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.

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. [

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 [

Assume that

The cumulative distribution function (CDF) of

See Johnson et al. [

The remainder of this article is structured as follows. The MAPTIW distribution is introduced and its mixed representation is derived in

We may define the CDF of the MAPTIW distribution in the following form by inserting the CDF of the IW distribution obtained by

The PDF that corresponds to the CDF in

MAPTIW distribution has the following special sub-models. If

The various plots of the PDF of the MAPTIW distribution with

The PDF in

When

On the other hand, the expansion of the CDF in

Hence

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

One can employ

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

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

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. [

The Shannon entropy can also be calculated as

Let

The beta function is represented as

Using

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

Using

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

Based on

Let

Using the series expansion in

Using the series expansion in

Using the series expansion in

Using the series expansion in

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.

Given a random sample of size

The maximum likelihood estimates (MLEs) denoted by

To get the MLEs of

Therefore, the

In this part, we use a simulation with variable sample sizes

Establish the sample size and parameter initial values.

Create a random sample of size

Compute the MLEs of

Compute the MSEs of

Get the CLs of

Redo Steps 2–5, 1000 times.

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

Parameter | ||||||||
---|---|---|---|---|---|---|---|---|

AE(MSE) | CL | AE(MSE) | CL | AE(MSE) | CL | |||

0.5 | 1.5 | 1.5 | 0.55408(0.65817) | 1.59758 | 1.52803(0.29439) | 0.51286 | 1.53482(0.21841) | 0.46001 |

3 | 3 | 0.53070(0.48201) | 1.25144 | 3.04586(0.21344) | 0.31218 | 3.63060(0.19166) | 0.27550 | |

5 | 5 | 0.53997(0.52713) | 1.29063 | 5.04692(0.22020) | 0.41108 | 5.76559(0.17896) | 0.29195 | |

1.5 | 1.5 | 1.5 | 1.57570(0.58324) | 1.36819 | 1.56092(0.23711) | 0.38703 | 1.51871(0.21711) | 0.28688 |

3 | 3 | 1.58239(0.32943) | 0.95184 | 3.05714(0.12040) | 0.23631 | 3.11317(0.09571) | 0.24144 | |

5 | 5 | 1.58197(0.43798) | 1.16554 | 5.05323(0.16283) | 0.24246 | 5.27621(0.14872) | 0.24504 | |

3 | 1.5 | 1.5 | 2.69997(0.69001) | 1.63578 | 3.33097(0.24510) | 0.40800 | 1.51947(0.21104) | 0.28400 |

3 | 3 | 2.93792(0.49867) | 1.20014 | 2.89746(0.13105) | 0.32173 | 3.17920(0.28563) | 0.24903 | |

5 | 5 | 2.83930(0.42533) | 1.19646 | 5.09245(0.18548) | 0.38321 | 5.59062(0.32245) | 0.25649 |

Parameter | ||||||||
---|---|---|---|---|---|---|---|---|

AE(MSE) | CL | AE(MSE) | CL | AE(MSE) | CL | |||

0.5 | 1.5 | 1.5 | 0.52704(0.30408) | 0.79879 | 1.51401(0.14719) | 0.25643 | 1.51741(0.10920) | 0.23001 |

3 | 3 | 0.51535(0.24101) | 0.62857 | 3.02293(0.10672) | 0.15609 | 3.31531(0.09830) | 0.13775 | |

5 | 5 | 0.51998(0.26356) | 0.64531 | 5.02346(0.11010) | 0.20554 | 5.38279(0.08948) | 0.14597 | |

1.5 | 1.5 | 1.5 | 1.53785(0.29162) | 0.68409 | 1.53046(0.11855) | 0.19351 | 1.509356(0.10855) | 0.14344 |

3 | 3 | 1.54116(0.16471) | 0.47592 | 3.02857(0.06020) | 0.11815 | 3.05658(0.04785) | 0.12072 | |

5 | 5 | 1.54098(0.12899) | 0.58277 | 5.02661(0.08141) | 0.12123 | 5.13810(0.07436) | 0.12252 | |

3 | 1.5 | 1.5 | 2.84998(0.34500) | 0.81789 | 1.51548(0.12255) | 0.20400 | 1.50973(0.10567) | 0.14200 |

3 | 3 | 2.96895(0.24933) | 0.60007 | 2.94862(0.06552) | 0.17108 | 3.08960(0.14281) | 0.24903 | |

5 | 5 | 2.91965(0.21266) | 0.59823 | 5.04622(0.09274) | 0.19105 | 5.29531(0.16122) | 0.12824 |

Parameter | ||||||||
---|---|---|---|---|---|---|---|---|

AE(MSE) | CL | AE(MSE) | CL | AE(MSE) | CL | |||

0.5 | 1.5 | 1.5 | 0.51352(0.15204) | 0.39939 | 1.50701(0.07359) | 0.12821 | 1.50871(0.05476) | 0.11503 |

3 | 3 | 0.50767(0.12050) | 0.31428 | 3.01146(0.05336) | 0.07804 | 3.15765(0.04915) | 0.06887 | |

