We introduce a new two-parameter lifetime model, referred to alpha power transformed inverted Topp-Leone, derived by combining the alpha power transformation-G family with the inverted Topp-Leone distribution. Structural properties of the proposed distribution are implemented like; quantile function, residual and reversed residual life, Rényi entropy measure, moments and incomplete moments. The maximum likelihood, weighted least squares, maximum product of spacing, and Bayesian methods of estimation are considered. A simulation study is worked out to evaluate the restricted sample properties of the proposed distribution. Numerical results showed that the Bayesian estimates give more accurate results than the corresponding other estimates in the majority of the situations. The flexibility of the suggested model is demonstrated given some applications related to reliability, medicine, and engineering. A real data set is used to illustrate the potentiality of the alpha power transformed inverted Topp-Leone distribution compared to inverted Topp-Leone, inverse Weibull, alpha power inverse Weibull, inverse Lomax, alpha power inverse Lomax, inverse exponential, and alpha power exponential distributions. Criteria measures and their results showed that the suggested distribution is the best candidate for the considered data sets. The alpha power transformed inverted Topp-Leone distribution operates well for lifetime modeling.

In recent times, probability distributions play a significant role in modeling naturally occurring phenomena. In fact, the statistics literature contains hundreds of continuous univariate distributions and their successful applications. However, there still remain many real-world phenomena involving data, which do not follow any of the traditional probability distributions. So, several attempts are introduced by many researchers to provide more flexibility to a family of distributions. Mahdavi et al. [

Relevant works have been provided based on the AP method, for instance; AP Weibull distribution [

The inverted distributions were suggested in the literature using the inverse transformation of probability distributions. These distributions display different features in the behavior of the density and hazard rate shapes. They allow applicability to the phenomenon in many fields such as; biological sciences, life testing problems, survey sampling, and engineering sciences. Inverted distributions and their applications were discussed by several authors (see [

In Hassan et al. [

and,

In this paper, we propose a new two-parameter related to the ITL distribution depending on the AP family. We call it alpha power inverted Topp Leone (APITL) distribution. The basic motivations to introduce the APITL model are (i) Generalizing a new useful version of the ITL distribution based on the APT method along with deriving its statistical properties, (ii) Providing flexible PDF with right-skewed and uni-modal shapes, (iii) Modeling decreasing, increasing, upside-down hazard rate function (HRF), and (iv) Introducing some real applications in some areas.

This paper is constructed as follows. Section 2 describes the APITL distribution. Section 3 gives some structural properties of the APITL distribution. Section 4 gives the maximum likelihood (ML), the weighted least squares (WLS), the maximum product of spacing (MPS), and Bayesian estimators. Section 5 examines the effectiveness of the proposed estimates through a numerical illustration. Data analyses and some concluding remarks are employed, consequently, in Sections 6 and 7.

In this section, based on the AP family we introduce a new probability distribution related to the ITL distribution. We define the PDF, CDF, HRF and cumulative HRF of the APITL distribution.

A random variable

The PDF related to

Descriptive PDF plots of the APITL distribution for some choices of parameters are represented in

The reliability function and the HRF of

and

An illustration of the HRF plots for the APITL distribution, for some choices of

Here, we give some statistical properties.

The APITL distribution is simulated by inverting CDF

The ^{th} quantile for the APITL random variable is obtained by solving

where, ^{th}, 50^{th}, and 75^{th} percentiles for the random variable

where

Skewness and kurtosis plots of the APITL model, based on the quantiles, are exhibited in

Moments of a probability distribution are crucial to deduce its properties such as measures of central tendency, dispersion, skewness, and kurtosis. The ordinary r^{th} moment of the APITL distribution is derived. The r^{th} moment of the APITL distribution is obtained from

Since the power series representation is written as:

Using the power series expansion

Using the binomial expansion in

After using binomial expansion, then the r^{th} moment is given by:

where,

The first four moments, for ^{th} central moment

Values of mean

(3,4.5) | 0.458 | 0.684 | 4.339 | 89.518 |

(3,6) | 0.347 | 0.335 | 2.936 | 21.996 |

(3,10) | 0.226 | 0.121 | 2.053 | 9.306 |

(10,4.5) | 0.564 | 0.932 | 3.866 | 72.824 |

(10,6) | 0.421 | 0.446 | 2.588 | 17.809 |

(10,10) | 0.269 | 0.157 | 1.785 | 7.526 |

We concluded from

The class of probability-weighted moments (PWMs), denoted by

Substituting

where,

The ^{th} moment of the residual life (RLe), say ^{th} moment of the residual life of

Therefore, the ^{th} moment of RLe for APITL distribution is obtained by substituting PDF

where ^{th} moment of the reversed RLe (RRLe) for the APITL distribution is given by:

where,

The mean of RRLe serves as the waiting time elapsed since the failure of an item on the condition that this failure had occurred.

