This paper introduces a modified form of the inverse Lomax distribution which offers more flexibility for modeling lifetime data. The new three-parameter model is provided as a member of the truncated Lomax-G procedure. The new modified distribution is called the truncated Lomax inverse Lomax distribution. The density of the new model can be represented as a linear combination of the inverse Lomax distribution. Expansions for quantile function, moment generating function, probability weighted moments, ordinary moments, incomplete moments, inverse moments, conditional moments, and Rényi entropy measure are investigated. The new distribution is capable of monotonically increasing, decreasing, reversed J-shaped and upside-down shaped hazard rates. Maximum likelihood estimators of the population parameters are derived. Also, the approximate confidence interval of parameters is constructed. A simulation study framework is established to assess the accuracy of estimates through some measures. Simulation outcomes show that there is a great agreement between theoretical and empirical studies. The applicability of the truncated Lomax inverse Lomax model is illustrated through two real lifetime data sets and its goodness-of-fit is compared with that of the recent models. In fact, it provides a better fit to these data than the other competitive models.

The inverse Lomax (IL) distribution is very flexible in analyzing situations of different types of hazard rate function. It is a member of the inverted family of distributions, that is, the IL distribution is the reciprocal of the well-known Lomax distribution. Kleiber et al. [

where,

Recent studies about the generalization and extensions of the IL distribution have been proposed by several authors. Hassan et al. [

A classical strategy to generate families of probability distributions consists of adding parameter(s) to baseline distributions. These families have the ability to improve the desirable properties of the probability distributions as well as to extract more information from the several data applied in many areas like engineering, economics, biological studies and environmental sciences. Another useful generator that works with the truncated random variable. The notable studies about the truncated-G families in this regard, are the truncated Fréchet-G [

Hassan et al. [

where

The main purpose of this work is to suggest a more flexible and enhance model called the truncated Lomax IL (TLIL) distribution. The key motivations of the TLIL distribution in the practice are (i) to enhance the flexibility of the IL distribution by using TL-G, (ii) to introduce the modified form of the IL distribution whose density can be expressed as a linear combination of the IL distributions, (iii) to develop a new model with increasing, decreasing, reversed J-shaped and upside-down shaped hazard functions and (iv) to provide better fits than the competing models. Further, we study various statistical properties and estimate the model parameters by using the maximum likelihood (ML) method. Finally, simulation studies as well as applications to real data are given.

This work can be arranged as follows. In Section 2, we give the model description of the TLIL distribution and provide a linear representation of its density and distribution functions. Section 3 provides some structural properties of the TLIL distribution. Estimation of parameters along with simulation study is considered in Section 4. Applications are given in Section 5 followed by comments and conclusions.

In this section, we define and provide the density and distribution function of the TLIL distribution. This model is yielded by taking the base-line

where,

Expression of hazard rate function (hrf) is given by:

Graphic features of the pdf and hrf plots of

The quantile function of

where

Here, we provide the representation of cdf

where

where,

Furthermore, Hassan et al. [

where

The probability-weighted moments (PWMs) are less sensitive to outliers. They are utilized to study characteristics of the probability distributions. They are sometimes used when maximum likelihood estimates are unavailable or difficult to compute. The PWMs, denoted by

where,

where;

Here, we provide an infinite sum representation for the

The first four moments are obtained by setting

Based on

The

The harmonic mean of the TLIL distribution can be obtained by using the first inverse moment.

Additionally, the

where, B(.,.,

The conditional moments and the mean residual lifetime function are of interest for lifetime models to be obtained. The

The

where B(.,.,

Moreover, the reversed residual life (RRL) is defined as the conditional random variable 𝑡−

The

The mean of RRL (mean waiting time) represents the waiting time elapsed since the failure of an item on condition that this failure had occurred. For,

The entropy affords a great tool to evaluate the amount of information (or uncertainty) exists in a random observation relating to its parent distribution. A small value of entropy provides the smaller uncertainty in the data. Hassan et al. [

where

So, after some manipulation, the Rényi entropy of the TLIL distribution is given by

where, B(.,.) is the beta function.

In this section, the ML estimators of the model parameters are derived in the case of complete samples. Also, approximate confidence intervals (CIs) are obtained. Furthermore, a numerical study is conducted.