5 | 5 | 0.50999(0.13178) | 0.32265 | 5.01173(0.05505) | 0.10277 | 5.01913(0.04474) | 0.07298 | |

1.5 | 1.5 | 1.5 | 1.51892(0.14581) | 0.34204 | 1.51526(0.08981) | 0.09675 | 1.50467(0.05427) | 0.07172 |

3 | 3 | 1.52058(0.08235) | 0.23796 | 3.01429(0.03010) | 0.05907 | 3.02829(0.02392) | 0.06036 | |

5 | 5 | 1.52049(0.10949) | 0.29138 | 5.01330(0.04070) | 0.06061 | 5.06905(0.03718) | 0.06126 | |

3 | 1.5 | 1.5 | 2.92499(0.17250) | 0.40894 | 1.50774(0.06127) | 0.10200 | 1.50486(0.05283) | 0.07100 |

3 | 3 | 2.98447(0.12466) | 0.30003 | 2.97431(0.03276) | 0.08554 | 3.04480(0.07140) | 0.12451 | |

5 | 5 | 2.95982(0.10633) | 0.29911 | 5.02311(0.04637) | 0.09552 | 5.14765(0.08061) | 0.06412 |

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. [

Distribution | |
---|---|

ILo | |

APILo | |

APIW | |

APILi | |

We |

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. [

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. [

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.

Model | Estimates | AIC | CAIC | BIC | HQIC | K.S | |||
---|---|---|---|---|---|---|---|---|---|

MAPTIW | 289.17 | 289.50 | 296.16 | 291.96 | 0.0739 | 0.4997 | 0.0608 | 0.9413 | |

ILo | 289.48 | 289.65 | 294.14 | 291.35 | 0.0935 | 0.6173 | 0.0741 | 0.7969 | |

APTILo | 294.13 | 294.46 | 301.12 | 296.92 | 0.1260 | 0.8505 | 0.1128 | 0.2880 | |

IW | 294.34 | 294.50 | 299.00 | 296.20 | 0.1516 | 1.0210 | 0.1021 | 0.4059 | |

APIW | 292.04 | 292.37 | 299.03 | 294.83 | 0.1093 | 0.7248 | 0.0796 | 0.7208 | |

APTILi | 301.30 | 301.46 | 305.96 | 303.16 | 0.1780 | 1.1423 | 0.1498 | 0.0658 | |

We | 287.50 | 287.66 | 292.16 | 289.36 | 0.1133 | 0.7653 | 0.0805 | 0.7074 | |

Model | Estimates | AIC | CAIC | BIC | HQIC | K.S | |||
---|---|---|---|---|---|---|---|---|---|

MAPTIW | 829.74 | 829.93 | 838.29 | 833.21 | 0.0510 | 0.3675 | 0.0398 | 0.9872 | |

ILo | 853.35 | 853.44 | 859.05 | 855.66 | 0.2232 | 1.4660 | 0.1184 | 0.0549 | |

APTILo | 868.78 | 868.97 | 877.33 | 872.25 | 0.4545 | 2.8593 | 0.1017 | 0.1410 | |

IW | 892.00 | 892.09 | 897.70 | 894.31 | 0.7443 | 4.5464 | 0.1407 | 0.0125 | |

APIW | 860.26 | 860.45 | 868.81 | 863.73 | 0.4020 | 2.5320 | 0.0957 | 0.1910 | |

APTILi | 866.94 | 867.04 | 872.64 | 869.26 | 0.5242 | 3.2391 | 0.0924 | 0.2242 | |

We | 832.17 | 832.26 | 837.87 | 834.49 | 0.1313 | 0.7864 | 0.0699 | 0.5575 | |

Model | Estimates | AIC | CAIC | BIC | HQIC | K.S | |||
---|---|---|---|---|---|---|---|---|---|

MAPTIW | 395.64 | 396.78 | 399.30 | 396.65 | 0.0475 | 0.2955 | 0.0980 | 0.9506 | |

ILo | 403.57 | 404.11 | 406.00 | 404.24 | 0.1414 | 0.8349 | 0.2359 | 0.1042 | |

APTILo | 405.94 | 407.08 | 409.60 | 406.95 | 0.1136 | 0.6681 | 0.2532 | 0.0669 | |

IW | 401.71 | 402.25 | 404.14 | 402.38 | 0.1677 | 0.9896 | 0.1755 | 0.3796 | |

APIW | 399.49 | 400.63 | 403.15 | 400.50 | 0.1170 | 0.6805 | 0.1387 | 0.6709 | |

APTILi | 403.66 | 404.20 | 406.10 | 404.33 | 0.1269 | 0.7475 | 0.2460 | 0.0808 | |

We | 396.96 | 397.51 | 399.40 | 397.639 | 0.0477 | 0.4254 | 0.1414 | 0.6993 | |

Data | |||
---|---|---|---|

1 | (0, 0.1381) | (0.9506, 1.5563) | (0, 121.3253) |

2 | (0, 0.0132) | (1.4539, 1.9253) | (0, 1086.253) |

3 | (126.53, 3926.65) | (1.6933, 3.1842) | (0, 5122.63) |

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.

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.