In this paper, we suggested and studied the inverse length biased Maxell distribution (ILBMD) as a new continuous distribution of one parameter. The ILBMD is obtained by considering the inverse transformation technique of the Maxwell length biased distribution. Statistical characteristics of the ILBMD such as the moments, moment generating function, mode, quantile function, the coefficient of variation, coefficient of skewness, Moors and Bowley measures of kurtosis and skewness , stochastic ordering, stress-strength reliability, and mean deviations are obtained. In addition, the Bonferroni and Lorenz curves, Gini index, the reliability function, the hazard rate function, the reverse hazard rate function, the odds function, and the distributions of order statistics for the ILBMD, are presented. The ILBMD parameter is estimated using the maximum likelihood method, the method of moments, the maximum product of spacing technique, the ordinary and weight least square procedures, and the Cramer-Von-Mises methods. The Fishers information, as well as the Rényi and q-entropies, are derived. To investigate the usefulness of the proposed lifetime distribution and to illustrate the purpose of the study, a real dataset of the relief times of 20 patients receiving an analgesic is used.

A random variable

where

and a cumulative distribution function defined as

The mode and median of the LBMD are

The rest of this paper is organized as follows: In Section 2, we present the derivation of the suggested distribution. Section 3 deals with the main statistical properties of the ILBMD. Different methods of estimation for the distribution parameter are given in Section 4. In Section 5, a simulation study is conducted to investigate the distribution. An application of real data is presented in Section 6 and the paper is concluded in Section 7.

If a random variable

and

Plots of the pdf and cdf of the ILBMD are presented in

In this section, the main properties if the proposed model are presented.

The reliability is a well-known in engineering where it gives the probability for surviving at least time t of a product operate, while the hazard function shows the nature of failure rate related to the product. Generally, the reliability and hazard functions are fundamental to study the characteristics of the time to event data.

To determine the shape of the HR function we followed the technique of [

Based on

In this section, the mode of the ILBMD is derived. Since the pdf

The derivative of

In this section, we derived the various moments of the suggested ILBMD as

Let

If

where

From

The variance of the ILBMD is

The degree of long-tail is measured by skewness (

The coefficient of variation of the ILBMD is

which is a very small value.

Let

Based on

If

The Moors and Bowley measures of kurtosis and skewness, respectively, are given by

The first derivative of this function with respect to

Again differentiate the last equation with respect to

Now, take the expectation of

This information is very helpful in determining the variance of estimator or lower bound of an estimator.

Let

and the corresponding cdf is defined as

The stochastic ordering can be considered to compare the behavior of two random variables. Let

1) Mean residual life order

2) Likelihood ratio order

3) Hazard rate order

4) Stochastic order

Based on these relations, we have

where its logarithm is

The first derivative of this equation with respect to

Now, if

This section, introduced the mean and median deviations of the ILBMD,

where

where Erfc is the complementary function of the error function erfc, and

Since the median of the ILBMD is

The _{q} of the ILBMD can be found by solving the equation

Let

The Gini index for the ILBMD is given by

It is clear that the Gini index value is small and it is about 0.24. The Bonferroni curve for the ILBMD is defined as

The Lorenz curve for the ILBMD is defined as

Let

The Rényi entropy is defined as

Let

The q-entropy, say

In this section, we discuss different estimation procedures for estimating the unknown suggested model parameter.

Let

The log likelihood function is given by

The derivative of

The mean of the ILBMD random variable is

Let

For the proposed ILBMD, the

with respect to

Let

For the ILBMD, the LS estimator,

with respect to

with respect to

with respect to

The maximum product of spacing (MPS) method as suggested by Cheng et al. [

Let uniform spacing’s of a random sample of size

where

Or, equivalently, by maximizing the function

Now, the MPS estimate of the ILBMD parameter

In this section, we compared the various suggested estimators of the model parameters. We selected the values of the parameters

