1. Introduction
Entropy is one of the most popular measure of uncertainty. As former mathematical work, Reference [
1] proposed a theory on the concept of entropy, with numerical indicators as well. This theory was enhanced by numerous other entropy-like measures, arising from various applied fields. In this regard, a complete survey can be found in Reference [
2]. Here, we focus our attention on two of the most famous entropy measures—the Rényi entopy by Reference [
3] and the
q-entropy by Reference [
4] (also called Tsallis entropy). The Rényi entropy finds its source in the information theory and the
q-entropy comes from the statistical physics, with a plethora of applications in their respective fields. For a random variable
X having the probability density function (pdf)
, where
represents the corresponding parameters, these two entropy measures are, respectively, defined by
where
and
, and
where
and
. In particular, the Rényi entropy contains several well-known entropy measures, such as the Hartley entropy given by
, the Shannon entropy obtained as
and the collision entropy given as
.
In most of the observed phenomena, at least
is unknown, and the entropy as well. For this reason, the theoretical or practical statistical treatment of the entropy have been the object of all the attentions, in various settings. Among the notable studies in this regard, we may refer the reader to Reference [
5] discussing the entropy of ordered sequences and order statistic, Reference [
6] focusing on the entropy of upper record values, Reference [
7] proposing the entropy of hybrid censoring schemes, Reference [
8] discussing the entropy of progressively censored samples, Reference [
9] investigating the estimation of the entropy for the Weibull distribution under the progressive censoring scheme, Reference [
10] using the maximum likelihood and Bayes estimators via doubly-generalized Type II hybrid censored samples to estimate the entropy for the Rayleigh distribution, Reference [
11] studying the Bayes estimation of the entropy for the Weibull distribution under the generalized progressive hybrid censoring scheme, Reference [
12] providing the maximum likelihood and Bayes estimators of the entropy for the Weibull distribution under a generalized progressive hybrid censoring scheme, Reference [
13] studying the estimation of the entropy for the generalized exponential distribution based on record values, Reference [
14] discussing the Shannon entropy for the Lomax distribution based on the generalized progressively hybrid censoring scheme and Reference [
15] investigating point and interval estimation of the Shannon entropy for the inverse Weibull distribution under multiple censored data.
This paper provides a contribution to the estimation of the Rényi and
q-entropies for the inverse Lomax distribution when multiple censored data are observed. Let us now motivate the consideration of the inverse Lomax distribution in this setting, as well as the multiple censored data. First and foremost, the inverse Lomax (IL) distribution is a lifetime distribution, defined as the distribution of the reciprocal of a random variable following the famous Lomax (L) distribution. Mathematically, the L distribution is defined by the cumulative distribution function (cdf) and pdf given as
respectively. Here,
is a shape parameter and
is a scale parameter. The essential of the L distribution can be found in References [
16,
17,
18]. Thus, for a random variable
X having the L distribution, the random variable
follows the IL distribution with cdf and pdf given by
respectively. Among the advantages of the IL distribution, the corresponding probability functions are tractable, it is parsimonious in parameters and possesses a non-monotonic hazard rate function; it possesses decreasing and upside-down bathtub shapes. The practical usefulness of the related model is illustrated in Reference [
19] for its application to the analyses of geophysical data and in Reference [
20] for its application in economics and actuarial sciences. Also, we refer to Reference [
21] for the estimation of the reliability parameter via Type II censoring samples, in Reference [
22] for the estimation of the parameters based on hybrid censored samples, and in Reference [
23] for the Bayesian estimation of the two-component mixture of the IL distribution under the Type I censoring scheme. Also, recent studies have proposed extensions of the IL distributions for further purposes. In this regard, let us cite Reference [
24] for the inverse power Lomax (or power IL) distribution, Reference [
25] for the Weibull IL distribution, Reference [
26] for the alpha power transformed IL distribution, Reference [
27] for the Marshall-Olkin IL distribution and Reference [
28] for the odd generalized exponentiated IL distribution.
However, as far as we know, despite its interest, the estimation of entropy measures for the IL distribution, such as the Rényi and
q-entropies, remains an unexplored aspect. This study fills this gap by considering this problem under the realistic scenario of multiple censored data. This scenario commonly occurs where several censoring levels logically exist, which is often the case for many applications in life testing and survival analysis. We refer the reader to References [
29,
30,
31], as well as the recent estimation studies of Reference [
32] and Reference [
15]. In our statistical framework, after investigating the maximum likelihood estimates of
and
, estimates for the Rényi and
q-entropies are derived. Then, two-sided approximate confidence intervals of the Rényi and
q-entropies are discussed. A complete numerical study is performed, showing the favorable behavior of the obtained estimates at various sample sizes. In particular, the mean squared errors, approximate confidence intervals along with the corresponding average lengths are used as benchmarks. Our numerical investigations show that, when the sample size increases, the values of the mean squared errors and average lengths decrease. Also, when the censoring level decreases, the considered of Rényi and
q-entropies estimates approach the true value. Two real life data sets, one physiological data set and one economic data set, are used to illustrate the findings.
The rest of the article is arranged as follows. The Rényi and
q-entropies for the IL distribution are expressed in
Section 2.
Section 3 studies their estimation under multiple censored data. Simulation and numerical results are given in
Section 4. An application to real data sets is presented in
Section 5. The article ends with some concluding remarks in
Section 6.
5. Application
In this Section, two real life data sets are used to illustrate the finding, both described below.
The first data set is a physiological data set extracted from Reference [
33]. It concerns twenty Duchenne patients (6–18 years age) with classical type of the muscular dystrophy. The Electrocardiography of these 20 patients based on the heart rate is given below in
Table 9.
Then, based on the data, adopting the multiple censored data scheme, we apply
and
to estimate
and
where
X denotes the considered random variables of interest, assuming to follow the IL distribution. Different values for CL,
and
q are considered. The obtained numerical results are displayed in
Table 11 and
Table 12 for the first and second data sets, respectively.
Thus,
Table 11 and
Table 12 show some numerical values of estimated entropies in a concrete scenario, following the multiple censored data scheme. We see that the results depends on the entropy parameters (
or
q), and also, the value for CL, beyond the standard complete standard (which can be obtained by taking CL
).