In this paper, a modified form of the traditional inverse Lomax distribution is proposed and its characteristics are studied. The new distribution which called modified logarithmic transformed inverse Lomax distribution is generated by adding a new shape parameter based on logarithmic transformed method. It contains two shape and one scale parameters and has different shapes of probability density and hazard rate functions. The new shape parameter increases the flexibility of the statistical properties of the traditional inverse Lomax distribution including mean, variance, skewness and kurtosis. The moments, entropies, order statistics and other properties are discussed. Six methods of estimation are considered to estimate the distribution parameters. To compare the performance of the different estimators, a simulation study is performed. To show the flexibility and applicability of the proposed distribution two real data sets to engineering and medical fields are analyzed. The simulation results and real data analysis showed that the Anderson-Darling estimates have the smallest mean square errors among all other estimates. Also, the analysis of the real data sets showed that the traditional inverse Lomax distribution and some of its generalizations have shortcomings in modeling engineering and medical data. Our proposed distribution overcomes this shortage and provides a good fit which makes it a suitable choice to model such data sets.

Inverse Lomax (IL) distribution is a very important lifetime distribution which can be used as a good alternative to the well known distributions such as gamma, inverse Weibull, Weibull and Lomax distributions. It can be considered as a member of generalized beta family of distributions. It has different applications in modelling various types of data including economics and actuarial sciences data because its hazard rate can be decreasing and upside down bathtub shaped. The random variable

The cumulative distribution function (CDF) of

where

The main objective of this paper is to propose a new form of the IL distribution by adding a new shape parameter to the CDF in

and

respectively, with

To develop different shapes for the PDF and hazard rate function.

To increase the flexibility of the traditional IL distribution in modelling different phenomenons.

To model skewed data which can not be modeled by other traditional models.

To increase the flexibility of the traditional IL distribution properties like mean, variance, skewness and kurtosis.

Two applications showed that the MLTIL distribution provides a better fit than the traditional IL distribution and some of its generalizations.

Another motivation to this article is to use six classical estimation methods to estimate the parameters in order to recommend which method provide the best estimates based on mean square error criteria and via a simulation study.

The hazard rate function of the MLTIL distribution can has decreasing or upside-down shapes depending on its shape parameters which makes the distribution is quite effectively in modelling lifetime data. It can be used as an alternative to IL and inverse Weibull distributions. Simulation results reveal that the Anderson-Darling (AD) estimators perform better than other estimators in terms of minimum mean-squared errors. Finally, the analysis of engineering and medical data sets show the ability of the MLTIL distribution to provide a better fit than some other competitive models. The rest of the paper is organized as follows: In the next Section we describe the MLTIL distribution and a mixture representation of its density. Some of its statistical properties are discussed in Section 3. Six classical estimation methods are considered in Section 4. A simulation study is conducted in Section 5. Two applications are considered in Section 6. In Section 7, the paper is concluded.

In this section we introduce the MLTIL distribution. Let the random variable

and its CDF is

The survival function (SF) is given by

and the hazard rate functions is

For some selected values of the parameters

Now, we can obtain an useful representation for the PDF and CDF of the MLTIL distribution. Using the series representation in the form

Applying

where

Different structural properties of the MLTIL distribution can be determined using this representation. By integrating

where

In this section some statistical properties of the MLTIL distribution are obtained including quintile function, moments, incomplete moments, conditional moments and entropies.

For the MLTIL distribution the quantile function, say _{MLTIL}

We can easily generate

The

In particular,

The

The Sk and Ku measures can be computed using the following expressions:

The following propositions are a description of three different types of moments such as incomplete moments, moment generating function (mgf) and conditional moment.

where,

The first incomplete moment, _{1}(

Entropy has been used in areas like in physics (sparse kernel density estimation), medicin (molecular imaging of tumors) and engineering (measure the randomness of systems). The entropy is a measure of variation of the uncertainty. The R

For the MLTIL distribution in

The

Shannon’s entropy (SE) is defined as

Let

where

From

Particularly, we can obtain the PDF of the first and last order statistics from

and

The

Based on

where

The

Based on ^{n}, we have

The

Based on ^{n}, we can write

Let _{1} represents stress and _{2} represents strength, then the stress-strength parameter, denoted by

Using series expansion in the last equation, we obtain

In this section, six estimation methods are considered to estimate the MLIIL distribution parameters.

