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Bayesian and non-Bayesian estimation of dynamic cumulative residual Tsallis entropy for moment exponential distribution under progressive censored type II

  • Salem A. Alyami , Amal S. Hassan EMAIL logo , Ibrahim Elbatal , Mohammed Elgarhy EMAIL logo and Ahmed R. El-Saeed
Published/Copyright: July 18, 2023
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Open Physics
From the journal Open Physics Volume 21 Issue 1

Abstract

The dynamic cumulative residual (DCR) entropy is a helpful randomness metric that may be used in survival analysis. A challenging issue in statistics and machine learning is the estimation of entropy measures. This article uses progressive censored type II (PCT-II) samples to estimate the DCR Tsallis entropy (DCRTE) for the moment exponential distribution. The non-Bayesian and Bayesian approaches are the recommended estimating strategies. We obtain the DCRTE Bayesian estimator using the gamma and uniform priors via symmetric and asymmetric (linear exponential and general entropy) loss functions (LoFs). The Metropolis–Hastings algorithm is employed to generate Markov chain Monte Carlo samples from the posterior distribution. The accuracy of different estimates for various sample sizes, is implemented via Monte Carlo simulations. Generally, we note based on the simulation study that, in the majority of cases, the DCRTE Bayesian estimates under general entropy followed by linear exponential LoFs are preferable to the others. The accuracy measure of DCRTE Bayesian estimates using a gamma prior has smaller values than the others using a uniform prior. As sample sizes grow, the Bayesian estimates of the DCRTE are closer to the true value. Finally, analysis of the leukemia data confirmed the proposed estimators.

Keywords: Tsallis entropy; moment exponential distribution; squared error loss function; Markov chain Monte Carlo; maximum likelihood; general entropy loss function

1 Introduction

The concept of Tsallis ( T ) entropy, which can be employed to calculate the uncertainty in a random observation, was first proposed in a study by Tsallis [1]. It has been widely employed in the research of quantum communication protocols, quantum systems, and quantum correlations [2,3]. The entropy (T) of order q is provided via

(1) T ( q ) = ( q − 1 ) − 1 1 − ∫ 0 ∞ g q ( z ) d z ,

where q > 0 , q ≠ 1 , and g ( z ) is the probability density function (PDF). A number of authors have recently examined the estimation of the entropy measures utilizing various statistical distributions and sampling techniques (see, for instance, [4–14]). Recently, several researchers have become interested in alternative uncertainty measures of probability distributions, particularly in reliability and survival analysis research. This is why [15] suggested cumulative residual (CR) entropy. The following is the definition for the dynamic form of CR entropy, known as dynamic cumulative residual Tsallis entropy (DCRTE), as proposed by Sunoj and Linu [16]

(2) γ ( q , t ) = ( q − 1 ) − 1 1 − ∫ t ∞ G ¯ q ( z ) G ¯ q ( t ) d z ,

where q > 0 , q ≠ 1 , and G ¯ ( t ) = 1 − G ( t ) is the survival function (SF) and for t = 0, the DCRTE leads to CR Tsallis entropy (CRTE). Only a few studies from the literature were considered in the case of inferential approaches to dynamic CR (DCR) entropy for probability distributions. Kamari [17] demonstrated the properties of the DCR entropy derived from order statistics. The CR has been established in a study by Kundu et al. [18] as extensions of the cumulative entropies for truncated random variables. The DCR entropy of the Pareto model was studied by the Bayesian technique using a range of sampling approaches in a few studies [19–23]. Almarashi et al. [24] used Bayesian inference to estimate the DCR entropy of the Lindley distribution.

A flexible strategy for analyzing lifetime datasets is typically appealing to researchers. The moment exponential (MExp) distribution (or length-biased exponential distribution) was established in ref. [25] by weighting the exponential distribution in accordance with Fisher’s (1934). Dara and Ahmad in their study [25] offered many characteristics and applications of the MExp distribution. The PDF of the MExp distribution is

(3) g ( z ; λ ) = z λ 2 e − z λ ; z , λ > 0 .

The cumulative distribution function (CDF) and the SF of the MExp model are as follows:

(4) G ( z ; λ ) = 1 − e − z λ 1 + z λ ; z , λ > 0 ,

and

(5) G ¯ ( z ; λ ) = e − z λ 1 + z λ ; z , λ > 0 .

Due to the MExp model’s versatility, it attracted a lot of attention, and as a result, other authors investigated and further generalized it for more complicated datasets. For instance, the generalized exponentiated MExp distribution [26], the Marshall–Olkin length-biased exponential distribution [27], the Topp–Leone MExp distribution [28], the Kumaraswamy MExp distribution [29], the Marshall–Olkin Kumaraswamy MExp distribution [30], the Burr XII-MExp distribution [31], and the alpha power MExp distribution [32] and references therein.

In reliability experiments, investigators seek to know how long it takes for units to malfunction. However, investigators are unable to observe the life of all units due to time and money constraints, as well as a number of other issues. This leads to the availability of data that have been censored. Censoring types I (CT-I) and II (CT-II) are the two most reasonably available methods. But in some situations, like medical/engineering survival analysis, units might be taken out at intermediate stages for a number of reasons that are beyond the experimenter’s control. Since it allows for the removal of surviving items prior to the test’s conclusion, a progressive censoring (PC) system is a suitable censoring method in this context. The PC type-I (PCT-I) takes place when the percentage of survivors falls to predetermined levels, whereas the PC type-II (PCT-II) takes place when the percentage of survivors falls to specific values.

A PCT-II sample is prepared in the manner described below: A life-testing experiment that has n units and the PC strategy r i , i = 1 , 2 , … , m is implemented. Units are picked at random from the remaining n − 1 surviving units at the moment of the first failure z ( 1 ) . After the second failure z ( 2 ) , units from the remaining n − 2 − r 1 are also randomly removed. The test goes on until the mth failure happens, at which point all remaining n − m − r 1 − r 2 − … − r m − 1 units are discarded. The number of failures m in addition to the PC design r 1 , r 2 , … , r m is predetermined and fixed. Assume that z ( 1 ) , z ( 2 ) , … , z ( m ) indicate a PCT-II sample with ( r 1 , r 2 , … , r m ) being the PC. A useful description and good summary of PC can be found in in a study by Balakrishnan and Aggrawala [33]. It should be noted that, when r 1 = r 2 = … = r m − 1 = 0 and r m = n − m , this censoring reduces to CT-II. In addition, for m = n and r i = 0 , i = 1 , 2 , … , n , it reduces to a complete sample. Figure 1 represents this PC strategy.

Figure 1 PCT-II scheme.
Figure 1

PCT-II scheme.

This article looks into the DCRTE estimators for the MExp model in the presence of PCT-II data. We construct analytical formulas for the proposed DCRTE metric. The Bayesian estimator (BE) of the DCRTE measure is derived, along with its maximum likelihood estimator (MLE). The BE is regarded under squared error (SE), linear exponential (LIN), and general entropy (GE) loss functions (LoFs). Using various sample sizes and censoring strategies, Monte Carlo simulations are utilized to assess the performance of the DCRTE measure. Using actual data, the inferential techniques described in this study were also shown.

The article is broken down into six sections. Section 2 presents the formula for the DCRTE for the MExp distribution. PCT-II is used in Sections 3 and 4 to give DCRTE estimators by using the proposed approaches to estimation. In Section 5, data analysis and the numerical study of the produced estimators are covered. The study’s summary and conclusions are included in Section 6.

2 Formula of the DCRTE for MExp distribution

The DCRTE formula for MExp distribution is shown in this section. The MExp distribution’s DCRTE is determined by inserting Eq. (5) into Eq. (2):

(6) γ ( q , t ) = 1 ( q − 1 ) 1 − 1 G ¯ q ( t ) ∫ t ∞ e − q z λ 1 + z λ q d z = 1 ( q − 1 ) 1 − 1 G ¯ q ( t ) I ,

where, I = ∫ t ∞ e − q z λ 1 + z λ q d z . To obtain I , we utilize the transformation y = 1 + z λ , which gives us z = λ ( y − 1 )

(7) I = ∫ 1 + t λ ∞ e − q λ ( y − 1 ) λ λ y q d y = λ e q ∫ 1 + t λ ∞ y q e − q y d y .