Suppose that _{1}, _{2},…,_{n} be the observed values from TLIL distribution whose lifetimes have the pdf

where,

For interval estimation of the parameters, it is known that under regularity conditions, the asymptotic distribution of ML estimators of elements of unknown parameters for

where,

where

In this sub-section, we perform a simulation study to evaluate the behavior of the ML estimates (MLEs). The attitude of the different estimates is checked in terms of their mean square errors (MSEs), standard errors (SEs) and average lengths (ALs) of the CIs. 10000 random samples of sizes 30, 50, 75 and 100 are generated from TLIL distribution. Four sets of parameter values are chosen as;

Measures | (i) | (ii) | |||||
---|---|---|---|---|---|---|---|

2.0307 | 1.8101 | 0.7903 | 1.3591 | 0.4968 | 0.3187 | ||

0.2181 | 0.1476 | 0.0633 | 0.1649 | 0.0603 | 0.0346 | ||

0.8551 | 0.5785 | 0.2483 | 0.6463 | 0.2363 | 0.1357 | ||

1.8644 | 1.3712 | 0.7645 | 1.3516 | 0.3468 | 0.3089 | ||

0.1635 | 0.0866 | 0.0380 | 0.1309 | 0.0267 | 0.0206 | ||

0.6411 | 0.3396 | 0.1491 | 0.5132 | 0.1047 | 0.0808 | ||

1.8632 | 0.9086 | 0.7416 | 1.3083 | 0.3037 | 0.3012 | ||

0.1318 | 0.0411 | 0.0273 | 0.1040 | 0.0166 | 0.0137 | ||

0.5165 | 0.1612 | 0.1070 | 0.4079 | 0.0652 | 0.0537 | ||

1.6464 | 0.8515 | 0.7291 | 1.3006 | 0.2941 | 0.2981 | ||

0.1166 | 0.0320 | 0.0251 | 0.0958 | 0.0124 | 0.0117 | ||

0.4570 | 0.1255 | 0.0984 | 0.3755 | 0.0486 | 0.0459 | ||

1.5751 | 0.7546 | 0.6981 | 1.2597 | 0.2764 | 0.2949 | ||

0.0923 | 0.0195 | 0.0160 | 0.0753 | 7.3890* | 7.9447* | ||

0.3619 | 0.0765 | 0.0626 | 0.2950 | 0.0290 | 0.0311 |

Note: * Indicate that the value multiply

Measures | (iii) | (iv) | |||||
---|---|---|---|---|---|---|---|

0.8450 | 0.1031 | 0.1109 | 0.4564 | 3.1732* | 0.3244 | ||

0.1057 | 0.0273 | 0.0316 | 0.1189 | 0.0101 | 0.1032 | ||

0.4144 | 0.1071 | 0.1238 | 0.4660 | 0.0397 | 0.4045 | ||

0.8080 | 0.0776 | 0.1006 | 0.4435 | 1.6097* | 0.0669 | ||

0.0814 | 0.0136 | 0.0180 | 0.0894 | 5.6538* | 0.0364 | ||

0.3190 | 0.0534 | 0.0705 | 0.3506 | 0.0222 | 0.1426 | ||

0.7385 | 0.0684 | 0.0952 | 0.3987 | 1.1765* | 0.0273 | ||

0.0653 | 8.6751* | 0.0122 | 0.0702 | 3.9515* | 0.0191 | ||

0.2558 | 0.0340 | 0.0479 | 0.2752 | 0.0155 | 0.0747 | ||

0.7243 | 0.0649 | 0.0938 | 0.3672 | 0.7097* | 0.0221 | ||

0.0562 | 6.4092* | 9.8561* | 0.0591 | 2.6644* | 0.0148 | ||

0.2204 | 0.0251 | 0.0386 | 0.2315 | 0.0104 | 0.0579 | ||

0.6873 | 0.0633 | 0.0932 | 0.3567 | 0.5559* | 0.0123 | ||

0.0454 | 4.1558* | 6.4674* | 0.0477 | 1.9260* | 9.0701* | ||

0.1778 | 0.0163 | 0.0254 | 0.1870 | 7.5499* | 0.0356 |

Note: * Indicate that the value multiply

Based on the above tables, we conclude the following

It is clear that MSEs, SEs and ALs decrease as sample size increases for all estimates.

The MSEs for

The MSEs for

As the value of

As the value of

The SEs and ALs for

In this section, we provide the applicability of TLIL distribution by using two real data sets. These data have been used by several authors to show the superiority of other competing models. We also provide a formative evaluation of the goodness of-fit of the models and make comparisons with other distributions. The measures of goodness of fit including; the Akaike information criterion (AIC), Consistent AIC (CAIC), Hannan-Quinn information criterion (HQIC), Anderson-Darling(A^{∗}) and Cramér- von Mises (W^{∗}) are calculated to compare the fitted models. Generally, the best fit to the data that is correspond to the lowest values of these statistics.