The entropy of a random variable measures the amounts of information (or uncertainty) contained in a random observation; i.e., large value of entropy indicates higher uncertainty in the data. Rényi entropy of

Substituting PDF

From

where,

In this section, we deal with parameter estimators of the APITL distribution based on ML, WLS, MPS, and Bayesian estimation methods.

Let _{1},…, _{n} e the observed values from the APITL distribution with parameters

Then the log-likelihood function, say

Therefore, the ML equations are given by:

and,

Solving the non-linear equations

Let _{1},…, _{n} be a simple random sample from the APITL distribution and let _{(1)}< _{(2)}<…<_{(n)} be the associated order statistics. The WLS estimators of

where

The MPS method is an alternative procedure of the ML method which provides a parameter estimate of continuous distribution. The MPS estimators of

Solving the non-linear equations

and,

Equating

Here, we get the Bayesian estimator of the APITL parameters. The Bayesian estimator is regarded under symmetric (squared error loss function (SELF)) which is defined as

To elicit the hyper-parameters of the informative priors, the ML estimator for

where,

From

To obtain the Bayesian estimators, we can use the Markov Chain Monte Carlo (MCMC) approach. A useful sub-class of the MCMC techniques is the Gibbs sampling and more general Metropolis within Gibbs samplers. The Metropolis-Hastings (MH) algorithm along with the Gibbs sampling are the two most popular examples of the MCMC method. We use the MH within Gibbs sampling steps to generate random samples from conditional posterior densities of α and

and

The Bayesian estimators are obtained via SELF and LINEX loss function (for more information see [

A Monte-Carlo simulation study was conducted to evaluate and compare the behavior of the different estimates based on mean square errors (MSEs) and biases. Generate 10000 random samples of sizes

We calculated the ML estimate (MLE), WLS estimate (WLSE), MPS estimate (MPSE), and Bayesian estimate (BE) of

The bias and MSE for all estimates decrease as

As values of

For a fixed value of

For a fixed value of

The measures of WLSEs are better than MLEs and MPSEs with decreasing sample size.

The measures of MPSEs are preferable to MLEs and WLSEs with sample sizes.

The BEs under the LINEX loss function is preferable to the other estimates.

ML | WLS | MPS | SELF | LINEX (0.5) | LINEX (1.5) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | ||||

50 | 0.75 | 0.5 | −0.0548 | 0.9675 | −0.0215 | 0.1834 | −0.0550 | 0.3838 | 0.12695 | 0.08138 | 0.09019 | 0.06391 | 0.02555 | 0.04295 | |

−0.0716 | 0.0227 | −0.0282 | 0.0148 | −0.0718 | 0.0295 | 0.00794 | 0.00117 | 0.00648 | 0.00113 | 0.00358 | 0.00108 | ||||

1.2 | 0.9620 | 1.0917 | 1.1126 | 0.7401 | 0.9194 | 1.0035 | 0.13249 | 0.08978 | 0.09817 | 0.07064 | 0.03771 | 0.04739 | |||

0.0192 | 0.0634 | −0.0667 | 0.0547 | −0.1264 | 0.0797 | 0.00529 | 0.00043 | 0.00317 | 0.00041 | −0.00103 | 0.00039 | ||||

3 | −0.0465 | 0.8156 | 0.2324 | 0.7940 | −0.0467 | 0.3477 | 0.12012 | 0.09971 | 0.09288 | 0.08479 | 0.04281 | 0.06343 | |||

−0.3475 | 0.7940 | −0.0114 | 0.6331 | −0.3481 | 0.7195 | 0.00191 | 0.00012 | −0.00039 | 0.00011 | −0.00498 | 0.00014 | ||||

2 | 0.5 | −0.2319 | 0.6502 | −0.1322 | 0.2170 | −0.2317 | 0.4993 | 0.06612 | 0.02823 | 0.00341 | 0.02121 | −0.10808 | 0.02961 | ||

−0.0332 | 0.0072 | −0.0134 | 0.0053 | −0.0333 | 0.0074 | 0.00422 | 0.00034 | 0.00360 | 0.00034 | 0.00236 | 0.00033 | ||||

1.2 | 0.4612 | 1.0682 | −0.1374 | 0.7401 | −0.3306 | 1.0032 | 0.24996 | 0.29895 | 0.06710 | 0.17261 | −0.20445 | 0.15091 | |||

