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In this paper, we introduce a class of Lindley and Weibull distributions (LW) that are useful for modeling lifetime data with a comprehensive mathematical treatment. The new class of generated distributions includes some well-known distributions, such as exponential, gamma, Weibull, Lindley, inverse gamma, inverse Weibull, inverse Lindley, and others. We provide closed-form expressions for the density, cumulative distribution, survival function, hazard rate function, moments, moments generating function, quantile, and stochastic orderings. Moreover, we discuss maximum likelihood estimation and the algorithm for computing the parameters estimates. Some sub models are discussed as an illustration with real data sets to show the flexibility of this class.

The survival analysis is imperative aspect for statisticians, engineers, and personnel in other scientific fields, such as public health, actuarial science, biomedical studies, demography, and industrial reliability. Several lifetime distributions have been suggested in statistics literature for modeling survival data. Of these distributions two types grabbed the attention of the researchers for fitting lifetime data: Weibull distributions and Lindley distributions. The choice between the two types is due to the nature of hazard rate. Extensive research, Bagheri et al. [

The remainder of this paper is organized as follows: In Section 2, we define the class of Lindley and Weibull (LW) distributions and show that many existing distributions belong to this class. The LW properties, such as survival function, hazard rate function, moments, moment generating function, quantile, and stochastic orderings, are discussed in Section 3. In Section 4, some special cases of the LW class are introduced to show the flexibility of this class in generating existing distributions. Section 5 contains the maximum likelihood estimates of the LW class and the relevant asymptotic confidence interval. Two real data sets are introduced in Section 6 to show the applicability of the LW class. In Section 7, we introduce a conclusion to summarize the contribution of this paper.

In this section, we introduce simple forms of cumulative distribution function (cdf) and probability distribution function (pdf) for the LW class.

Definition. Let

nonnegative parameter vector

The corresponding pdf becomes

And for

Many Lindley types and Weibull types of distributions are members of the LW class, depending on the choice of the function

The pdf(2) can be shown as a mixture of two distributions, as follows:

where

depend on the type of

For any non-decreasing function

and the associate hazard rate function is given by

For

and

Distribution | References | ||||
---|---|---|---|---|---|

Exponential | - | Johnson et al. [ | |||

Rayleigh | - | Rayleigh [ | |||

Weibull | Johnson et al. [ | ||||

Modified Weibull | Lai et al. [ | ||||

Weibull extension | Xie et al. [ | ||||

Gompertz | Gompertz [ | ||||

Exponential power | Smith & Bain [ | ||||

Chen | b | Chen [ | |||

Pham | Pham [ | ||||

Lindley | - | Lindley [ | |||

Inverse Lindley | - | Sharma et al. [ | |||

Power Lindley | Ghitany et al. [ | ||||

Generalized inverse Lindley | Sharma et al. [ | ||||

Two parameters Lindley | - | Shanker et al. [ | |||

Extended power Lindley | Alkarni [ | ||||

Extended inverse Lindley | Alkarni [ |

The

Using the series expansion

As a special case, if we let

and, hence, the mean and the variance are

For

The mean and the variance, then, are

Theorem 1. Let X be a random variable with pdf as in (2), the quantile function, say

where

Proof: We have

is complete.

Note that one can use the same proof above to obtain

Stochastic ordering of positive continuous random variables is an important tool for judging the comparative behavior. A random variable X is said to be smaller than a random variable Y in the following contests:

1) Stochastic order

2) Hazard rate order

3) Mean residual life order

4) Likelihood ratio order

The following implications (Shaked & Shanthikumar, [

The following theorem shows that all members of the LW class are ordered with respect to “likelihood ratio” ordering.

Theorem 2. Suppose

1) If

2) If

Proof. We have

and

Thus,

Case 1) If

then

Case 2) If

The original Lindley distribution (L), proposed by Lindley [

The associated pdf using (2) is given by

It can be seen that this distribution is a mixture of exponential

and

A direct substitution in (9) and (10), with

The mean and the variance from (11) and (12) are

Power Lindley distribution (PL), introduced by Ghitany et al. [

The associated pdf using (2) is given by

The PL distribution is a mixture distribution of the Weibull distribution (with shape parameters

The sf and hrf of the PL distribution are obtained from (5) and (6),

Therefore, the mean and the variance of PL distribution are obtained by direct substitution in (11) and (12),

Extended power Lindley distribution (EPL), introduced by Alkarni [

The associated pdf using (2) is given by

We see that the EPL is a two-component mixture of the Weibull distribution (with shape

The sf and hrf of the EPL distribution are obtained as a direct substitution in (5) and (6),

Using (11) and (12), the mean and the variance of the EPL distribution are given, respectively, by

Inverse Lindley (IL) distribution, proposed by Sharma et al. [

The associated pdf using (4) is given by

We see that the IL is a two-component mixture of the Weibull distribution (with shape

The sf and hrf of the IL distribution are obtained as a direct substitution in (7) and (8),