Let x = q y , then, (7) can be expressed as follows:

(8) I = λ e q ∫ q 1 + t λ ∞ x q q e − x 1 q d x = λ e q [ q ] q + 1 Γ q + 1 , q 1 + t λ ,

where Γ ( . , x ) stands for incomplete gamma function. The DCRTE for MExp distribution is expressed, as follows, after putting Eq. (8) into Eq. (6):

(9) γ ( q , t ) = 1 q − 1 1 − λ e A ( λ , t , q ) Γ ( q + 1 , A ( λ , t , q ) ) q ( A ( λ , t , q ) ) q ,

where A ( λ , t , q ) = q 1 + t λ .

The plots of the DCRTE for the MExp distribution at various values of q and λ are shown in Figure 2. This figure shows that when the value of q is less than 1, the DCRTE for the MExp distribution takes increasing and then decreasing shapes. On the other hand, when the value of q is greater than 1, the DCRTE for the MExp distribution has rising forms. In addition, we see that when q is less than 1 and as the value of λ increases, the DCRTE for the MExp distribution exhibits increasing behavior.

Figure 2 Plots of DCRTE for the MExp distribution.
Figure 2

Plots of DCRTE for the MExp distribution.

3 Maximum likelihood estimation

Suppose z ( 1 ) ≤ z ( 2 ) ≤ … ≤ z ( m ) represent a PCT-II sample of size m from a sample of size n with PDF (3), CDF (4), and censoring scheme r 1 , r 2 , … , r m . The likelihood function, under the PCT-II sample, is as follows:

(10) L ( λ ) = c ∏ i = 1 m f ( z ( i ) ) [ 1 − F ( z ( i ) ) ] r i = c ∏ i = 1 m z ( i ) λ 2 e − ( r i + 1 ) z ( i ) λ 1 + z ( i ) λ r i ,

where c = n ( n − r 1 − 1 ) ( n − r 1 − r 2 − 2 ) … n − m + 1 − ∑ i = 1 m − 1 r i . As a result, the constant is the number of different ways in which the m PCT-II order statistics might arise if the observed failure times are z ( 1 ) , z ( 2 ) , … , z ( m ) . The log-likelihood function of Eq. (10) is supplied by

(11) ln L ( λ ) = ln c − 2 m ln λ + ∑ i = 1 m ln z ( i ) − ∑ i = 1 m ( r i + 1 ) z ( i ) λ + ∑ i = 1 m r i ln 1 + z ( i ) λ .

Form Eq. (11), we derive the likelihood equation for λ as follows:

(12) d ln L ( λ ) d λ = − 2 m λ + ∑ i = 1 m ( r i + 1 ) z ( i ) λ 2 − 1 λ ∑ i = 1 m r i z ( i ) λ + z ( i ) .

By solving the nonlinear Eq. (12) after setting it to 0, the MLE of λ can be found using the numerical technique. We can acquire the MLE of the DCRTE measure specified in Eq. (9) after the MLE of λ , say λ ˆ , is computed. As a result, the invariance property is used to generate the MLE of DCRTE, designated by γ ˆ ( q , t ) , by putting in Eq. (9) as shown below:

(13) γ ˆ ( q , t ) = 1 q − 1 1 − λ ˆ e A ( λ ˆ , q , t ) Γ ( q + 1 , A ( λ ˆ , q , t ) ) q ( A ( λ ˆ , q , t ) ) q ,

where A ( λ ˆ , q , t ) = q 1 + t λ ˆ .

The theoretical findings presented above can be further specialized in two situations. First, the MLE λ ˆ and γ ˆ ( q , t ) are yielded when r 1 = r 2 = … = r m − 1 = 0 and r m = n − m via CT-II. Second, we obtain the recommended MLE of λ and γ ( q , t ) for r 1 = r 2 = … = r m = 0 .

4 Bayesian estimation

The BE of the DCRTE for MExp distribution under the PCT-II scheme will be covered in this section. Different LoFs, such as the SE, LIN, and GE, can be taken into consideration for Bayesian estimation. For an informative prior (IP) distribution of the parameter λ of the MExp distribution, we can suggest using independent gamma prior for λ having PDF

π ( λ ) ∝ λ a e − b λ λ > 0 , a > 0 , b > 0 ,

where the parameters a and b are chosen to represent the prior knowledge about the unknown parameter. The corresponding posterior density given the observed data, z = ( z ( 1 ) , z ( 2 ) , … , z ( m ) ) , is given as follows:

π ( λ ∣ z ) = π ( λ ) L ( λ ) ∫ 0 ∞ π ( λ ) L ( λ ) d λ ∝ π ( λ ) L ( λ ) .

As a result, the posterior density function may be expressed as follows:

(14) π ( λ ∣ z ) ∝ λ − 2 m + a e − b λ ∏ i = 1 m z ( i ) e − ( r i + 1 ) z ( i ) λ 1 + z ( i ) λ r i .

The BE of γ ( q , t ) , denoted by γ ˆ BE SE ( q , t ) , under the SE LoF, is given as follows:

(15) γ ˆ BE SE ( q , t ) = E [ γ ( q , t ) ∣ z ] = ∫ 0 ∞ γ ( q , t ) π ( λ ∣ z ) d λ .

Based on LIN LoF, the Bayes estimator of γ ( q , t ) , say γ ˆ BE LIN ( q , t ) , is as follows:

(16) γ ˆ BE LIN ( q , t ) = − 1 v log E [ e { − v γ ( q , t ) } ∣ z ] = − 1 v log ∫ 0 ∞ e { − v γ ( q , t ) } π ( λ ∣ z ) d λ .

Furthermore, we consider the GE LoF where the Bayes estimator of γ ( q , t ) , say γ ˆ BE G E ( q , t ) , is as follows:

(17) γ ˆ BE G E ( q , t ) = [ E ( ( γ ( q , t ) ) − τ ∣ z ) ] − 1 ∕ τ = ∫ 0 ∞ ( γ ( q , t ) ) − τ π ( λ ∣ z ) d λ − 1 ∕ τ .

It is seen that the estimates given by Eqs. (15)–(17) cannot be simplified into closed form expressions. Therefore, we next apply the Markov chain Monte Carlo (MCMC) approach and generate a posterior sample using Metropolis–Hasting (MH) algorithm to obtain the desired DCRTE under BEs. Note that, by selecting the hyper parameters a = b = 0 and using the same MCMC approach, the previous BE is obtained in the case of noninformative prior (Non-IP).

4.1 MH algorithm

In order to execute the MH algorithm for the DCRTE of MExp distribution, a suggested distribution and the initial values of the unknown parameter λ have to be specified. For the proposal distribution, a normal distribution will be taken into account, i.e., q ( λ ′ ∣ λ ) ≡ N ( λ , S λ ) , where S λ represents the variance–covariance matrix (V-CM). For the initial values, the MLE for λ is considered, i.e., λ ( 0 ) = λ ˆ MLE . The choice of S λ is thought to be the asymptotic V-CM, say I − 1 ( λ ˆ MLE ) , where I ( . ) is the Fisher information matrix. In this manner, the MH method uses the stages listed below to draw a sample from the posterior density provided by Eq. (14):

Step 1. Set initial value of λ as λ ( 0 ) = λ ˆ MLE .

Step 2. For i = 1 , 2 , … , M repeat the following steps:

  • Set λ = λ ( i − 1 ) .

  • Create a new candidate parameter value using N ( λ , S λ ) .

  • Compute the formula δ = π ( λ ′ ∣ z ) π ( λ ∣ z ) , where π ( ⋅ ) is the posterior density (Eq. (14)).

  • Create a sample u from the uniform U ( 0 , 1 ) distribution.

  • Accept or reject the new candidate λ ′

    If u ≤ δ put λ ( i ) = λ ′ elsewhere put λ ( i ) = λ .

Eventually, a portion of the initial samples can indeed be removed (burn-in), and the remaining samples can be used to calculate Bayes estimates using random samples of size M drawn from the posterior density. The BE of γ ( q , t ) is assessed by employing MCMC with SE, LIN, and GE LoFs as follows:

(18) γ ˆ BE SE ( q , t ) = 1 M − l B ∑ i = l B M γ ( q , t ) ( i ) ,

(19) γ ˆ BE LIN ( q , t ) = − 1 v log 1 M − l B ∑ i = l B M e { − v γ ( q , t ) ( i ) } ,

(20) γ ˆ BE G E ( q , t ) = 1 M − l B ∑ i = l B M ( γ ( q , t ) ( i ) ) − τ − 1 ∕ τ ,

where l B represents the number of burn-in samples.

