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# A New Three-Parameter Inverse Weibull Distribution with Medical and Engineering Applications

1 Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh, 11671, Saudi Arabia

2 Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

3 Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, 11884, Cairo, Egypt

4 Department of Statistics, Faculty of Commerce, Al-Azhar University, Nasr City, 11884, Cairo, Egypt

5 Department of Statistics, Faculty of Commerce, Zagazig University, Zagazig, 44519, Egypt

* Corresponding Authors: Mazen Nassar. Email: ,

(This article belongs to the Special Issue: New Trends in Statistical Computing and Data Science)

*Computer Modeling in Engineering & Sciences* **2023**, *135*(2), 1255-1274. https://doi.org/10.32604/cmes.2022.022623

**Received** 17 March 2022; **Accepted** 13 June 2022; **Issue published** 27 October 2022

## Abstract

The objective of this article is to provide a novel extension of the conventional inverse Weibull distribution that adds an extra shape parameter to increase its flexibility. This addition is beneficial in a variety of fields, including reliability, economics, engineering, biomedical science, biological research, environmental studies, and finance. For modeling real data, several expanded classes of distributions have been established. The modified alpha power transformed approach is used to implement the new model. The data matches the new inverse Weibull distribution better than the inverse Weibull distribution and several other competing models. It appears to be a distribution designed to support decreasing or unimodal shaped distributions based on its parameters. Precise expressions for quantiles, moments, incomplete moments, moment generating function, characteristic generating function, and entropy expression are among the determined attributes of the new distribution. The point and interval estimates are studied using the maximum likelihood method. Simulation research is conducted to illustrate the correctness of the theoretical results. Three applications to medical and engineering data are utilized to illustrate the model’s flexibility.## Keywords

Recently, many statistical distributions have been proposed by statisticians. The necessity to develop new distributions appear either due to practical investigations or theoretical concerns or both. Many applications in domains including dependability analysis, finance and risk modelling, insurance, and biological sciences, among others, have indicated in recent years that data sets that follow standard distributions are more often the exception than the usual. Because modified distributions are necessary, significant progress has been made in the modification of several classic distributions and their efficient utilization to challenges in these fields. Lately, specifically since 1980, the direction of statisticians to develop new distributions is on adding parameter(s) to some existing distributions or integrating classical distributions, see Lee et al. [1]. This addition of parameter(s) has been demonstrated valuable in analysing tail properties and also for enhancing the goodness of fit of the family under consideration. For more details about the methods of adding parameters, one can refer to Mudholkar et al. [2], Marshall et al. [3] and Mahdavi et al. [4].

The inverse-Weibull (IW) distribution has many vital applications in life testing and reliability studies. The IW distribution bears some other names like Fréchet distribution. The IW distribution is viewed as reciprocal to the usual Weibull distribution, see Drapella [5] and Mudholkar et al. [6], it is used to describe the degradation of mechanical components in diesel engines, such as the crankshaft and pistons, see Keller et al. [7]. Several academics proposed several modified variations of the IW distribution to increase its flexibility. Khan [8] looked at the beta IW distribution, for example. The generalized IW distribution was proposed by de Gusmao et al. [9], the modified IW distribution was proposed by Khan et al. [10], the Kumaraswamy generalized IW distribution was proposed by Oluyede et al. [11], and the Kumaraswamy modified IW distribution was proposed by Aryal et al. [12]. In addition, Okasha et al. [13] suggested the Marshall-Olkin IW distribution, Basheer [14] proposed the alpha power IW distribution, Dey et al. [15] discussed the logarithmic transformed IW distribution, and Afify et al. [16] suggested the logarithmic transformed IW distribution.