The first data set was proposed by Murthy et al. [

Distribution | MLEs | ||||
---|---|---|---|---|---|

2.459 (0.251) | 47.134 (14.289) | 1151 (0.044) | |||

826.196 (3505) | 785.944 (3343) | 4.471 (0.886) | |||

2.615 (1.343) | 100.276 (404.095) | 5.277 (37.988) | 78.677 (799.338) | ||

0.034 (0.048) | 0.379 (0.025) | −0.354 (0.815) | 26.430 (40.252) | ||

34.660 (17.527) | 81.846 (52.014) | 14.433 (27.095) | 0.204 (0.042) | ||

17.686 (6.222) | 33.639 (19.994) | 1.940 (1.011) | 0.306 (0.045) | 16.721 (9.622) | |

34.180 (14.838) | 11.496 (6.730) | 1.360 (1.002) | 0.298 (0.060) | ||

200.747 (87.275) | 1.952 (0.125) | 0.102 (0.017) | −0.869 (0.101) |

Distribution | AIC | CAIC | HQIC | A* | W* |
---|---|---|---|---|---|

267.323 | 267.623 | 270.255 | 0.6702 | 0.0639 | |

268.515 | 269.022 | 271.447 | 0.708 | 0.072 | |

270.296 | 270.802 | 280.019 | 0.868 | 0.097 | |

269.975 | 270.481 | 279.700 | 0.786 | 0.085 | |

281.434 | 281.941 | 291.158 | 1.506 | 0.185 | |

283.899 | 284.669 | 296.053 | 1.591 | 0.199 | |

305.028 | 305.534 | 314.751 | 3.220 | 0.465 | |

309.472 | 309.978 | 319.195 | 2.404 | 0.320 |

It is clear from

This data set describes there mission times (in months) of a random sample of 128 bladder cancer patients studied by Lee et al. [^{∗} and W^{∗} as compared to the other competitive models. Therefore, TLIL distribution provides the best fit for the cancer patient data. More information can be found in

Distribution | MLEs | ||||
---|---|---|---|---|---|

1.429 (0.151) | 41.006 (10.109) | 12.589 (0.053) | |||

2.677 (1.005) | 7.235 (7.009) | 1.32 (1.855) | |||

22.186 (21.956) | 20.277 (17.296) | 0.224 (0.144) | 1.780 (1.076) | 1.306 (1.079) | |

2.327 (0.369) | 0.0002 (0.0002) | 17.931 (7.385) | 0.543 (0.042) | 0.001 (0.0003) | |

12.526 (24.469) | 33.342 (36.348) | 27.753 (71.507) | 0.169 (0.104) | ||

7.376 (5.389) | 0.047 (0.004) | 0.118 (0.260) | 0.049 (0.036) | ||

0.0002 (0.011) | 0.1208 (0.024) | 0.8955 (0.626) | 0.407 ( 0.407) | ||

0.00003 (0.0061) | 1.0065 (0.035) | 0.1139 (0.032) | 0.9722 (0.125) | −0.1630 (0.280) |

Distribution | AIC | CAIC | HQIC | A* | W* |
---|---|---|---|---|---|

825.827 | 826.02 | 828.99 | 0.1514 | 0.023 | |

827.465 | 827.659 | 830.942 | 0.340 | 0.048 | |

841.268 | 841.760 | 855.528 | 0.900 | 0.134 | |

839.824 | 840.316 | 854.085 | 2.618 | 0.410 | |

842.965 | 843.290 | 854.373 | 1.121 | 0.168 | |

866.350 | 866.675 | 877.758 | 2.361 | 0.398 | |

836.450 | 836.775 | 847.858 | 3.125 | 0.760 | |

838.478 | 838.970 | 852.739 | 3.113 | 0.703 |

As seen from

We provide a new three-parameter lifetime distribution depends on the recent truncated Lomax-G family. The new truncated Lomax inverse Lomax model offers flexibility for modeling lifetime data. Expressions of density and distribution functions are obtained as a linear combination of the inverse Lomax distribution. Several mathematical properties of the new model are derived like; probability weighted moments, quantile function, moment generating function, ordinary and incomplete moments, inverse moments, conditional moments, and Rényi entropy. The maximum likelihood method of estimation is employed to obtain the point and approximate confidence interval of population parameters. We assess the accuracy of estimates viz simulation study. Applications to two real data are utilized to illustrate the importance and usefulness of the new model compared to some models.

We would like to thank all four reviewers and the academic editor for their interesting comments on the article, greatly improving it in this regard.