0.0192 | 0.0634 | −0.0667 | 0.0547 | −0.1279 | 0.0799 | 0.01179 | 0.00326 | 0.00667 | 0.00310 | −0.00341 | 0.00296 | ||||

3 | −0.0302 | 0.9561 | 0.0334 | 0.9424 | −0.3011 | 0.9032 | 0.36851 | 0.63164 | 0.12254 | 0.35078 | −0.21574 | 0.24823 | |||

−0.2719 | 0.4773 | −0.1700 | 0.4296 | −0.2723 | 0.3900 | 0.03683 | 0.02339 | 0.00285 | 0.02093 | −0.06248 | 0.02296 | ||||

100 | 0.75 | 0.5 | −0.0825 | 0.3758 | −0.0312 | 0.1116 | −0.0826 | 0.2327 | 0.12948 | 0.06681 | 0.09599 | 0.05252 | 0.03725 | 0.03545 | |

−0.0550 | 0.0124 | −0.0214 | 0.0081 | −0.0551 | 0.0179 | 0.00833 | 0.00116 | 0.00712 | 0.00113 | 0.00471 | 0.00108 | ||||

1.2 | 1.1799 | 0.6729 | 1.1556 | 0.6264 | 0.9860 | 0.6087 | 0.11617 | 0.06892 | 0.09008 | 0.05684 | 0.04303 | 0.04026 | |||

0.0055 | 0.0371 | −0.0456 | 0.0322 | −0.0798 | 0.0367 | 0.00533 | 0.00058 | 0.00334 | 0.00056 | −0.00064 | 0.00054 | ||||

3 | −0.0781 | 0.6382 | 0.1700 | 0.5510 | −0.0782 | 0.2098 | 0.08784 | 0.06171 | 0.06821 | 0.05404 | 0.03166 | 0.04277 | |||

−0.2766 | 0.4818 | −0.0039 | 0.4971 | −0.2771 | 0.4474 | 0.00182 | 0.00017 | −0.00044 | 0.00017 | −0.00496 | 0.00019 | ||||

2 | 0.5 | −0.1630 | 0.6384 | −0.0451 | 0.0974 | −0.1629 | 0.2653 | 0.07283 | 0.03301 | 0.01439 | 0.02512 | −0.08987 | 0.02942 | ||

−0.0201 | 0.0044 | −0.0023 | 0.0027 | −0.0202 | 0.0036 | 0.00229 | 0.00037 | 0.00179 | 0.00036 | 0.00080 | 0.00036 | ||||

1.2 | −0.0701 | 0.7286 | −0.0944 | 0.6264 | −0.2640 | 0.6087 | 0.22919 | 0.26288 | 0.07037 | 0.15881 | −0.16927 | 0.13156 | |||

0.0055 | 0.0371 | −0.0456 | 0.0322 | −0.0798 | 0.0371 | 0.00925 | 0.00343 | 0.00511 | 0.00333 | −0.00309 | 0.00322 | ||||

3 | −0.2413 | 0.7090 | 0.0892 | 0.6865 | −0.2411 | 0.5432 | 0.35177 | 0.45415 | 0.14393 | 0.31028 | −0.15169 | 0.20198 | |||

−0.1764 | 0.2568 | −0.0968 | 0.2831 | −0.1767 | 0.1923 | 0.02796 | 0.02341 | 0.00021 | 0.02178 | −0.05398 | 0.02321 |

ML | WLS | MPS | SELF | LINEX (0.5) | LINEX(1.5) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | ||||

150 | 0.75 | 0.5 | −0.0499 | 0.24593 | 0.0021 | 0.10933 | −0.0495 | 0.15739 | 0.11137 | 0.06125 | 0.08280 | 0.04947 | 0.03186 | 0.03468 | |

−0.0337 | 0.00768 | −0.0123 | 0.00590 | −0.0337 | 0.00955 | 0.00761 | 0.00109 | 0.00658 | 0.00106 | 0.00451 | 0.00103 | ||||

1.2 | 0.1631 | 0.32274 | 0.0962 | 0.27620 | −0.0499 | 0.17235 | 0.07791 | 0.04444 | 0.05725 | 0.03753 | 0.01968 | 0.02845 | |||

0.0257 | 0.02466 | −0.0102 | 0.06043 | −0.0844 | 0.05781 | 0.00754 | 0.00074 | 0.00562 | 0.00071 | 0.00179 | 0.00067 | ||||

3 | −0.0498 | 0.32287 | 0.1404 | 0.38843 | −0.0494 | 0.13954 | 0.07904 | 0.04635 | 0.06217 | 0.04073 | 0.03098 | 0.03273 | |||