5 Data analysis and simulation study

This section’s goal is to examine the behavior of the suggested DCRTE estimators for the MExp distribution under the PCT-II scheme that was addressed in earlier parts. For demonstration purposes, we investigate an actual dataset. Furthermore, we conducted a simulation study to investigate the behavior of the proposed approaches and evaluate the estimates’ performance under various censoring schemes. We used R, a statistical programming language, for calculation. Furthermore, one can utilize b b m l e package to compute MLEs in R-language.

5.1 Data analysis

A real dataset is analyzed for illustrative purposes as well as to assess the statistical performances of the MLE and BEs for DCRTE in the case of the MExp distribution under different PCT-II schemes.

According to the International Bone Marrow Transplant Registry, 101 patients with advanced acute myelogenous leukemia are linked to the following datasets (see [34]). Fifty-one of these patients have received an autologous bone marrow transplant. Fifty-one patients had an allogeneic bone marrow transplant. The 51 autologous transplant patients’ leukemia-free survival periods (in months) are shown in Table 1.

Table 1

The leukemia-free survival times (in months) dataset

0.658 0.822 1.414 2.500 3.322 3.816 4.737 4.836 4.934 5.033 5.757
5.855 5.987 6.151 6.217 6.447 8.651 8.717 9.441 10.329 11.480 12.007
12.007 12.237 12.401 13.059 14.474 15.000 15.461 15.757 16.480 16.711 17.204
17.237 17.303 17.664 18.092 18.092 18.750 20.625 23.158 27.730 31.184 32.434
35.921 42.237 44.638 46.480 47.467 48.322 56.086

We begin by determining if the MExp distribution is appropriate for evaluating this dataset. To assess the quality of fit, we provide the MLE of λ and the Kolmogorov–Smirnov (K–S) test statistic value. The estimated K–S distance for the MExp distribution between the empirical and fitted distributions is 0.13301, and its p -value is 0.3276 where λ ˆ = 8.36519 , indicating that this distribution may be deemed an appropriate model for the current dataset. For graphically fitting the given dataset, the empirical CDF (ECDF), the empirical PDF (EPDF), the empirical SF (ECDF), and probability–probability (PP)-plots are represented in Figure 3 for the MExp distribution. This figure also indicates that the MExp distribution provides a good fit for this dataset.

Figure 3 The EPDF, ECDF, ESF, and PP plots for MExp model.
Figure 3

The EPDF, ECDF, ESF, and PP plots for MExp model.

For convergence of BEs, we divide real data by 100, from the original data, hence one can generate, e.g., four schemes, namely, Scheme 1 (Sch.1), Scheme 2 (Sch.2), Scheme 3 (Sch.3), and Scheme 4 (Sch.4) from PCT-II samples with the number of stages m = 26 and eliminated items r i are presumed to be the following:

  1. r 1 = r 2 = … = r 25 = 0 , r 26 = 25

  2. r 1 = r 2 = … = r 25 = 1 , r 26 = 0

  3. r 1 = 26 , r 2 = … = r 25 = 0

  4. r 1 = r 2 = … = r 26 = 0 , n = m = 51 .

Note that: Sch.1 represents the usual CT-II and Sch.4 represents a complete sampling case. Two distinct values of the constant t are used, 0.5 and 1.5 given q = 2 .

In Table 2, the MLEs of λ have been investigated and then plugged into DCRTE measures at the proposed schemes for PCT-II samples as in the suggested real dataset. In addition, BEs can be computed employing the MH algorithm under Non-IP (uniform prior), i.e., the hyper-parameter values are taken as a = b = 0 . Different LoFs, including SE, LIN-1 ( v = 0.5 ), LIN-2 ( v = − 0.5 ), GE-1 ( τ = − 0.5 ), and GE-2 ( τ = 0.5 ) are assumed. It is indicated that, while generating samples from the posterior distribution utilizing the MH algorithm, initial values of λ are considered as λ ( 0 ) = λ ˆ MLE . Eventually, among the total 10,000 samples generated by the posterior density, 2,000 burn-in samples were removed, and a DCRTE estimate was derived.

Table 2

The estimate values of ( γ ( q , t ) ) for MExp distribution under different PCT-II schemes for leukemia-free survival times data

t Sch. ML BE-MCMC: Non-IP
SE LIN-1 LIN-2 GE-1 GE-2
0.5 Sch.1 0.84347 0.84124 0.84142 0.84106 0.84103 0.84059
Sch.2 0.93009 0.92865 0.92868 0.92861 0.92861 0.92854
Sch.3 0.95671 0.95574 0.95576 0.95573 0.95573 0.95570
Sch.4 0.95176 0.95116 0.95117 0.95115 0.95115 0.95113
1.5 Sch.1 0.84347 0.84739 0.84750 0.84729 0.84727 0.86702
Sch.2 0.93772 0.93660 0.93662 0.93658 0.93658 0.93653
Sch.3 0.96017 0.95939 0.95940 0.95938 0.95937 0.95935
Sch.4 0.95176 0.95543 0.95543 0.95542 0.95542 0.95541

From Table 2, one can infer that the behavior of Sch.3 is better for estimating γ ( q , t ) . In addition, as t increases, the estimates of γ ( q , t ) increase. The convergence (scatter plot, histogram, and cumulative mean) of MCMC estimation for λ and γ ( q , t ) is shown in Figure 4 in the case of complete sampling. From Figure 4, the normality of MCMC estimates can be observed.

Figure 4 Convergence of MCMC estimates for γ ( q , t ) \gamma \left(q,t) using MH algorithm. (a) Graphs of MCMC estimates for γ ( q = 2 , t = 0.5 ) \gamma \left(q=2,t=0.5) , (b) Graphs of MCMC estimates for γ ( q = 2 , t = 1.5 ) \gamma \left(q=2,t=1.5) .
Figure 4

Convergence of MCMC estimates for γ ( q , t ) using MH algorithm. (a) Graphs of MCMC estimates for γ ( q = 2 , t = 0.5 ) , (b) Graphs of MCMC estimates for γ ( q = 2 , t = 1.5 ) .

5.2 Simulation study

In this part, we use a Monte Carlo simulation analysis to assess the effectiveness of estimate techniques, specifically ML and Bayesian using MCMC, for MExp distribution under the PCT-II scheme. We produce 1,000 MLEs from the MExp distribution with the following principles:

  1. The parameter of the MExp distribution λ is assumed to be 0.4 , 0.8 , and 1.

  2. The value of q is equal to 2, and the constant value of t is equal 0.5 , 1.5 .

  3. The true value of γ ( q , t ) , say γ 1 for simplified form, at t = 0.5 , q = 2 , and λ = 0.4 is γ 1 = 0.69135 . The true value of γ ( q , t ) , say γ 2 for simplified form, at t = 1.5 , q = 2 , and λ = 0.4 , is γ 2 = 0.75346 .

  4. The true value of γ ( q , t ) , say γ 3 for simplified form, at t = 0.5 , q = 2 , and λ = 0.8 is γ 3 = 0.27811 . The true value of γ ( q , t ) , say γ 4 for simplified form, at t = 1.5 , q = 2 , and λ = 0.8 is γ 4 = 0.43667 .

  5. The true value of γ ( q , t ) , say γ 5 for simplified form, at t = 0.5 , q = 2 , and λ = 1 is γ 5 = 0.05556 . The true value of γ ( q , t ) , say γ 6 for simplified forms, at t = 1.5 , q = 2 , and λ = 1 is γ 6 = 0.26000 .

  6. The sample sizes are n = 30 and n = 60 .

  7. The number of stages of the PCT-II scheme are m = 10 , 20 , and 40.

  8. Removed items r i are assumed at n and m values as shown in Table 3 where r m = n − ∑ i = 1 m − 1 r i + m and r is the number of failure items.