MLE | CV | ||||||
---|---|---|---|---|---|---|---|

10 | 1 | 0.995924 | 0.011847 | 0.991184 | 0.012989 | 0.980571 | 0.013476 |

↑ | 2 | 1.981047 | 0.046646 | 1.971264 | 0.051318 | 1.973994 | 0.053759 |

↑ | 3 | 2.977570 | 0.103597 | 2.954931 | 0.134482 | 2.954117 | 0.124594 |

20 | 1 | 0.997368 | 0.005989 | 0.989289 | 0.007110 | 0.991652 | 0.005833 |

↑ | 2 | 1.998237 | 0.025302 | 1.982299 | 0.025994 | 1.978287 | 0.027209 |

↑ | 3 | 3.001566 | 0.054498 | 2.963013 | 0.064315 | 2.970309 | 0.059139 |

40 | 1 | 0.999047 | 0.003172 | 0.991716 | 0.003276 | 0.991394 | 0.003234 |

↑ | 2 | 1.998907 | 0.012212 | 1.993159 | 0.013333 | 1.983139 | 0.012917 |

↑ | 3 | 2.992754 | 0.026943 | 2.971542 | 0.032232 | 2.982549 | 0.030891 |

60 | 1 | 0.997824 | 0.002377 | 0.994213 | 0.002429 | 0.996004 | 0.002219 |

↑ | 2 | 2.002695 | 0.008112 | 1.991705 | 0.009028 | 1.985409 | 0.008619 |

↑ | 3 | 2.990985 | 0.018646 | 2.982247 | 0.020553 | 2.980146 | 0.019963 |

80 | 1 | 0.999361 | 0.001616 | 0.997906 | 0.001802 | 0.996385 | 0.001648 |

↑ | 2 | 1.997114 | 0.006034 | 1.994376 | 0.006188 | 1.993271 | 0.006873 |

↑ | 3 | 2.997377 | 0.013123 | 2.994897 | 0.014442 | 2.986911 | 0.014781 |

100 | 1 | 1.000008 | 0.001280 | 0.997574 | 0.001243 | 0.997230 | 0.001419 |

↑ | 2 | 1.995360 | 0.004856 | 1.998231 | 0.005106 | 1.993223 | 0.005391 |

↑ | 3 | 2.998770 | 0.010524 | 2.985528 | 0.012388 | 2.988051 | 0.010962 |

200 | 1 | 0.998602 | 0.000652 | 0.996814 | 0.000705 | 0.998669 | 0.000628 |

↑ | 2 | 2.002684 | 0.002847 | 1.995579 | 0.002652 | 1.996316 | 0.002650 |

↑ | 3 | 2.996923 | 0.005726 | 2.993280 | 0.006111 | 2.995057 | 0.006072 |

400 | 1 | 0.999881 | 0.000312 | 0.999214 | 0.000362 | 0.999735 | 0.000308 |

↑ | 2 | 1.999506 | 0.001259 | 1.997887 | 0.001297 | 1.999012 | 0.001315 |

↑ | 3 | 3.001122 | 0.002856 | 2.997404 | 0.003018 | 2.997452 | 0.003068 |

LS | WLE | MPS | |||||
---|---|---|---|---|---|---|---|

10 | 1 | 1.007925 | 0.015105 | 1.006955 | 0.014096 | 1.130497 | 0.033357 |

↑ | 2 | 2.012742 | 0.060097 | 2.005761 | 0.056658 | 2.247161 | 0.124619 |

↑ | 3 | 3.016037 | 0.143091 | 3.038500 | 0.133337 | 3.411764 | 0.314966 |

20 | 1 | 0.999859 | 0.007603 | 1.005104 | 0.006716 | 1.066338 | 0.011475 |

↑ | 2 | 2.006540 | 0.030309 | 2.006146 | 0.027890 | 2.137371 | 0.046024 |

↑ | 3 | 3.011123 | 0.062426 | 2.997492 | 0.059193 | 3.195782 | 0.104628 |

40 | 1 | 1.005808 | 0.003683 | 1.001200 | 0.003192 | 1.038072 | 0.004547 |

↑ | 2 | 2.001974 | 0.014647 | 1.995240 | 0.012789 | 2.075144 | 0.019910 |

↑ | 3 | 3.001535 | 0.033451 | 2.987103 | 0.030327 | 3.118552 | 0.046022 |

60 | 1 | 1.002426 | 0.002372 | 0.997947 | 0.002258 | 1.024475 | 0.002672 |

↑ | 2 | 2.000718 | 0.009925 | 1.998646 | 0.008446 | 2.047890 | 0.010844 |

↑ | 3 | 2.997560 | 0.022334 | 3.002908 | 0.020097 | 3.080202 | 0.024972 |

80 | 1 | 0.999352 | 0.001885 | 1.001006 | 0.001669 | 1.019037 | 0.001871 |

↑ | 2 | 1.998633 | 0.007305 | 1.995840 | 0.007117 | 2.037926 | 0.007875 |

↑ | 3 | 3.005454 | 0.017804 | 2.998899 | 0.015568 | 3.056238 | 0.018755 |

100 | 1 | 1.000232 | 0.001459 | 0.998625 | 0.001338 | 1.015747 | 0.001519 |

↑ | 2 | 2.001972 | 0.005673 | 2.002987 | 0.004788 | 2.030212 | 0.006199 |

↑ | 3 | 3.003503 | 0.014207 | 2.999699 | 0.011628 | 3.044901 | 0.013372 |

200 | 1 | 0.998634 | 0.000707 | 1.000297 | 0.000622 | 1.009054 | 0.000715 |

↑ | 2 | 2.001705 | 0.002883 | 1.996734 | 0.002457 | 2.014366 | 0.002870 |