Using a random sample of size

where

It is observed that these equations cannot be solved analytically for

Kao et al. [

where _{(j)} is the ordered observation of _{j}_{j}

Cheng et al. [

To obtain the MPS estimates (MPSEs) of

with respect to

and

where

Swain et al. [

with respect to

The AD method of estimation is a type of minimum distance estimator which obtained by minimizing an AD statistic. The AD estimates (ADEs) of

We cannot compare the performance of the different proposed estimators theoretically, therefore a simulation study is done in order to show the behavior of the various estimators in terms of mean square error (MSE) criteria. To conduct the simulation study, we choose two sets of the parameters values; Set I:

Par | MLEs | PEs | MPSEs | LSEs | WLSEs | ADEs | |
---|---|---|---|---|---|---|---|

20 | 1.1604 | 1.4832 | 2.2886 | 1.6726 | 1.7763 | 1.6367 | |

2.0005 | 1.7900 | 3.0197 | 2.0821 | 2.1711 | 1.7594 | ||

0.8496 | 1.5604 | 0.6775 | 0.6466 | 0.6219 | 0.6995 | ||

0.5289 | 3.4964 | 0.4734 | 0.2771 | 0.2380 | 0.2503 | ||

1.1459 | 1.9455 | 0.6493 | 1.0761 | 1.0240 | 0.9762 | ||

1.5818 | 6.4506 | 0.5248 | 1.4903 | 1.3346 | 0.5239 | ||

50 | 1.3942 | 1.7430 | 2.1332 | 1.5095 | 1.5987 | 1.4844 | |

1.5724 | 1.7501 | 2.2973 | 1.4051 | 1.3911 | 1.3195 | ||

0.6236 | 1.5054 | 0.5288 | 0.5814 | 0.5782 | 0.5922 | ||

0.0631 | 3.3498 | 0.0350 | 0.0780 | 0.2102 | 0.0315 | ||

0.9662 | 2.0991 | 0.7091 | 0.8991 | 0.8654 | 0.8724 | ||

1.0591 | 5.7922 | 0.4954 | 0.7303 | 0.7310 | 0.4146 | ||

100 | 1.4189 | 1.5875 | 1.8143 | 1.4026 | 1.4560 | 1.3970 | |

1.5217 | 1.1934 | 1.6017 | 1.0632 | 1.1859 | 1.0429 | ||

0.5878 | 1.2454 | 0.5277 | 0.5654 | 0.5675 | 0.5694 | ||

0.0368 | 2.3354 | 0.0217 | 0.0256 | 0.0287 | 0.0259 | ||

0.8933 | 2.0128 | 0.7322 | 0.8138 | 0.8228 | 0.8032 | ||

0.6525 | 4.4985 | 0.3852 | 0.5253 | 0.4945 | 0.3550 | ||

150 | 1.4509 | 1.4629 | 1.7503 | 1.4787 | 1.4674 | 1.4642 | |

0.9781 | 0.7618 | 0.8992 | 0.6851 | 0.7495 | 0.6072 | ||

0.5434 | 1.2519 | 0.5050 | 0.5466 | 0.5383 | 0.5454 | ||

0.0143 | 1.8435 | 0.0101 | 0.0155 | 0.0116 | 0.0100 | ||

0.7426 | 1.6745 | 0.6484 | 0.7334 | 0.7068 | 0.7283 | ||

0.3446 | 3.1477 | 0.2432 | 0.3190 | 0.2813 | 0.2344 | ||

200 | 1.4645 | 1.4999 | 1.5109 | 1.4550 | 1.4883 | 1.4380 | |

0.5327 | 0.3899 | 0.4489 | 0.3856 | 0.4176 | 0.4334 | ||

0.5324 | 1.1448 | 0.5076 | 0.5249 | 0.5270 | 0.5256 | ||

0.0057 | 1.3608 | 0.0047 | 0.0056 | 0.0050 | 0.0045 | ||

0.5978 | 1.4511 | 0.5545 | 0.5793 | 0.5919 | 0.5840 | ||

0.0489 | 2.1426 | 0.0366 | 0.0498 | 0.0499 | 0.0356 | ||

250 | 1.4734 | 1.4186 | 1.5640 | 1.4916 | 1.4830 | 1.4618 | |

0.3037 | 0.1825 | 0.2893 | 0.2405 | 0.2665 | 0.2273 | ||

0.5137 | 1.0331 | 0.4957 | 0.5208 | 0.5183 | 0.5168 | ||

0.0031 | 0.7968 | 0.0027 | 0.0065 | 0.0042 | 0.0016 | ||

0.5482 | 1.0652 | 0.5314 | 0.5476 | 0.5514 | 0.5470 | ||

0.0304 | 1.0735 | 0.0260 | 0.0367 | 0.0347 | 0.0325 |

Par | MLEs | PEs | MPSEs | LSEs | WLSEs | ADEs | |
---|---|---|---|---|---|---|---|