Table 3

Different patterns for removing items from life test at different number of stages

n m Censoring schemes
Sch.1 Sch.2 Sch.3 Sch.4
30 10 ( 20 , 0 ∗ 9 ) ( 10 , 0 ∗ 8 , 10 ) ( 0 ∗ 4 , 10 , 10 , 0 ∗ 4 ) ( 0 ∗ 9 , 20 )
20 ( 10 , 0 ∗ 19 ) ( 5 , 0 ∗ 18 , 5 ) ( 0 ∗ 9 , 5 , 5 , 0 ∗ 9 ) ( 0 ∗ 19 , 10 )
60 20 ( 40 , 0 ∗ 19 ) ( 20 , 0 ∗ 18 , 20 ) ( 0 ∗ 9 , 20 , 20 , 0 ∗ 9 ) ( 0 ∗ 19 , 40 )
40 ( 20 , 0 ∗ 39 ) ( 10 , 0 ∗ 38 , 10 ) ( 0 ∗ 19 , 10 , 10 , 0 ∗ 19 ) ( 0 ∗ 39 , 20 )

Here, ( 1 ( 3 ) , 0 ) , for example, means that the censoring scheme employed is ( 1 , 1 , 1 , 0 ) .

MLEs are determined employing PCT-II based on the generated data and the preceding assumptions. When calculating MLEs, the initial estimate values are assumed to be the same as the actual parameter values. These values, known as MLEs, are then employed to compute the DCRTE ( γ ( q , t ) ) given q and t .

For the Bayesian method, BEs using the MH algorithm using Non-IP and IP under different LoFs are computed. Thus:

  1. For Non-IP, the hyper-parameter values are a = b = 0 , thus π ( λ ) = 1 λ .

  2. For IP, the hyper-parameter values are taken as a = 0.5 and b = 1.5 .

  3. Different LoFs, including SE, LIN-1 ( v = 0.5 ), LIN-2 ( v = − 0.5 ), GE-1 ( τ = − 0.5 ), and GE-2 ( τ = 0.5 ) are assumed.

These values are then employed to determine the estimated values. When using the MH technique, the MLEs take into consideration the related V-CM S λ of λ ˆ MLE as initial estimate values. Finally, 2,000 burn-in samples are removed from the total 10,000 samples created by the posterior density, and the estimates of DCRTE ( γ ( q , t ) ) are derived.

The mean squared error (MSEr) for all DCRTE estimates is reported in Tables 1, 2, 3, 4, 5, and 6 covering all inputs of Monte Carlo simulation. From tabulated result values, it can be noted that:

  1. For MLEs, higher values of n and m lead to a decrease in MSEr and convergence to the initial values of γ ( q , t ) as expected.

  2. The MSEr of Bayes estimates under IP gradually decreases as n and m increase.

  3. As the true value of γ ( q , t ) increases, the values of ( γ ˆ ( q , t ) ) approach to true values implying that there is more information.

  4. As can be seen in Tables 4 and 6, in the majority of cases, the precision measure of DCRTE under GE-1 and LIN-1 for IP and Non-IP are more effective than those under GE-2 and LIN-2 for all schemes (Table 7).

  5. At true value γ 5 = 0.05556 , the Bayes estimates of γ ( q , t ) under LIN-1 and GE-2 are preferable to the corresponding estimates under LIN-2 and GE-1 in both IP and Non-IP (see Tables 8 and 9) for all schemes.

  6. In both IP and Non-IP, Bayes estimates of γ ( q , t ) under LIN-1 and GE-2 are commonly chosen over similar estimates under those LIN-2 and GE-1 at true value γ 6 = 0.26000 for all schemes (see Tables 8 and 9).

  7. Bayes estimate under Non-IP has a decreasing pattern as compared to the others under IP. However, Bayesian estimates under Non-IP provide larger estimate values.

  8. The MSEr of all estimates based on Sch.3 has the smallest values, at true value γ 2 = 0.75346 , when compared to others at m = 20 , 40 , and n = 60 (see Table 5). In addition, at γ 1 = 0.69135 , as seen in Table 5, the MSEr of all estimates based on Sch.3 also obtains the smallest values in comparison to other estimates at m = 40 , and n = 60 .

  9. As shown in Table 6, the MSEr of all estimates based on Sch.3 typically has the fewest values, when compared to other estimates, in the majority of cases, for m = 10 , and n = 30 with a real value of γ 4 = 0.43667 .

  10. The MSEr of all estimates based on Sch.2 generally yields the smallest values when compared to others, in the majority of cases, at m = 20 and n = 30 with a real value of γ 3 = 0.27811 , as seen in Table 6.

  11. The MSEr of all estimates based on Sch.2 typically produces the smallest values when compared to other estimates at m = 10 and n = 30 , in most of the situations, as seen in Table 8, when γ 5 = 0.05556 and γ 6 = 0.26000 . Also, we observe that estimates for Sch.4 provide the least MSEr compared to others at m = 20 and n = 30 , when γ 5 = 0.05556 and γ 6 = 0.26000 .

  12. The MSEr of all estimates based on Sch.4 generally yields the smallest values when compared to other estimates at m = 10 , 20 and n = 30 , as seen in Table 4.

  13. The smallest MSEr results are often produced for all estimates in Sch.2 when compared to other estimates, in a majority of situations at m = 10 , as provided in Table 9.

  14. Figure 5 demonstrates that the precision measure of the Bayesian estimates in the Non-IP case has the biggest values when compared to the other estimates in the IP case. Also, as the value of m increases, the MSEr for all estimates decreases.

  15. As t increases, a slight decrease in most estimates (MLE, SE, LIN-1, and LIN-2) and a slight increase in GE estimates were observed.

Table 4

MSEr of DCRTE estimates based on MExp distribution at λ = 0.4 under different PCT-II schemes at n = 30

m Sch. MLE BE-MCMC: IP BE-MCMC: Non-IP
SE LIN-1 LIN-2 GE-1 GE-2 SE LIN-1 LIN-2 GE-1 GE-2
t = 0.5, γ 1 = 0.69135
10 Sch.1 0.00562 0.00691 0.00669 0.00735 0.00718 0.00814 0.00759 0.00716 0.00785 0.00903 0.01026
Sch.2 0.00499 0.00594 0.00579 0.00643 0.00628 0.00704 0.00661 0.00610 0.00681 0.00738 0.00853
Sch.3 0.00493 0.00583 0.00568 0.00625 0.00615 0.00688 0.00644 0.00600 0.00666 0.00719 0.00847
Sch.4 0.00431 0.00490 0.00479 0.00546 0.00511 0.00589 0.00562 0.00503 0.00580 0.00590 0.00706
20 Sch.1 0.00281 0.00302 0.00298 0.00310 0.00311 0.00325 0.00314 0.00306 0.00319 0.00333 0.00350
Sch.2 0.00273 0.00295 0.00291 0.00294 0.00303 0.00307 0.00298 0.00299 0.00303 0.00321 0.00332
Sch.3 0.00267 0.00289 0.00285 0.00289 0.00296 0.00302 0.00293 0.00293 0.00297 0.00316 0.00323
Sch.4 0.00261 0.00286 0.00282 0.00298 0.00295 0.00313 0.00303 0.00291 0.00308 0.00314 0.00335
t = 1.5, γ 2 = 0.75346
10 Sch.1 0.00318 0.00388 0.00380 0.00433 0.00404 0.00467 0.00445 0.00397 0.00457 0.00441 0.00520
Sch.2 0.00267 0.00321 0.00315 0.00344 0.00334 0.00369 0.00352 0.00328 0.00362 0.00366 0.00410
Sch.3 0.00282 0.00347 0.00339 0.00389 0.00363 0.00417 0.00399 0.00355 0.00408 0.00403 0.00466
Sch.4 0.00262 0.00319 0.00312 0.00331 0.00331 0.00353 0.00338 0.00325 0.00346 0.00360 0.00396
20 Sch.1 0.00171 0.00189 0.00187 0.00194 0.00192 0.00200 0.00196 0.00191 0.00198 0.00199 0.00208
Sch.2 0.00160 0.00167 0.00166 0.00177 0.00169 0.00182 0.00179 0.00168 0.00181 0.00174 0.00188
Sch.3 0.00161 0.00177 0.00175 0.00170 0.00179 0.00174 0.00172 0.00178 0.00173 0.00186 0.00180
Sch.4 0.00157 0.00166 0.00165 0.00170 0.00168 0.00174 0.00171 0.00167 0.00173 0.00173 0.00180
Table 5