↑ | 3 | 2.994524 | 0.006424 | 3.001192 | 0.005964 | 3.028472 | 0.006385 |

400 | 1 | 1.007925 | 0.015105 | 0.999890 | 0.000336 | 1.130497 | 0.033357 |

↑ | 2 | 2.012742 | 0.060097 | 2.000354 | 0.001277 | 2.247161 | 0.124619 |

↑ | 3 | 3.016037 | 0.143091 | 3.000917 | 0.002992 | 3.411764 | 0.314966 |

The bias of the suggested estimators is very small and goes to zero for all cases considered in this study. Also, as the samples sizes increase the MSE of all proposed estimators decreases.

In this section, we use lifetime data set to compare the fit of the suggested ILBMD distribution with four competitors distributions: Rani, length-biased Maxwell distribution, Rama, and exponential defined as

1) Rani distribution (Rn) suggested by Shanker et al. [

2) Length-biased Maxwell distribution (LBM),

3) Rama distribution (Rm) suggested by Gross et al. [

4) Exponential distribution (Exp),

The data set given in this section represents the relief times of 20 patients receiving an analgesic. This data set was taken from [

In order to compare the two models, we consider the Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Hannan-Quinn Information Criterion (HQIC), and Bayesian Information Criterion (BIC). The generic formulas for finding AIC, CAIC, HQIC, and BIC are respectively, given as

ILBM | Rm | Exp | Rn | LBM | |
---|---|---|---|---|---|

0.30091 | 1.52130 | 0.52617 | 1.71928 | 1.01012 | |

0.02379 | 0.15231 | 0.11766 | 0.12778 | 0.07986 | |

35.43390 | 61.70660 | 67.67416 | 67.30852 | 40.61917 | |

35.65611 | 61.92882 | 67.89638 | 67.53074 | 40.84139 | |

36.42962 | 62.70233 | 68.66989 | 68.30425 | 41.61490 | |

35.62827 | 61.90097 | 67.86853 | 67.50290 | 40.81354 | |

16.71695 | 29.85330 | 32.83708 | 32.65426 | 19.30958 | |

0.15809 | 0.35667 | 0.43942 | 0.35355 | 0.17891 | |

0.69950 | 0.01233 | 0.00088 | 0.01348 | 0.54400 |

Hence, we can deduce that the inverse length biased Maxwell distribution leads to a better fit than the Rama, Rani, length biased Maxwell and exponential distribution. The Kolmogorov Smirnov p-value suggests that inverse length biased Maxwell distribution fits statistically better than other distributions considered in this example to the 20 patients data set. Plots of the fitted densities and the histogram are given in

In this article, we introduced and studied the ILBMD. Some statistical properties of the ILBMD are derived and discussed. The reliability and hazard functions of the distribution are analyzed. Also, the distribution of order statistics, mode, harmonic mean, Fisher's information, the stochastic ordering and the mean deviations about the mean and median are presented. The distribution parameter is estimated using different estimation methods includes the maximum likelihood estimation, method of moments, maximum product of spacing, ordinary and weight least square procedures, and the Cramer-Von-Mises methods. The q and Rényi entropies are derived as well as the stress strength reliability is obtained. A real data sets is considered to support the paper objectives. It is revealed that the ILBMD is more power than its competitors used in this study. As a future works the distribution parameter can be estimated based on ranked set sampling method, see [

The authors are grateful to the Editor and anonymous reviewers for their valuable comments and suggestions.