In recent years, there has been an increased interest among the researchers to propose new families of distributions to provide the best fit to lifetime data with monotonic (increasing, decreasing, constant) and non-monotonic (unimodal, modified unimodal, bathtub) hazard functions. We further carry this area of research and propose a new family of lifetime distributions called a new logarithmic family via the T-

Speaking broadly, statistical distributions are frequently used for modeling real phenomena in many applied areas including engineering, medical sciences, actuarial, environmental studies, economics, finance, and insurance. Among these distributions, the exponential, Rayleigh and Weibull are some of the most useful models used quite effectively in real-life data modeling. Unfortunately, these distributions have a limited range of capability and thus cannot be applied in all situations to get a better description of the phenomena under consideration. For example, the exponential model is often used for real-life data modeling, but its hazard function is only constant. Whereas, the Rayleigh model is another promising model, but it has an increasing hazard function only. However, the Weibull is a more flexible model offering the features of both the exponential and Rayleigh distributions and additionally offering data modeling with decreasing hazard function. But, the problem with the Weibull model is that it is not capable of modeling data with non-monotonic hazard function.

To provide an adequate fit to data having non-monotonic hazard function, there is a clear need for the generalized versions of these distributions. This fact motivated the researchers to propose new extended distributions. This has been done either introducing the modified versions of the existing models or introducing new families of distributions to obtain flexible model capable of modeling data with non-monotonic hazard function.

In the recent advances in distribution theory, researchers have shown a deep interest in proposing new methods to expand the family of lifetime distributions. This has been done through many different approaches by introducing new generators. Some of the well-known generators include a new generalized class of distributions [

In this article, a new family of lifetime distributions called a new logarithmic (NL) family of distributions is introduced by adopting the T-

where,

The survival function (sf), hazard rate function (hrf) and cumulative hazard rate function (chrf) of the NL family are given, respectively, by

and

The new pdf is most tractable when

A very simple and convenient method of adding additional parameters to modify the existing distributions.

To improve the characteristics and flexibility of the existing distributions.

To introduce the extended version of the baseline distribution having a closed form for cdf, sf as well as hrf.

To provide better fits than the other modified models.

The rest of this article is organized as follows: In Section 2, a special sub-model of the proposed family is discussed. Some mathematical properties are obtained in Section 3. The maximum likelihood estimates of the model parameters are obtained in Section 4. A Monte Carlo simulation study is conducted in Section 5. Section 6 is devoted to analyzing two real-life applications. Finally, concluding remarks are provided in Section 7.

In this section, we define a special sub-model of the proposed family, called a new logarithmic Weibull (NLW) distribution. Let

The pdf corresponding to

The sf and hrf of the NLW distribution are given, respectively, by

and

For

In this section, some statistical properties of the proposed family are derived.

Let

where, ^{−1}(.) is the inverse function of

Moments are very important and play an essential role in statistical analysis, especially in the applications. It helps to capture the important features and characteristics of the distribution (e.g., central tendency, dispersion, skewness and kurtosis). The

where,

Furthermore, a general expression for moment generating function (mgf) of the NL random variable

The residual life offers wider applications in reliability theory and risk management. The residual lifetime of

Additionally, the reverse residual life of the NL random variable denoted by

Order statistics are among the essential tools in inferential and non-parametric statistics. The applications of these statistics appear in the study of reliability and life testing. Let

In this section, the estimation of the unknown parameters of the NL family via the method of maximum likelihood is discussed. Let

Obtaining the partial derivatives of

Setting

In order to assess the performances of the maximum likelihood parameters of the proposed distribution, a small simulation study is carried out. The process is carried out as follows: The number of Monte Carlo replications was made 1000 times each with sample size