MSEr for DCRTE estimates based on MExp distribution at λ = 0.4 under different PCT-II schemes at n = 60

m Sch. MLE BE-MCMC: IP BE-MCMC: Non-IP
SE LIN-1 LIN-2 GE-1 GE-2 SE LIN-1 LIN-2 GE-1 GE-2
t = 0.5, γ 1 = 0.69135
20 Sch.1 0.00260 0.00291 0.00287 0.00307 0.00301 0.00324 0.00313 0.00296 0.00319 0.00327 0.00351
Sch.2 0.00249 0.00264 0.00262 0.00265 0.00269 0.00273 0.00268 0.00267 0.00270 0.00281 0.00287
Sch.3 0.00247 0.00271 0.00268 0.00269 0.00278 0.00278 0.00272 0.00275 0.00275 0.00296 0.00293
Sch.4 0.00238 0.00257 0.00254 0.00257 0.00263 0.00265 0.00260 0.00260 0.00263 0.00276 0.00279
40 Sch.1 0.00163 0.00168 0.00168 0.00166 0.00170 0.00168 0.00166 0.00169 0.00167 0.00173 0.00171
Sch.2 0.00133 0.00134 0.00134 0.00133 0.00135 0.00134 0.00133 0.00134 0.00134 0.00136 0.00136
Sch.3 0.00136 0.00135 0.00135 0.00138 0.00136 0.00140 0.00139 0.00136 0.00139 0.00138 0.00142
Sch.4 0.00141 0.00137 0.00137 0.00141 0.00138 0.00142 0.00141 0.00138 0.00142 0.00140 0.00145
t = 1.5, γ 2 = 0.75346
20 Sch.1 0.00173 0.00191 0.00189 0.00199 0.00195 0.00206 0.00202 0.00193 0.00204 0.00203 0.00215
Sch.2 0.00136 0.00148 0.00147 0.00152 0.00151 0.00156 0.00153 0.00150 0.00155 0.00156 0.00162
Sch.3 0.00129 0.00137 0.00136 0.00141 0.00140 0.00145 0.00142 0.00139 0.00144 0.00144 0.00150
Sch.4 0.00136 0.00148 0.00147 0.00147 0.00150 0.00150 0.00148 0.00150 0.00150 0.00155 0.00155
40 Sch.1 0.00089 0.00091 0.00090 0.00090 0.00091 0.00091 0.00091 0.00091 0.00091 0.00092 0.00092
Sch.2 0.00080 0.00081 0.00080 0.00080 0.00081 0.00081 0.00080 0.00081 0.00081 0.00082 0.00081
Sch.3 0.00074 0.00076 0.00075 0.00079 0.00076 0.00079 0.00079 0.00076 0.00079 0.00077 0.00081
Sch.4 0.00078 0.00078 0.00078 0.00079 0.00078 0.00080 0.00079 0.00078 0.00080 0.00079 0.00081
Table 6

MSEr for DCRTE estimates based on MExp distribution at λ = 0.8 under different PCT-II schemes at n = 30

m Sch. MLE BE-MCMC: IP BE-MCMC: Non-IP
SE LIN-1 LIN-2 GE-1 GE-2 SE LIN-1 LIN-2 GE-1 GE-2
t = 0.5, γ 3 = 0.27811
10 Sch.1 0.02942 0.03725 0.03627 0.04669 0.01988 0.01905 0.04843 0.03834 0.05032 0.02386 0.02461
Sch.2 0.02481 0.02967 0.02897 0.03516 0.01682 0.01682 0.03653 0.03048 0.03808 0.02079 0.02262
Sch.3 0.02613 0.03250 0.03167 0.03965 0.01723 0.01718 0.04108 0.03346 0.04265 0.02117 0.02269
Sch.4 0.02436 0.03032 0.02950 0.03558 0.01746 0.01659 0.03685 0.03127 0.03825 0.02184 0.02162
20 Sch.1 0.01589 0.01940 0.01911 0.02021 0.01331 0.01308 0.02065 0.01973 0.02114 0.01725 0.01796
Sch.2 0.01488 0.01709 0.01688 0.01828 0.01195 0.01179 0.01865 0.01734 0.01906 0.01553 0.01623
Sch.3 0.01503 0.01763 0.01739 0.01742 0.01285 0.01193 0.01774 0.01792 0.01809 0.01670 0.01641
Sch.4 0.01464 0.01638 0.01620 0.01756 0.01233 0.01198 0.01792 0.01659 0.01832 0.01630 0.01692
t = 1.5, γ 4 = 0.43667
10 Sch.1 0.01941 0.02356 0.02305 0.02801 0.01948 0.02036 0.02889 0.02411 0.02985 0.02683 0.02852
Sch.2 0.01540 0.01999 0.01956 0.02278 0.01687 0.01775 0.02350 0.02046 0.02429 0.02368 0.02626
Sch.3 0.01444 0.01844 0.01793 0.02171 0.01554 0.01709 0.02250 0.01900 0.02334 0.02239 0.02560
Sch.4 0.01522 0.01807 0.01766 0.02261 0.01517 0.01814 0.02326 0.01853 0.02396 0.02134 0.02630
20 Sch.1 0.01014 0.01156 0.01140 0.01223 0.01146 0.01191 0.01247 0.01175 0.01273 0.01535 0.01620
Sch.2 0.00890 0.01013 0.00998 0.01138 0.01007 0.01126 0.01163 0.01031 0.01190 0.01369 0.01570
Sch.3 0.00896 0.01053 0.01039 0.01097 0.01038 0.01081 0.01120 0.01068 0.01147 0.01342 0.01528
Sch.4 0.00861 0.00975 0.00962 0.01053 0.00945 0.01040 0.01075 0.00990 0.01099 0.01239 0.01430
Table 7

MSEr for DCRTE estimates based on MExp distribution at λ = 0.8 under different PCT-II schemes at n = 60

m Sch. MLE BE-MCMC: IP BE-MCMC: Non-IP
SE LIN-1 LIN-2 GE-1 GE-2 SE LIN-1 LIN-2 GE-1 GE-2
t = 0.5, γ 3 = 0.27811
20 Sch.1 0.01483 0.01623 0.01604 0.01756 0.01200 0.01185 0.01792 0.01647 0.01834 0.01595 0.01663
Sch.2 0.01267 0.01496 0.01473 0.01518 0.01080 0.01065 0.01549 0.01522 0.01584 0.01492 0.01550
Sch.3 0.01401 0.01657 0.01628 0.01784 0.01172 0.01170 0.01828 0.01690 0.01877 0.01605 0.01695
Sch.4 0.01183 0.01336 0.01313 0.01444 0.00978 0.00984 0.01480 0.01363 0.01521 0.01412 0.01515
40 Sch.1 0.00767 0.00793 0.00792 0.00833 0.00710 0.00728 0.00839 0.00795 0.00846 0.00944 0.01008
Sch.2 0.00771 0.00780 0.00779 0.00849 0.00707 0.00747 0.00855 0.00782 0.00861 0.00927 0.01029
Sch.3 0.00806 0.00862 0.00857 0.00882 0.00780 0.00787 0.00889 0.00867 0.00898 0.01060 0.01105
Sch.4 0.00719 0.00752 0.00749 0.00775 0.00691 0.00697 0.00779 0.00755 0.00785 0.00923 0.00968
t = 1.5, γ 4 = 0.43667
20 Sch.1 0.00909 0.01108 0.01090 0.01147 0.01074 0.01140 0.01172 0.01128 0.01201 0.01419 0.01638
Sch.2 0.00765 0.00862 0.00851 0.00951 0.00880 0.00952 0.00970 0.00874 0.00991 0.01153 0.01300
Sch.3 0.00835 0.00917 0.00904 0.01026 0.00905 0.01030 0.01046 0.00931 0.01068 0.01168 0.01397
Sch.4 0.00684 0.00746 0.00739 0.00890 0.00749 0.00898 0.00908 0.00755 0.00927 0.00945 0.01217
40 Sch.1 0.00468 0.00486 0.00485 0.00512 0.00497 0.00535 0.00516 0.00487 0.00520 0.00562 0.00628
Sch.2 0.00471 0.00478 0.00477 0.00523 0.00490 0.00541 0.00526 0.00479 0.00530 0.00545 0.00628
Sch.3 0.00493 0.00528 0.00526 0.00543 0.00543 0.00568 0.00547 0.00532 0.00552 0.00617 0.00656
Sch.4 0.00439 0.00461 0.00459 0.00477 0.00472 0.00498 0.00480 0.00463 0.00483 0.00527 0.00573
Table 8