−0.1777 | 0.28429 | 0.0135 | 0.38771 | −0.1773 | 0.25787 | 0.00289 | 0.00021 | 0.00065 | 0.00020 | −0.00382 | 0.00022 | ||||

2 | 0.5 | −0.0958 | 0.38265 | 0.0116 | 0.27172 | −0.0953 | 0.15978 | 0.06533 | 0.03428 | 0.01100 | 0.02764 | −0.08656 | 0.03170 | ||

−0.0118 | 0.00240 | −0.0008 | 0.00189 | −0.0117 | 0.00183 | 0.00173 | 0.00037 | 0.00130 | 0.00037 | 0.00045 | 0.00037 | ||||

1.2 | 0.2403 | 0.62056 | 0.0029 | 0.61890 | −0.1635 | 0.37899 | 0.23018 | 0.24647 | 0.08626 | 0.15308 | −0.13520 | 0.11692 | |||

0.0124 | 0.02279 | −0.0250 | 0.02251 | −0.0472 | 0.01905 | 0.00947 | 0.00338 | 0.00591 | 0.00328 | −0.00116 | 0.00317 | ||||

3 | −0.1492 | 0.59608 | 0.1421 | 0.55189 | −0.1485 | 0.33969 | 0.29826 | 0.24158 | 0.11988 | 0.24502 | −0.14413 | 0.16560 | |||

−0.1057 | 0.15444 | −0.0431 | 0.14067 | −0.1055 | 0.10120 | 0.03659 | 0.02650 | 0.01246 | 0.02303 | −0.03511 | 0.02274 | ||||

200 | 0.75 | 0.5 | −0.0541 | 0.15230 | 0.0255 | 0.10660 | −0.0544 | 0.12684 | 0.10701 | 0.06040 | 0.08064 | 0.04936 | 0.03347 | 0.03510 | |

−0.0324 | 0.00575 | −0.0070 | 0.00484 | −0.0325 | 0.00758 | 0.00738 | 0.00103 | 0.00646 | 0.00102 | 0.00462 | 0.00101 | ||||

1.2 | 0.1065 | 0.22339 | 0.0826 | 0.23848 | −0.0547 | 0.13868 | 0.07764 | 0.03694 | 0.05978 | 0.03163 | 0.02700 | 0.02435 | |||

0.0044 | 0.01361 | −0.0151 | 0.05126 | −0.0808 | 0.04575 | 0.00711 | 0.00081 | 0.00528 | 0.00078 | 0.00164 | 0.00074 | ||||

3 | −0.0525 | 0.22652 | 0.0986 | 0.27801 | −0.0528 | 0.11287 | 0.07426 | 0.03685 | 0.06035 | 0.03287 | 0.03426 | 0.02684 | |||

−0.1730 | 0.21711 | −0.0140 | 0.28686 | −0.1737 | 0.20680 | 0.00385 | 0.00024 | 0.00163 | 0.00023 | −0.00279 | 0.00023 | ||||

2 | 0.5 | −0.0900 | 0.16462 | −0.0461 | 0.14601 | −0.0902 | 0.13328 | 0.07360 | 0.03368 | 0.02149 | 0.02634 | −0.07237 | 0.02793 | ||

−0.0124 | 0.00144 | −0.0061 | 0.00165 | −0.0125 | 0.00136 | 0.00231 | 0.00035 | 0.00194 | 0.00035 | 0.00118 | 0.00034 | ||||

1.2 | −0.0190 | 0.46434 | −0.0108 | 0.41257 | −0.1521 | 0.31712 | 0.20131 | 0.23086 | 0.07490 | 0.15318 | −0.12522 | 0.11925 | |||

−0.0050 | 0.01542 | −0.0215 | 0.01560 | −0.0463 | 0.01419 | 0.00809 | 0.00329 | 0.00496 | 0.00321 | −0.00128 | 0.00312 | ||||

3 | −0.1362 | 0.40894 | 0.1118 | 0.39202 | −0.1365 | 0.28282 | 0.29218 | 0.39220 | 0.13049 | 0.24178 | −0.11010 | 0.16256 | |||

−0.1035 | 0.11330 | −0.0447 | 0.10584 | −0.1039 | 0.07466 | 0.03851 | 0.02562 | 0.01718 | 0.02244 | −0.02495 | 0.02136 |

Here, we fit the APITL distribution under three real data taken from fields of survival times of medicine, engineering, and reliability. The APITL model is compared to other competitive models as, ITL, inverse Weibull (IW), alpha power IW (APIW), inverse Lomax (ILo), alpha power ILo (APILo), inverse exponential (IEx), and alpha power inverse exponential (APIEx) distributions.