Assume that X is a random variable follows IW distribution with

The cumulative distribution function (CDF) of Eq. (1) is

See Johnson et al. [17] for further information on the IW distribution. In this study, we offer a new three-parameter IW distribution utilizing the same idea proposed by Alotaibi et al. [18]. The modified alpha power transformed IW (MAPTIW) distribution is the name given to the new distribution. Two shape parameters and one scale parameter are included. It includes some lifetime distributions, such as IW, inverted exponential and inverse Rayleigh distributions. Recently, The alpha power transformation approach was used by Alotaibi et al. [18] to suggest a new method for adding a new shape parameter to a baseline distribution. Modified alpha power transformation (MAPT) was the name given to the approach they proposed. The CDF and PDF of MAPT method are expressed as

and

where

The remainder of this article is structured as follows. The MAPTIW distribution is introduced and its mixed representation is derived in Section 2. In Section 3, we will look at some of the MAPTIW’s most important characteristics. Section 4 shows how to estimate unknown parameters using maximum likelihood estimation. We present the simulation results in Section 5 to show how well the estimations performed. Three real-life data sets are used to assess the importance of the MAPTIW distribution in Section 6. Section 7 presents the conclusion.

We may define the CDF of the MAPTIW distribution in the following form by inserting the CDF of the IW distribution obtained by Eq. (2) into the CDF of the MAPT generator given by Eq. (3)

The PDF that corresponds to the CDF in Eq. (5) can be expressed as follows:

MAPTIW distribution has the following special sub-models. If

and

respectively. Henceforth, we will use X to refer to the random variable that possesses the PDF in Eq. (6).

The various plots of the PDF of the MAPTIW distribution with

The PDF in Eq. (6) might be used to obtain various statistical characteristics of the MAPTIW distribution. Finding the PDF’s mixture structure in Eq. (5) is a simple method. Observe the power series below:

where y is a variable and

where y is a variable,

When Eqs. (9) and (10) are applied to the PDF in Eq. (6), the following is a suitable linear representation for the PDF

where

On the other hand, the expansion of the CDF in Eq. (5) may be provided as by integrating Eq. (11) as

Hence

3 MAPTIW Distribution’s Properties

The quantile function, moments and stress-strength parameter are some of the properties of the MAPTIW distribution that we derive in this part.

The quantile function for the MAPTIW distribution may be calculated as follows using Eq. (5):

One can employ Eq. (13) to simulate the random variable X by taking

Moments play an essential part in statistics. Numerous vital aspects of any statistical distribution can be explored via moments. For the MAPTIW distribution, the

where

and

The following formulae can also be used to obtain the variance (var), skewness (Sk), and kurtosis (Ku):

and

where

Prop 3.1. If

where

Prop 3.2. If

and

Prop 3.3. If

Prop 3.4. If

where

3.3 Entropies of MAPTIW Distribution

Entropy is used to measure uncertainty. It recreates a vital role in the area of engineering, probability, statistics, information theory and financial analysis. For example, Gençay et al. [19] delivered a relative analysis of the stock market during the 1987 and 2008 financial crises. Rashidi et al. [20] examined and simulated the heat transfer flow employing entropy generation in solar still. The Rényi entropy (RE) for the MAPTIW distribution may be calculated using Eq. (6) as follows:

The Shannon entropy can also be calculated as

Let

The beta function is represented as

where

Using Eq. (17), we can derive the properties of

3.5 Moments of Residual Life and Reversed Failure Rate Function

The mean residual lifetime has been studied by engineers and survival analysts. Given a feature or a system is of age

Using Eqs. (5) and (6), and the binomial expansion of

The mean residual lifetime can be obtained from the last equation by setting

Based on Eqs. (5), (6) and using the binomial expansion of

Let

Case one:

Using the series expansion in Eq. (10), it follows:

Using the series expansion in Eq. (9), we can obtain

where

and

Case two:

Using the series expansion in Eq. (10), we have

Using the series expansion in Eq. (9), we can obtain

In this section, we consider the maximum likelihood approach to estimate the unknown parameters of the MAPTIW distribution. Furthermore, the approximate confidence intervals (ACIs) of the unknown parameters are acquired.