Par | MLE | Bias | MSE | MLE | Bias | MSE | |
---|---|---|---|---|---|---|---|

30 | 0.5149 | 0.0132 | 0.0073 | 0.7711 | 0.0213 | 0.0165 | |

0.5269 | 0.0252 | 0.0188 | 0.5233 | 0.0234 | 0.0163 | ||

0.5151 | 0.0149 | 0.0079 | 0.5154 | 0.0152 | 0.0084 | ||

0.5331 | 0.0262 | 0.0181 | 0.5212 | 0.0231 | 0.0162 | ||

50 | 0.5073 | 0.0071 | 0.0042 | 0.7584 | 0.0112 | 0.0084 | |

0.5219 | 0.0222 | 0.0109 | 0.5121 | 0.0115 | 0.0097 | ||

0.5078 | 0.0080 | 0.0052 | 0.5075 | 0.0078 | 0.0048 | ||

0.5213 | 0.0243 | 0.0106 | 0.5092 | 0.0109 | 0.0087 | ||

100 | 0.5021 | 0.0030 | 0.0021 | 0.7593 | 0.0097 | 0.0039 | |

0.5128 | 0.0134 | 0.0051 | 0.5046 | 0.0050 | 0.0045 | ||

0.5021 | 0.0029 | 0.0037 | 0.5070 | 0.0071 | 0.0025 | ||

0.5187 | 0.0136 | 0.0053 | 0.5040 | 0.0097 | 0.0044 |

Par | MLE | Bias | MSE | MLE | Bias | MSE | |
---|---|---|---|---|---|---|---|

30 | 1.5489 | 0.0485 | 0.0669 | 1.5589 | 0.0586 | 0.1212 | |

0.5772 | 0.0771 | 0.0414 | 0.5129 | 0.0131 | 0.0151 | ||

0.5183 | 0.0185 | 0.0089 | 1.5083 | 0.0088 | 0.0206 | ||

0.5736 | 0.0769 | 0.0411 | 0.5120 | 0.0126 | 0.0147 | ||

50 | 1.5272 | 0.0273 | 0.0437 | 1.5221 | 0.0222 | 0.0615 | |

0.5689 | 0.0690 | 0.0248 | 0.5040 | 0.0040 | 0.0087 | ||

0.5109 | 0.0106 | 0.0056 | 1.4996 | 0.0065 | 0.0109 | ||

0.5674 | 0.0675 | 0.0243 | 0.5039 | 0.0039 | 0.0081 | ||

100 | 1.5174 | 0.0172 | 0.0181 | 1.5166 | 0.0169 | 0.0293 | |

0.5531 | 0.0531 | 0.0121 | 0.4932 | 0.0037 | 0.0040 | ||

0.5063 | 0.0057 | 0.0028 | 1.5021 | 0.0029 | 0.0059 | ||

0.5527 | 0.0530 | 0.0239 | 0.4924 | 0.0031 | 0.0032 |

The empirical results are given in

In this section, we provide two applications of the proposed model to the real data sets. We compare the fits of the proposed distribution to those of the three-parameter exponentiated Weibull (EW) Marshall-Olkin Weibull (MOW) and beta Weibull (BW) distributions. The goodness-of-fit measures such as Anderson-Darling (AD), Cramer–von Mises (CM), Kolmogorov-Smirnov (KS) statistic and the corresponding p-value are considered to compare the proposed method with the fitted models. In general, a model with smaller values of these analytical measures and high p-value indicates better fit to the data. All the required computations have been carried out in the R-language using “BFGS” algorithm.

Dist. | ||||
---|---|---|---|---|

NLW | 0.218 | 2.645 | 0.679 | 0.400 |

MOW | 11.829 | 0.564 | 0.877 | |

EW | 4.332 | 0.541 | 0.720 | |

BW | 3.196 | 1.143 | 0.609 | 0.486 |

Dist. | KS | CM | AD | |
---|---|---|---|---|

NLW | 0.044 | 0.043 | 0.2879 | 0.9619 |

MOW | 0.075 | 0.150 | 0.884 | 0.451 |

EW | 0.046 | 0.046 | 0.324 | 0.940 |

BW | 0.945 | 1.592 | 1.576 | 2.2e-16 |

From the results given in

Dist. | ||||
---|---|---|---|---|

NLW | 1.680 | 1.364 | 1.470 | 0.375 |

MOW | 0.336 | 1.807 | 0.200 | |

EW | 1.950 | 0.937 | 1.040 | |

BW | 1.810 | 0.433 | 1.120 | 1.758 |

Dist. | KS | CM | AD | |
---|---|---|---|---|

NLW | 0.063 | 0.019 | 0.142 | 0.961 |

MOW | 0.075 | 0.022 | 0.151 | 0.915 |

EW | 0.083 | 0.027 | 0.165 | 0.899 |

BW | 0.633 | 0.955 | 5.367 | 7.063e-11 |

In this article, a new method is adopted to extend the existing distributions. This effort leads to a new family of lifetime distributions, called a new logarithmic family of distributions. General expressions for some of the mathematical properties of the new family are investigated. Maximum likelihood estimates are also obtained. There are certain advantages of using the proposed method like its cdf has a closed form solution and facilitating data modeling with monotonic and non-monotonic failure rates. A special sub-model of the new family, called a new logarithmic Weibull distribution is considered and two real applications are analyzed. In simulation study, the consistency and proficiency of the maximum likelihood estimators of the proposed model are also illustrated. The practical applications of the proposed model reveal better fits to real-life data than the other well-known competitors. It is hoped that the proposed method will attract wider applications in reliability engineering and biomedical sciences.