20 | 1.2809 | 1.3543 | 0.9743 | 1.3234 | 1.2492 | 1.1243 | |

2.0139 | 2.4882 | 1.2214 | 1.9640 | 1.8293 | 1.1965 | ||

3.3311 | 3.2602 | 2.5432 | 2.3978 | 2.5560 | 2.6517 | ||

3.8339 | 4.3882 | 2.4643 | 2.0164 | 2.2287 | 2.0110 | ||

1.4935 | 2.6633 | 2.3092 | 2.3737 | 2.2743 | 2.2358 | ||

1.4649 | 3.9371 | 1.5313 | 1.9404 | 1.5747 | 1.4051 | ||

50 | 1.1317 | 1.3206 | 0.8494 | 0.9740 | 0.9650 | 0.9242 | |

1.0138 | 1.3307 | 0.6400 | 0.7398 | 0.7357 | 0.6187 | ||

2.7140 | 2.6147 | 2.1473 | 2.2559 | 2.3143 | 2.2754 | ||

1.5975 | 2.8690 | 0.8521 | 1.1503 | 1.1863 | 0.8238 | ||

1.6998 | 2.5816 | 2.2930 | 2.1570 | 2.0636 | 2.0997 | ||

1.3562 | 3.6776 | 1.2410 | 1.2698 | 1.1682 | 1.0776 | ||

100 | 0.8234 | 1.1222 | 0.6592 | 0.8100 | 0.7650 | 0.7797 | |

0.4265 | 0.8565 | 0.2745 | 0.4146 | 0.3906 | 0.2487 | ||

2.2291 | 2.6479 | 1.9685 | 2.0719 | 2.0772 | 2.1111 | ||

0.3826 | 1.9660 | 0.2800 | 0.3953 | 0.3289 | 0.2660 | ||

1.9395 | 3.0038 | 2.3626 | 2.0385 | 2.0961 | 2.0336 | ||

0.9087 | 2.3214 | 1.0649 | 0.5601 | 0.6380 | 0.5142 | ||

150 | 0.7437 | 0.8321 | 0.6380 | 0.7068 | 0.7616 | 0.7487 | |

0.2403 | 0.3895 | 0.1988 | 0.2219 | 0.2553 | 0.1844 | ||

2.1763 | 2.5740 | 1.9825 | 1.9978 | 2.0643 | 2.0714 | ||

0.2268 | 1.1945 | 0.1585 | 0.2610 | 0.2258 | 0.1472 | ||

1.9431 | 2.6172 | 2.2111 | 2.0253 | 1.9159 | 1.9129 | ||

0.5857 | 1.2741 | 0.6777 | 0.3739 | 0.3871 | 0.2687 | ||

200 | 0.5175 | 0.7575 | 0.4761 | 0.5723 | 0.5499 | 0.5341 | |

0.1186 | 0.3659 | 0.0951 | 0.1494 | 0.1359 | 0.0943 | ||

2.1460 | 2.3308 | 2.0230 | 2.0886 | 2.0479 | 2.0688 | ||

0.1172 | 0.4712 | 0.0895 | 0.1548 | 0.1142 | 0.1085 | ||

2.0274 | 2.1949 | 2.2052 | 2.0030 | 2.0005 | 2.0426 | ||

0.3106 | 0.4493 | 0.3647 | 0.1699 | 0.1822 | 0.1401 | ||

250 | 0.5489 | 0.8514 | 0.5335 | 0.6153 | 0.5820 | 0.5787 | |

0.0747 | 0.3122 | 0.0729 | 0.1174 | 0.0926 | 0.0642 | ||

2.0506 | 2.1351 | 1.9710 | 2.0067 | 2.0270 | 2.0271 | ||

0.0645 | 0.1379 | 0.0575 | 0.0745 | 0.0693 | 0.0570 | ||

1.9786 | 2.1614 | 2.0857 | 1.9838 | 1.9876 | 1.9758 | ||

0.0970 | 0.1514 | 0.1222 | 0.0749 | 0.0816 | 0.0649 |

To show the applicability of the proposed distribution and the different estimators derived in the previous sections two real data sets are analyzed. The first data set considered by [