^{th} survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli. These data were analyzed in Refs. [

Model | KS | P-Value | AIC | CAIC | HQIC | |||
---|---|---|---|---|---|---|---|---|

204.3711 | 4.5486 | – | 0.082 | 0.7179 | 191.1228 | 191.2967 | 192.9355 | |

(186.05) | (0.4218) | |||||||

416.1724 | 1.5890 | 0.2231 | 0.1449 | 0.09728 | 217.6459 | 217.9988 | 220.3649 | |

(377.35) | (0.1117) | (0.0431) | ||||||

0.0026 | 12.3172 | 0.2714 | 0.1141 | 0.3056 | 209.0081 | 209.361 | 211.7271 | |

(0.0018) | (3.5600) | (0.0906) | ||||||

0.0199 | 2.3775 | – | 0.1582 | 0.05434 | 229.4396 | 229.6135 | 231.2523 | |

(0.0195) | (0.3039) |

Model | KS | P-Value | AIC | CAIC | HQIC | |||
---|---|---|---|---|---|---|---|---|

1.2533 | 0.14799 | 0.3450 | 188.9506 | 189.0559 | 189.5613 | |||

(0.1982) | ||||||||

6.9168 | 1.8963 | |||||||

(6.6745) | (0.3783) | |||||||

3.7381 | 0.4900 | 0.11064 | 0.7092 | 189.0165 | 189.3408 | 190.2378 | ||

(2.1288) | (0.3351) | |||||||

10.9198 | 23.9997 | 0.0293 | 0.1123 | 0.6945 | 191.5962 | 192.2629 | 193.4282 | |

(9.0519) | (5.6358) | (0.0444) |

According to tables and

Furthermore, the suggested methods of estimation (see Section 4) for the APITL parameters were considered based on the three data.

Model | KS | P-Value | AIC | CAIC | HQIC | |||
---|---|---|---|---|---|---|---|---|

301.696 | 0.8271 | 0.150 | 0.5082 | 311.0474 | 311.4918 | 311.9439 | ||

(339.011) | (0.1006) | |||||||

0.7238 | 6.9663 | 0.159 | 0.4304 | 314.2288 | 314.6733 | 315.1253 | ||

(0.0927) | (1.8023) | |||||||

42.7187 | 0.8576 | 3.1712 | 0.158 | 0.4906 | 312.3953 | 313.3184 | 313.7401 | |

(70.859) | (0.1391) | (1.8568) | ||||||

11.1799 | 0.233 | 0.07706 | 320.1239 | 320.2668 | 320.5722 | |||

(2.0412) | ||||||||

23.767 | 5.1597 | 0.1591 | 0.4598 | 311.4365 | 311.8809 | 312.333 | ||

(23.4501) | (1.6219) |

Data | ML | WLS | SELF | LINEX (1.5) | LINEX( -1.5) | ||
---|---|---|---|---|---|---|---|

204.3711 | 230.2754 | 319.536 | 321.5280 | 303.5203 | |||

86.0516 | 49.2177 | 83.123 | 82.9465 | 82.9622 | |||

4.5486 | 4.5407 | 4.6113 | 4.5071 | 4.7201 | |||

0.4218 | 0.0895 | 0.3089 | 0.3092 | 0.2081 | |||

6.9168 | 5.7518 | 7.1372 | 2.6879 | 16.9063 | |||

6.6745 | 1.2317 | 4.2794 | 2.8905 | 2.2560 | |||

1.8963 | 1.7875 | 1.8793 | 1.8011 | 1.9589 | |||

0.3783 | 0.0953 | 0.3248 | 0.2595 | 0.1925 | |||

301.6959 | 364.1921 | 555.4224 | 517.0317 | 519.0317 | |||

33. 9011 | 34.264 | 33. 7555 | 24. 9852 | 26. 3894 | |||

0.8271 | 0.8195 | 0.8457 | 0.8386 | 0.8504 | |||

0.1006 | 0.1248 | 0.0973 | 0.0935 | 0.0897 |

The convergence of the MCMC estimation of

We proposed and studied the alpha power transformed inverted Topp-Leone distribution. Some structural properties of the APITL distribution were provided. Bayesian and non-Bayesian methods of estimation were considered. We obtained the ML, WLS, and MPS estimators of the population parameters. The Bayesian estimator was deduced under LINEX and SELF. The Monte Carlo simulation study was worked out to assess the behavior of estimates. Generally, we concluded that the Bayes estimates are preferable to the corresponding other estimates in approximately most of the situations. We proved empirically that the APITL model reveals its superiority over other competitive models for different real data.

Thank a lot for every one help full.