an independent and identically distributed random sample of
n units are set on a life test, the required number of failures
is prefixed and
are also predetermined; however, some values of
may adjust during the experimentation. Let
, where
, be two thresholds specified based on the dependability information on the product of interest. Let
and
be the number of failures occur before times
and
, respectively, where
. At the time of the first failure
,
units are randomly withdrawn from
live items. Likewise, at the time of the second failure
,
of
items are randomly withdrawn from the experiment, and so on. If
occurs first before time
, i.e.,
(Case-I: PT-IIC scheme), the experiment stops at
with censoring scheme
. If
(Case-II: APT-IIC scheme), where
and
, the experiment stops at
with censoring scheme
, where
, then no live units will be withdrawn from the test by placing
for
and at the time of the
failure all staying units are extracted. Finally, if
is not failed before time
, i.e.,
(Case-III: IAPT-IIC scheme), the test stops at
with censoring scheme
for
, and at
all the remaining items are withdrawn, i.e.,
. An experiment considering the IAPT-IIC scheme has three outputs, as shown in
Table 1.
Due to the facts that IW distribution is flexible in modeling real datasets, and the IAP-TIIC scheme’s efficiency in data acquisition, mainly when the experimental units are of a high degree of reliability, this study concentrates on estimating the entropy of the IW distribution utilizing samples obtained via IAP-TIIC plans. The main idea that motivated this study is comparing the samples acquired based on the PT-IIC, APT-IIC, and IAPT-IIC schemes based on the amount of information they provided. This comparison interests many researchers in selecting the appropriate censoring scheme when collecting the required data. Another motivation for this work is to compare the efficiency of two classical estimation methods, namely maximum likelihood and maximum product of spacing (MPS), and four confidence interval estimation methods, to see which estimation method is suitable for estimating the considered entropy measures assuming IW lifetimes. The main objectives of this work are: (1) To investigate point and interval estimations of the three entropy indices, namely RE, QE, and SE, using maximum likelihood and MPS techniques. (2) To compare the approximate confidence intervals (ACIs) with two parametric bootstrap confidence intervals of the entropy measures. (3) To examine the effectiveness of the various approaches using a variety of scenarios of sample sizes, progressive censoring techniques, and thresholds using simulation research. (4) To make clear the usage of the outlined methodologies by analyzing a pair of real datasets. It is crucial to note that the two selected datasets were utilized for the practical investigation when the parent distribution is the IW model, which does not necessarily suggest that additional datasets of this sort have the same connection.
The remainder of this paper is organized as follows.
Section 2 explores the maximum likelihood estimators (MLEs) and ACIs for RE, WE, and SE. The MPS estimators (MPSEs) and ACIs of the entropy measure are acquired in
Section 3.
Section 4 covers two parametric bootstrap confidence intervals for the entropy measures.
Section 5 reports the outcomes of Monte Carlo simulations, while
Section 6 provides the outcomes of the analyses for two actual datasets. In
Section 7, the paper is concluded with a discussion and future research directions.