MSEr for DCRTE estimates based on MExp distribution at λ = 1 under different PCT-II schemes at n = 30

m Sch. MLE BE-MCMC: IP BE-MCMC: Non-IP
SE LIN-1 LIN-2 GE-1 GE-2 SE LIN-1 LIN-2 GE-1 GE-2
t = 0.5, γ 5 = 0.05556
10 Sch.1 0.04913 0.06328 0.06216 0.07966 0.03555 0.03164 0.08217 0.06460 0.08494 0.02255 0.01960
Sch.2 0.04143 0.05078 0.05001 0.05995 0.02982 0.02630 0.06189 0.05171 0.06410 0.01862 0.01586
Sch.3 0.04361 0.05532 0.05434 0.06697 0.03016 0.02681 0.06910 0.05650 0.07147 0.01863 0.01630
Sch.4 0.04070 0.05226 0.05123 0.06142 0.03050 0.02649 0.06330 0.05350 0.06541 0.01975 0.01616
20 Sch.1 0.02655 0.03405 0.03363 0.03589 0.02076 0.01851 0.03664 0.03455 0.03750 0.01277 0.01073
Sch.2 0.02562 0.03206 0.03153 0.03572 0.01875 0.01645 0.03669 0.03269 0.03778 0.01134 0.00964
Sch.3 0.02368 0.02824 0.02774 0.03289 0.01627 0.01410 0.03386 0.02883 0.03493 0.00942 0.00765
Sch.4 0.02111 0.02530 0.02491 0.02821 0.01591 0.01460 0.02889 0.02578 0.02967 0.00914 0.00799
t = 1.5, γ 6 = 0.26000
10 Sch.1 0.03169 0.04137 0.04060 0.05345 0.02259 0.02097 0.05506 0.04223 0.05678 0.02526 0.02507
Sch.2 0.02667 0.03313 0.03258 0.03986 0.01885 0.01846 0.04109 0.03375 0.04244 0.02162 0.02285
Sch.3 0.02819 0.03628 0.03561 0.04489 0.01938 0.01877 0.04625 0.03704 0.04773 0.02222 0.02295
Sch.4 0.02615 0.03409 0.03339 0.04102 0.01968 0.01793 0.04222 0.03489 0.04354 0.02283 0.02199
20 Sch.1 0.01704 0.02208 0.02178 0.02353 0.01459 0.01389 0.02401 0.02241 0.02454 0.01774 0.01812
Sch.2 0.01647 0.02085 0.02051 0.02358 0.01353 0.01303 0.02418 0.02125 0.02483 0.01727 0.01776
Sch.3 0.01530 0.01838 0.01805 0.02178 0.01211 0.01181 0.02237 0.01874 0.02301 0.01616 0.01695
Sch.4 0.01357 0.01640 0.01614 0.01847 0.01125 0.01170 0.01890 0.01670 0.01937 0.01474 0.01667
Table 9

MSEr for DCRTE estimates based on MExp distribution at λ = 1 under different PCT-II schemes at n = 60

m Sch. MLE BE-MCMC: IP BE-MCMC: Non-IP
SE LIN-1 LIN-2 GE-1 GE-2 SE LIN-1 LIN-2 GE-1 GE-2
t = 0.5, γ 5 = 0.05556
20 Sch.1 0.02692 0.03384 0.03326 0.03810 0.01965 0.01730 0.03916 0.03452 0.04035 0.01194 0.01018
Sch.2 0.02112 0.02494 0.02453 0.02827 0.01453 0.01252 0.02906 0.02543 0.02994 0.00834 0.00671
Sch.3 0.02000 0.02383 0.02347 0.02635 0.01501 0.01380 0.02697 0.02427 0.02768 0.00861 0.00761
Sch.4 0.02109 0.02608 0.02569 0.02613 0.01530 0.01385 0.02669 0.02655 0.02733 0.00908 0.00778
40 Sch.1 0.01380 0.01543 0.01542 0.01546 0.01061 0.00935 0.01561 0.01547 0.01579 0.00632 0.00525
Sch.2 0.01258 0.01422 0.01419 0.01373 0.00937 0.00833 0.01382 0.01428 0.01395 0.00520 0.00452
Sch.3 0.01405 0.01519 0.01516 0.01551 0.01016 0.00948 0.01562 0.01525 0.01577 0.00603 0.00547
Sch.4 0.01218 0.01348 0.01346 0.01368 0.00906 0.00833 0.01377 0.01352 0.01388 0.00525 0.00465
t = 1.5, γ 6 = 0.26000
20 Sch.1 0.01731 0.02202 0.02164 0.02518 0.01404 0.01351 0.02584 0.02245 0.02655 0.01771 0.01825
Sch.2 0.01364 0.01621 0.01594 0.01866 0.01113 0.01095 0.01914 0.01651 0.01966 0.01513 0.01605
Sch.3 0.01285 0.01545 0.01521 0.01724 0.01077 0.01126 0.01762 0.01572 0.01805 0.01426 0.01622
Sch.4 0.01355 0.01694 0.01669 0.01709 0.01154 0.01110 0.01744 0.01724 0.01783 0.01497 0.01567
40 Sch.1 0.00885 0.00991 0.00989 0.01000 0.00857 0.00819 0.01010 0.00995 0.01022 0.01109 0.01144
Sch.2 0.00808 0.00917 0.00914 0.00887 0.00753 0.00737 0.00894 0.00922 0.00902 0.00973 0.01029
Sch.3 0.00902 0.00977 0.00974 0.01003 0.00833 0.00829 0.01010 0.00982 0.01020 0.01074 0.01120
Sch.4 0.00781 0.00867 0.00865 0.00883 0.00755 0.00758 0.00889 0.00870 0.00897 0.00986 0.01039
Figure 5 The MSEr for different estimates of DCRTE when λ = 0.8 \lambda =0.8 under Sch.1. (a) MSEr for n = 30 n=30 and t = 0.5 t=0.5 , (b) MSEr for n = 30 n=30 and t = 1.5 t=1.5 , (c). MSEr for n = 60 n=60 and t = 0.5 t=0.5 , (d) MSEr for n = 60 n=60 and t = 1.5 t=1.5 .
Figure 5

The MSEr for different estimates of DCRTE when λ = 0.8 under Sch.1. (a) MSEr for n = 30 and t = 0.5 , (b) MSEr for n = 30 and t = 1.5 , (c). MSEr for n = 60 and t = 0.5 , (d) MSEr for n = 60 and t = 1.5 .

Figure 5 represents the MSEr for DCRTE at different sample sizes ( n ) and different number of stages ( m ) when λ = 0.8 under scheme Sch.1 as a PCT-II.

6 Concluding remarks

The PCT-II method is used in this study to provide Bayesian and non-Bayesian estimations of the DCRTE measure for a MExp distribution. In the case of IP and Non-IP functions for three LoFs, the BEs of the DCRTE measure are produced. The MH algorithm-based MCMC approach is used to compute the BEs. The accuracy of the DCRTE estimations for the MExp distribution is examined, and applications to the leukemia data and simulation issues are given. We deduce from the simulation results that, as sample sizes grow, the Bayesian estimate of DCRTE gets closer to the genuine value. DCRTE Bayesian estimates under GE LoF followed by LIN LoF are generally preferred to the other competing estimates. The DCRTE Bayesian estimates using IP have lower values than those using Non-IP in terms of their accuracy. Sch.3 is frequently the one that yields the lowest MSEr findings, followed by Sch.2, when compared to other estimates, in most cases. In future research, one may use the E-Bayesian technique to estimate additional metrics of uncertainty, such as DCR Shannon entropy. Bayesian and non-Bayesian estimation of the DCRTE in the presence of outliers can be considered [35]. In addition, other methods such as Tierney–Kadane approximations can be used along with the MCMC approach.

Abbreviations

BE

Bayesian estimator

CDF

cumulative distribution function

CR

cumulative residual

CT-I

censoring type I

CT-II

censoring type II

DCR

dynamic cumulative residual

DCRTE

DCR Tsallis entropy

ECDF

empirical cumulative distribution function

EPDF

empirical probability density function

ESF

empirical survival function

IP

informative prior

GE

general entropy

K–S

Kolmogorov–Smirnov

LIN

linear exponential

LoFs

loss functions

MCMC

Markov chain Monte Carlo

MLE

maximum likelihood estimator

MH

Metropolis–Hasting

MSEr

mean squared error

MExp

moment exponential

Non-IP

non-informative prior

PC

progressive censoring

PCT-I

PC type I

PCT-II

PC type II

PDF

probability density function

PP

probability–probability

Sch.

scheme

SE

squared error

SF

survival function

T

tsallis

V-CM

variance–covariance matrix

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University (IMSIU), Saudi Arabia, for funding this research work through Grant No. (221412056).