4.1 Maximum Likelihood Estimation

Given a random sample of size n taken from the MAPTIW distribution with PDF given by Eq. (6), we can express the log-likelihood function as

The maximum likelihood estimates (MLEs) denoted by

and

where

and

To get the MLEs of

where

and

where

and

Therefore, the

where

In this part, we use a simulation with variable sample sizes n and various values of the parameters

1. Establish the sample size and parameter initial values.

2. Create a random sample of size n from the MAPTIW distribution using Eq. (13).

3. Compute the MLEs of

4. Compute the MSEs of

5. Get the CLs of

6. Redo Steps 2–5, 1000 times.

7. Determine the average values of estimates (AEs), MSEs, and CLs as follows:

In this part, we illustrate the MAPTIW distribution’s flexibility by using three real data sets. The MAPTIW distribution is compared to other models such as, alpha power inverse Lomax (APILo) distribution by ZeinEldin et al. [21], inverse Lomax (ILo) distribution, alpha power inverse Weibull (APIW) distribution by Basheer [14], alpha power inverse Lindley (APILi) distribution by Dey et al. [22] and Weibull (We) distribution. Table 4 displays the PDFs of these distributions.

The first data (Data 1) describes the mortality rates due to the COVID-19 pandemic in the United Kingdom for 76 days, from 15 April to 30 June 2020. The data is originally investigated by Mubarak et al. [23]. The entire data set is as follows: 0.0587, 0.0863, 0.1165, 0.1247, 0.1277, 0.1303, 0.1652, 0.2079, 0.2395, 0.2751, 0.2845, 0.2992, 0.3188, 0.3317, 0.3446, 0.3553, 0.3622, 0.3926, 0.3926, 0.4110, 0.4633, 0.4690, 0.4954, 0.5139, 0.5696, 0.5837, 0.6197, 0.6365, 0.7096, 0.7193, 0.7444, 0.8590, 1.0438, 1.0602, 1.1305, 1.1468, 1.1533, 1.2260, 1.2707, 1.3423, 1.4149, 1.5709, 1.6017, 1.6083, 1.6324, 1.6998, 1.8164, 1.8392, 1.8721, 1.9844, 2.1360, 2.3987, 2.4153, 2.5225, 2.7087, 2.7946, 3.3609, 3.3715, 3.7840, 3.9042, 4.1969, 4.3451, 4.4627, 4.6477, 5.3664, 5.4500, 5.7522, 6.4241, 7.0657, 7.4456, 8.2307, 9.6315, 10.187, 11.1429, 11.2019, 11.4584.

The second data set (Data 2) describes the remission times (in months) of a random sample of 128 bladder cancer patients studied by Lee et al. [24] and recently investigated by Al-Zahrani et al. [25–27]. The complete data set of is: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.

The third data set (Data 3) consists of the times of breakdown of a sample of 25 devices at

Before studying these data sets, we foremost scheme the corresponding TTT plots. Fig. 3 displays the TTT plots of these data. It indicates that Data 1 has a decreasing failure rate function, while the TTT plots for Data 2 and 3 reveal an upside-down bathtub failure rate function. Therefore, we can infer that the MAPTIW distribution is reasonable to model these data sets. The MLEs and their standard errors (SEs) of the MAPTIW distribution and some other competitive distributions are displayed in Tables 5, 6 and 7 for Data 1, 2 and 3, respectively. In addition, some goodness of fit statistics, namely, Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan-Quinn information criterion (HQIC) statistics, Anderson-Darling (A), Cramér-Von Mises

As a new extension of the inverse Weibull model, we introduced a new three-parameter inverse Weibull distribution. Additionally, numerous theoretical properties of the distribution were explored in order to develop a more flexible model that includes the decreasing and unimodal shape for the hazard rate function. We described the method of maximum likelihood for estimating the parameters of the suggested distribution. A simulation study is also used to investigate the asymptotic behaviour of the maximum likelihood estimators. The model’s efficiency is demonstrated using three real data sets to show its applicability in real life. The proposed distribution is a better distributional model for fitting such data sets than many of its related models, as well as several newly produced distributions, using various information measures.

Acknowledgement: The authors convey their sincere appreciation to the reviewers and the editors for making some valuable suggestions and comments. The authors extend their appreciations to Princess Nourah bint Abdulrahman University Researchers Supporting Project No. (PNURSP2022R50), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Funding Statement: This research was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project No. (PNURSP2022R50), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

**APA Style**

*Computer Modeling in Engineering & Sciences*,

*135*

*(2)*, 1255-1274. https://doi.org/10.32604/cmes.2022.022623

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**IEEE Style**

*Comput. Model. Eng. Sci.*, vol. 135, no. 2, pp. 1255-1274. 2023. https://doi.org/10.32604/cmes.2022.022623