Data I | Data II | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

10 | 17 | 23 | 40 | 71 | 107 | 0.33 | 1.02 | 6.52 | 27.4 | 92.9 |

14 | 18 | 23 | 49 | 74 | 107 | 0.33 | 1.17 | 7.25 | 27.43 | |

14 | 20 | 24 | 51 | 75 | 116 | 0.5 | 1.72 | 8.58 | 31.93 | |

14 | 20 | 26 | 52 | 87 | 150 | 0.5 | 1.83 | 10.25 | 38.37 | |

14 | 20 | 30 | 60 | 96 | 0.5 | 3.2 | 11.58 | 40.02 | ||

14 | 20 | 30 | 61 | 105 | 0.95 | 4.35 | 13.83 | 62.77 | ||

15 | 20 | 31 | 67 | 107 | 1 | 5.25 | 15.93 | 88.27 |

We compare the results of the MLTIL distribution with IL distribution, inverse Weibull (IW) distribution, APIL distribution by [

We first obtain the MLEs of the competitive distributions. These estimates are presented in

Model | Estimates | K-S | |||
---|---|---|---|---|---|

IL( |
35.616 | 1.594 | 0.3708 | 0.0000 | |

IW( |
1.531 | 24.961 | 0.6187 | 0.0000 | |

MLTIL( |
0.00022 | 20.3948 | 6.8869 | 0.1172 | 0.6574 |

APIL( |
1.552 | 148.947 | 1.594 | 0.1321 | 0.5044 |

APIW( |
0.063 | 103.324 | 0.467 | 0.1250 | 0.5762 |

Model | Estimates | K-S | |||
---|---|---|---|---|---|

IL( |
3.308 | 0.352 | 0.3595 | 0.0011 | |

IW( |
0.634 | 2.195 | 0.1733 | 0.3486 | |

MLTIL( |
52.57694 | 41.521 | 0.01526 | 0.0958 | 0.9531 |

APIL( |
3.408 | 1.218 | 0.703 | 0.1083 | 0.8855 |

APIW( |
0.745 | 0.887 | 7.600 | 0.1217 | 0.7839 |

To see which estimation method provide a good fit to these data we compare the other estimation methods with the maximum likelihood methods based on K-S distance and its

Model | K-S | ||||
---|---|---|---|---|---|

MLEs | 0.00022 | 20.3948 | 6.8869 | 0.1172 | 0.6574 |

PEs | 0.0009 | 128.205 | 0.3341 | 0.5432 | 0.0000 |

MPSEs | 0.1975 | 137.030 | 0.3359 | 0.1621 | 0.2571 |

LSEs | 0.0774 | 191.701 | 0.2528 | 0.1221 | 0.6062 |

WLSEs | 0.0361 | 108.2163 | 0.5261 | 0.1150 | 0.6810 |

ADEs | 0.0009 | 20.8313 | 5.6234 | 0.1148 | 0.6827 |

Model | K-S | ||||
---|---|---|---|---|---|

MLEs | 52.57694 | 41.521 | 0.01526 | 0.0958 | 0.9531 |

PEs | 0.1265 | 51.1731 | 0.1911 | 0.3590 | 0.0011 |

MPSEs | 23.8235 | 2.175 | 0.5229 | 0.1337 | 0.6781 |

LSEs | 0.0097 | 1.035 | 49.1779 | 0.0974 | 0.9460 |

WLSEs | 0.0017 | 1.4282 | 42.379 | 0.0837 | 0.9872 |

ADEs | 0.0010 | 1.5226 | 42.4419 | 0.0833 | 0.9878 |

In this paper, we have considered and studied a new generalization of the traditional inverse Lomax distribution by adding a new shape parameter. We have used the logarithmic transformed method for this purpose and a new three parameters inverse Lomax distribution which called modified logarithmic transformed inverse Lomax distribution is introduced. The new distribution has different failure rate shapes, so it can be used in analyzing lifetime data. Some statistical properties of the new distribution are derived including quantiles, moments, probability weighted moments, entropies, residual life, stress-strength parameter and order statistics. To estimate the parameters of the proposed distribution, six classical methods are considered. To compare the efficiency of these methods a simulation study is performed and the performance of the different estimators is compared. To show the applicability of the new distribution, two real data sets are analyzed which indicate that our new distribution perform better than some other competitive distributions. Also, the numerical illustration revealed that the Anderson Darling estimation method is the best method to estimate the proposed distribution parameters. In this study, the proposed distribution shows its ability in modelling engineering and medical data sets where traditional and some recently proposed models cannot be used for this purpose. We hope that this model attract wider sets of applications in the other different fields.

The author would like to thank the editor and the two reviewers for their constructive comments which improve the quality of the paper.