  1. Funding information: The authors extend their appreciation to the Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University (IMSIU), Saudi Arabia, for funding this research work through Grant No. (221412056).

  2. Conflict of interest: The authors state no conflict of interest.

  3. Data availability statement: All data generated or analysed during this study are included in this published article.

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Received: 2022-11-29
Revised: 2023-06-06
Accepted: 2023-06-09
Published Online: 2023-07-18

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Dynamic properties of the attachment oscillator arising in the nanophysics
  3. Parametric simulation of stagnation point flow of motile microorganism hybrid nanofluid across a circular cylinder with sinusoidal radius
  4. Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach
  5. Behaviour and onset of low-dimensional chaos with a periodically varying loss in single-mode homogeneously broadened laser
  6. Ammonia gas-sensing behavior of uniform nanostructured PPy film prepared by simple-straightforward in situ chemical vapor oxidation
  7. Analysis of the working mechanism and detection sensitivity of a flash detector
  8. Flat and bent branes with inner structure in two-field mimetic gravity
  9. Heat transfer analysis of the MHD stagnation-point flow of third-grade fluid over a porous sheet with thermal radiation effect: An algorithmic approach
  10. Weighted survival functional entropy and its properties
  11. Bioconvection effect in the Carreau nanofluid with Cattaneo–Christov heat flux using stagnation point flow in the entropy generation: Micromachines level study
  12. Study on the impulse mechanism of optical films formed by laser plasma shock waves
  13. Analysis of sweeping jet and film composite cooling using the decoupled model
  14. Research on the influence of trapezoidal magnetization of bonded magnetic ring on cogging torque
  15. Tripartite entanglement and entanglement transfer in a hybrid cavity magnomechanical system
  16. Compounded Bell-G class of statistical models with applications to COVID-19 and actuarial data
  17. Degradation of Vibrio cholerae from drinking water by the underwater capillary discharge
  18. Multiple Lie symmetry solutions for effects of viscous on magnetohydrodynamic flow and heat transfer in non-Newtonian thin film
  19. Thermal characterization of heat source (sink) on hybridized (Cu–Ag/EG) nanofluid flow via solid stretchable sheet
  20. Optimizing condition monitoring of ball bearings: An integrated approach using decision tree and extreme learning machine for effective decision-making
  21. Study on the inter-porosity transfer rate and producing degree of matrix in fractured-porous gas reservoirs
  22. Interstellar radiation as a Maxwell field: Improved numerical scheme and application to the spectral energy density
  23. Numerical study of hybridized Williamson nanofluid flow with TC4 and Nichrome over an extending surface
  24. Controlling the physical field using the shape function technique
  25. Significance of heat and mass transport in peristaltic flow of Jeffrey material subject to chemical reaction and radiation phenomenon through a tapered channel
  26. Complex dynamics of a sub-quadratic Lorenz-like system
  27. Stability control in a helicoidal spin–orbit-coupled open Bose–Bose mixture
  28. Research on WPD and DBSCAN-L-ISOMAP for circuit fault feature extraction
  29. Simulation for formation process of atomic orbitals by the finite difference time domain method based on the eight-element Dirac equation
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  34. Chaotic characteristics and mixing performance of pseudoplastic fluids in a stirred tank
  35. Isomorphic shut form valuation for quantum field theory and biological population models
  36. Vibration sensitivity minimization of an ultra-stable optical reference cavity based on orthogonal experimental design
  37. Effect of dysprosium on the radiation-shielding features of SiO2–PbO–B2O3 glasses
  38. Asymptotic formulations of anti-plane problems in pre-stressed compressible elastic laminates
  39. A study on soliton, lump solutions to a generalized (3+1)-dimensional Hirota--Satsuma--Ito equation
  40. Tangential electrostatic field at metal surfaces
  41. Bioconvective gyrotactic microorganisms in third-grade nanofluid flow over a Riga surface with stratification: An approach to entropy minimization
  42. Infrared spectroscopy for ageing assessment of insulating oils via dielectric loss factor and interfacial tension
  43. Influence of cationic surfactants on the growth of gypsum crystals
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  47. Inspection of Couette and pressure-driven Poiseuille entropy-optimized dissipated flow in a suction/injection horizontal channel: Analytical solutions
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  50. Optimal interatomic potentials using modified method of least squares: Optimal form of interatomic potentials
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  116. Special Issue on Predicting pattern alterations in nature - Part I
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  118. Spectral analysis of variable-order multi-terms fractional differential equations
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https://doi.org/10.1515/phys-2022-0264
Keywords for this article
Tsallis entropy; moment exponential distribution; squared error loss function; Markov chain Monte Carlo; maximum likelihood; general entropy loss function
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Dynamic properties of the attachment oscillator arising in the nanophysics
  3. Parametric simulation of stagnation point flow of motile microorganism hybrid nanofluid across a circular cylinder with sinusoidal radius
  4. Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach
  5. Behaviour and onset of low-dimensional chaos with a periodically varying loss in single-mode homogeneously broadened laser
  6. Ammonia gas-sensing behavior of uniform nanostructured PPy film prepared by simple-straightforward in situ chemical vapor oxidation
  7. Analysis of the working mechanism and detection sensitivity of a flash detector
  8. Flat and bent branes with inner structure in two-field mimetic gravity
  9. Heat transfer analysis of the MHD stagnation-point flow of third-grade fluid over a porous sheet with thermal radiation effect: An algorithmic approach
  10. Weighted survival functional entropy and its properties
  11. Bioconvection effect in the Carreau nanofluid with Cattaneo–Christov heat flux using stagnation point flow in the entropy generation: Micromachines level study
  12. Study on the impulse mechanism of optical films formed by laser plasma shock waves
  13. Analysis of sweeping jet and film composite cooling using the decoupled model
  14. Research on the influence of trapezoidal magnetization of bonded magnetic ring on cogging torque
  15. Tripartite entanglement and entanglement transfer in a hybrid cavity magnomechanical system
  16. Compounded Bell-G class of statistical models with applications to COVID-19 and actuarial data
  17. Degradation of Vibrio cholerae from drinking water by the underwater capillary discharge
  18. Multiple Lie symmetry solutions for effects of viscous on magnetohydrodynamic flow and heat transfer in non-Newtonian thin film
  19. Thermal characterization of heat source (sink) on hybridized (Cu–Ag/EG) nanofluid flow via solid stretchable sheet
  20. Optimizing condition monitoring of ball bearings: An integrated approach using decision tree and extreme learning machine for effective decision-making
  21. Study on the inter-porosity transfer rate and producing degree of matrix in fractured-porous gas reservoirs
  22. Interstellar radiation as a Maxwell field: Improved numerical scheme and application to the spectral energy density
  23. Numerical study of hybridized Williamson nanofluid flow with TC4 and Nichrome over an extending surface
  24. Controlling the physical field using the shape function technique
  25. Significance of heat and mass transport in peristaltic flow of Jeffrey material subject to chemical reaction and radiation phenomenon through a tapered channel
  26. Complex dynamics of a sub-quadratic Lorenz-like system
  27. Stability control in a helicoidal spin–orbit-coupled open Bose–Bose mixture
  28. Research on WPD and DBSCAN-L-ISOMAP for circuit fault feature extraction
  29. Simulation for formation process of atomic orbitals by the finite difference time domain method based on the eight-element Dirac equation
  30. A modified power-law model: Properties, estimation, and applications
  31. Bayesian and non-Bayesian estimation of dynamic cumulative residual Tsallis entropy for moment exponential distribution under progressive censored type II
  32. Computational analysis and biomechanical study of Oldroyd-B fluid with homogeneous and heterogeneous reactions through a vertical non-uniform channel
  33. Predictability of machine learning framework in cross-section data
  34. Chaotic characteristics and mixing performance of pseudoplastic fluids in a stirred tank
  35. Isomorphic shut form valuation for quantum field theory and biological population models
  36. Vibration sensitivity minimization of an ultra-stable optical reference cavity based on orthogonal experimental design