The generalized inverse Lindley (GIL) distribution, proposed by Sharma et al. [

The associate pdf, using (4), is given by

The associate hrf, using (8), is given by

The mean and the variance of the generalized inverse Lindley distribution are given, respectively, by

The extended inverse Lindley (EIL) distribution, proposed by Alkarni [

The associated pdf, using (4), is given by

We see that the EIL is a two-component mixture of the inverse Weibull distribution (with shape

The hrf of the EIL distribution is given by

Therefore, the mean and the variance of the EIL distribution are given, respectively, by

Let

then the score function is given by

The maximum likelihood estimation (MLE) of

where the elements of

of the parameter space but not on the boundary, the asymptotic distribution of

fidence interval for the parameters and for the hazard rate and survival functions. An

where

standard normal distribution.

In this section, we introduce two data sets as applications of the LW class. For the first data set, we fit L, PL, and EPL models as well as the Two-parameter Lindley (TL) and the standard Weibull (W).

The first data set was introduced by Bader and Priest [

The MLEs of the parameters were obtained using the expectation-maximization (EM) algorithm. The MLEs, Kolmogorov-Smirnov statistic (K-S) with its respective p-value, the maximized log likelihood for the above distributions are listed in

1.312 | 1.314 | 1.479 | 1.552 | 1.700 | 1.803 | 1.861 | 1.865 | 1.944 | 1.958 | 1.966 |
---|---|---|---|---|---|---|---|---|---|---|

1.997 | 2.006 | 2.021 | 2.027 | 2.055 | 2.063 | 2.098 | 2.140 | 2.179 | 2.224 | 2.240 |

2.253 | 2.270 | 2.272 | 2.274 | 2.301 | 2.301 | 2.359 | 2.382 | 2.382 | 2.426 | 2.434 |

2.435 | 2.478 | 2.490 | 2.511 | 2.514 | 2.535 | 2.554 | 2.566 | 2.570 | 2.586 | 2.629 |

2.633 | 2.642 | 2.648 | 2.684 | 2.697 | 2.726 | 2.770 | 2.773 | 2.800 | 2.809 | 2.818 |

2.821 | 2.848 | 2.880 | 2.954 | 3.012 | 3.067 | 3.084 | 3.090 | 3.096 | 3.128 | 3.233 |

3.433 | 3.585 | 3.585 |

Distribution | K-S | p-value | ||||
---|---|---|---|---|---|---|

EPL | 0.0584 | 98.9 | 3.7313 | 0.0429 | 0.9996 | −48.9 |

PL | 0.0450 | - | 3.8678 | 0.0442 | 0.9993 | −49.06 |

W | 0.0100 | - | 4.8175 | 0.1021 | 0.4685 | −50.65 |

TL | 0.8158 | 4504.4 | - | 0.3614 | 0.000 | −105.7 |

L | 0.6545 | - | - | 0.4011 | 0.000 | −119.2 |

For the second data set, we demonstrate the applicability of the IL, GIL, and EIL, as well as the inverse Weibull (IW) and the generalized inverse Weibull (GIW) models.

The MLEs of the parameters, the Kolmogorov-Smirnov statistic (K-S) with its respective p-value, and the maximized log likelihood (logL) for the above distributions are given in

We define a new family of lifetime distributions, called the LW family of distributions, that generates Lindley and Weibull distributions. The LW class contains many lifetime subclasses and distributions. Various standard mathematical properties were derived, such as density and survival hazard functions, moments, moment generating function, and quantile function, and were introduced in flexible and useful forms. The maximum likelihood

0.654 | 0.613 | 0.315 | 0.449 | 0.297 |
---|---|---|---|---|

0.402 | 0.379 | 0.423 | 0.379 | 0.324 |

0.269 | 0.740 | 0.418 | 0.412 | 0.494 |

0.416 | 0.338 | 0.392 | 0.484 | 0.265 |

Distribution | K-S | p-value | ||||
---|---|---|---|---|---|---|

EIL | 0.1052 | 4.0439 | 2.9573 | 0.1395 | 0.8311 | 16.1475 |

GIL | 0.0899 | - | 3.0763 | 0.1445 | 0.7977 | 16.1475 |

IW | 0.0123 | - | 4.2873 | 0.1545 | 0.7263 | 16.096 |

GIW | 0.0302 | 4.3127 | 0.8071 | 0.1560 | 0.7150 | 16.097 |

IL | 0.6345 | - | - | 0.3556 | 0.0127 | −0.5854 |

method was used for parameter estimation using the EM algorithm. Finally, some special models were introduced and fitted to real datasets to show the flexibility and the benefits of the proposed class.

The author is highly grateful to the Deanship of Scientific Research at King Saud University, represented by the Research Center at the College of Business Administration, for supporting this research financially.

The author declares that there were no competing interests.

Said Hofan Alkarni, (2016) A Class of Lindley and Weibull Distributions. Open Journal of Statistics,06,685-700. doi: 10.4236/ojs.2016.64058