  37. Effect of dysprosium on the radiation-shielding features of SiO2–PbO–B2O3 glasses
  38. Asymptotic formulations of anti-plane problems in pre-stressed compressible elastic laminates
  39. A study on soliton, lump solutions to a generalized (3+1)-dimensional Hirota--Satsuma--Ito equation
  40. Tangential electrostatic field at metal surfaces
  41. Bioconvective gyrotactic microorganisms in third-grade nanofluid flow over a Riga surface with stratification: An approach to entropy minimization
  42. Infrared spectroscopy for ageing assessment of insulating oils via dielectric loss factor and interfacial tension
  43. Influence of cationic surfactants on the growth of gypsum crystals
  44. Study on instability mechanism of KCl/PHPA drilling waste fluid
  45. Analytical solutions of the extended Kadomtsev–Petviashvili equation in nonlinear media
  46. A novel compact highly sensitive non-invasive microwave antenna sensor for blood glucose monitoring
  47. Inspection of Couette and pressure-driven Poiseuille entropy-optimized dissipated flow in a suction/injection horizontal channel: Analytical solutions
  48. Conserved vectors and solutions of the two-dimensional potential KP equation
  49. The reciprocal linear effect, a new optical effect of the Sagnac type
  50. Optimal interatomic potentials using modified method of least squares: Optimal form of interatomic potentials
  51. The soliton solutions for stochastic Calogero–Bogoyavlenskii Schiff equation in plasma physics/fluid mechanics
  52. Research on absolute ranging technology of resampling phase comparison method based on FMCW
  53. Analysis of Cu and Zn contents in aluminum alloys by femtosecond laser-ablation spark-induced breakdown spectroscopy
  54. Nonsequential double ionization channels control of CO2 molecules with counter-rotating two-color circularly polarized laser field by laser wavelength
  55. Fractional-order modeling: Analysis of foam drainage and Fisher's equations
  56. Thermo-solutal Marangoni convective Darcy-Forchheimer bio-hybrid nanofluid flow over a permeable disk with activation energy: Analysis of interfacial nanolayer thickness
  57. Investigation on topology-optimized compressor piston by metal additive manufacturing technique: Analytical and numeric computational modeling using finite element analysis in ANSYS
  58. Breast cancer segmentation using a hybrid AttendSeg architecture combined with a gravitational clustering optimization algorithm using mathematical modelling
  59. On the localized and periodic solutions to the time-fractional Klein-Gordan equations: Optimal additive function method and new iterative method
  60. 3D thin-film nanofluid flow with heat transfer on an inclined disc by using HWCM
  61. Numerical study of static pressure on the sonochemistry characteristics of the gas bubble under acoustic excitation
  62. Optimal auxiliary function method for analyzing nonlinear system of coupled Schrödinger–KdV equation with Caputo operator
  63. Analysis of magnetized micropolar fluid subjected to generalized heat-mass transfer theories
  64. Does the Mott problem extend to Geiger counters?
  65. Stability analysis, phase plane analysis, and isolated soliton solution to the LGH equation in mathematical physics
  66. Effects of Joule heating and reaction mechanisms on couple stress fluid flow with peristalsis in the presence of a porous material through an inclined channel
  67. Bayesian and E-Bayesian estimation based on constant-stress partially accelerated life testing for inverted Topp–Leone distribution
  68. Dynamical and physical characteristics of soliton solutions to the (2+1)-dimensional Konopelchenko–Dubrovsky system
  69. Study of fractional variable order COVID-19 environmental transformation model
  70. Sisko nanofluid flow through exponential stretching sheet with swimming of motile gyrotactic microorganisms: An application to nanoengineering
  71. Influence of the regularization scheme in the QCD phase diagram in the PNJL model
  72. Fixed-point theory and numerical analysis of an epidemic model with fractional calculus: Exploring dynamical behavior
  73. Computational analysis of reconstructing current and sag of three-phase overhead line based on the TMR sensor array
  74. Investigation of tripled sine-Gordon equation: Localized modes in multi-stacked long Josephson junctions
  75. High-sensitivity on-chip temperature sensor based on cascaded microring resonators
  76. Pathological study on uncertain numbers and proposed solutions for discrete fuzzy fractional order calculus
  77. Bifurcation, chaotic behavior, and traveling wave solution of stochastic coupled Konno–Oono equation with multiplicative noise in the Stratonovich sense
  78. Thermal radiation and heat generation on three-dimensional Casson fluid motion via porous stretching surface with variable thermal conductivity
  79. Numerical simulation and analysis of Airy's-type equation
  80. A homotopy perturbation method with Elzaki transformation for solving the fractional Biswas–Milovic model
  81. Heat transfer performance of magnetohydrodynamic multiphase nanofluid flow of Cu–Al2O3/H2O over a stretching cylinder
  82. ΛCDM and the principle of equivalence
  83. Axisymmetric stagnation-point flow of non-Newtonian nanomaterial and heat transport over a lubricated surface: Hybrid homotopy analysis method simulations
  84. HAM simulation for bioconvective magnetohydrodynamic flow of Walters-B fluid containing nanoparticles and microorganisms past a stretching sheet with velocity slip and convective conditions
  85. Coupled heat and mass transfer mathematical study for lubricated non-Newtonian nanomaterial conveying oblique stagnation point flow: A comparison of viscous and viscoelastic nanofluid model
  86. Power Topp–Leone exponential negative family of distributions with numerical illustrations to engineering and biological data
  87. Extracting solitary solutions of the nonlinear Kaup–Kupershmidt (KK) equation by analytical method
  88. A case study on the environmental and economic impact of photovoltaic systems in wastewater treatment plants
  89. Application of IoT network for marine wildlife surveillance
  90. Non-similar modeling and numerical simulations of microploar hybrid nanofluid adjacent to isothermal sphere
  91. Joint optimization of two-dimensional warranty period and maintenance strategy considering availability and cost constraints
  92. Numerical investigation of the flow characteristics involving dissipation and slip effects in a convectively nanofluid within a porous medium
  93. Spectral uncertainty analysis of grassland and its camouflage materials based on land-based hyperspectral images
  94. Application of low-altitude wind shear recognition algorithm and laser wind radar in aviation meteorological services
  95. Investigation of different structures of screw extruders on the flow in direct ink writing SiC slurry based on LBM
  96. Harmonic current suppression method of virtual DC motor based on fuzzy sliding mode
  97. Micropolar flow and heat transfer within a permeable channel using the successive linearization method
  98. Different lump k-soliton solutions to (2+1)-dimensional KdV system using Hirota binary Bell polynomials
  99. Investigation of nanomaterials in flow of non-Newtonian liquid toward a stretchable surface
  100. Weak beat frequency extraction method for photon Doppler signal with low signal-to-noise ratio
  101. Electrokinetic energy conversion of nanofluids in porous microtubes with Green’s function
  102. Examining the role of activation energy and convective boundary conditions in nanofluid behavior of Couette-Poiseuille flow
  103. Review Article
  104. Effects of stretching on phase transformation of PVDF and its copolymers: A review
  105. Special Issue on Transport phenomena and thermal analysis in micro/nano-scale structure surfaces - Part IV
  106. Prediction and monitoring model for farmland environmental system using soil sensor and neural network algorithm
  107. Special Issue on Advanced Topics on the Modelling and Assessment of Complicated Physical Phenomena - Part III
  108. Some standard and nonstandard finite difference schemes for a reaction–diffusion–chemotaxis model
  109. Special Issue on Advanced Energy Materials - Part II
  110. Rapid productivity prediction method for frac hits affected wells based on gas reservoir numerical simulation and probability method
  111. Special Issue on Novel Numerical and Analytical Techniques for Fractional Nonlinear Schrodinger Type - Part III
  112. Adomian decomposition method for solution of fourteenth order boundary value problems
  113. New soliton solutions of modified (3+1)-D Wazwaz–Benjamin–Bona–Mahony and (2+1)-D cubic Klein–Gordon equations using first integral method
  114. On traveling wave solutions to Manakov model with variable coefficients
  115. Rational approximation for solving Fredholm integro-differential equations by new algorithm
  116. Special Issue on Predicting pattern alterations in nature - Part I
  117. Modeling the monkeypox infection using the Mittag–Leffler kernel
  118. Spectral analysis of variable-order multi-terms fractional differential equations
  119. Special Issue on Nanomaterial utilization and structural optimization - Part I
  120. Heat treatment and tensile test of 3D-printed parts manufactured at different build orientations
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