Inverse Length Biased Maxwell Distribution: Statistical Inference with an Application

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 CramerVon-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.


Introduction
A random variable W follows a Maxwell distribution with scale parameter a, if its probability density function (pdf) and cumulative distribution function (cdf), respectively, are given by f MD ðw; aÞ ¼ where Erf ðzÞ ¼ 2 ffiffiffi p p Z z 0 e À t 2 dt. In the literature of the probability distributions, uni-modal and skewed to the right characteristics are very important to the distribution of interest. One of these distributions is the Maxwell distribution, which is a well-known lifetime distribution in physics and statistical mechanics. Reference Iriarte et al. [1] suggested a gamma-Maxwell distribution. The tail behavior of the generalized Maxwell distribution is considered by Huang et al. [2]. Recently, Saghir et al. [3] suggested a lengthbiased Maxwell distribution (LBMD) as a modification of the base Maxwell distribution based on the weighted distribution suggested by Rao et al. [4], to obtain the probability density function given by: and a cumulative distribution function defined as F LBMD ðw; aÞ ¼ 1 À w 2 2a 2 þ 1 e À w 2 2a 2 ; 0 , w , 1; a > 0: (4) The mode and median of the LBMD are w M ¼ a ffiffi ffi 3 p and EðW Þ ¼ 3a 4 ffiffiffi 2 p r , respectively. For more information about the LBMD see [3]. Due to the large number of data in these times, a large number of distributions are suggested based several philosophies, assuming that the suggested distributions are more flexible in modeling data. For example, [5] introduced Marshall-Olkin length-biased Maxwell distribution. Reference Singh et al. [6] suggested length-biased weighted Maxwell distribution and [7] considered estimation of the inverse Maxwell distribution parameter. Reference Garaibah et al. [8] suggested size-biased Ishita distribution and [9] introduced transmuted Ishita distribution. The Marshall-Olkin length-biased exponential distribution is proposed by Shraa et al. [10]. A new mixture continuous Darna distribution is suggested by Al-Omari et al. [11,12] proposed length-biased Suja distribution. Reference Sharma et al. [13] studied the power size biased two-parameter Akash distribution with some statistical properties and real data applications. Reference Al-Omar et al. [14] proposed length and area biased Maxwell distributions. Reference Gharaibeh [15] suggested Top-Leone Mukherjee-Islam distribution and [16] proposed transmuted Aradhana distribution.
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.

Derivation of the Suggested Model
If a random variable W has a LBMD with pdf given in (3), then the random variable X ¼ 1 W is said to follow the inverse LBMD. The pdf and cdf of the inverse length-biased Maxwell distribution (ILBMD), respectively are given by and Plots of the pdf and cdf of the ILBMD are presented in Fig. 1 for various distribution parameter. Fig. 1, revealed that the pdf of the suggested distribution is skewed to the right and be more flatting as a values are increasing. Also, the pdf of the ILBMD can exhibit various behavior depending on the values of the parameter.

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

Reliability Analysis
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. Figs. 2 and 3 are the plots of the hazard, reliability reversed hazard, and the odds functions of the ILBMD for a ¼ 0:1; 0:2; 0:3; 0:4; 0:5. Hazard rate function: The hazard rate (HR) function is a very important property in characterizing any lifetime distribution. The HR of the ILBMD is given by To determine the shape of the HR function we followed the technique of [17] which is defined as  Fig. 2A. Therefore, the proposed ILBMD is useful in reliability data and medical fields due its skewness to the right with UBT shape of hazard rate function.
Reliability function: The reliability function of the ILBMD distribution is Fig. 2B shows that the reliability plots of the ILBMD intersect at the point f ðxÞ ¼ 1 for x ¼ 0, while as x goes to infinity, the reliability function decreases and goes to zero.
Reversed hazard function: The reversed hazard function of the ILBMD is defined as Odds function: The odds function of the ILBMD is given by Based on Fig. 3 it can be noted that the reversed hazard decreases with negative J-shaped distribution, while the odds function increases taking the J-shaped with more flatting for small amounts of the parameter a.

The Mode
In this section, the mode of the ILBMD is derived. Since the pdf f ILBMD ðx; aÞ of the model and its logarithm are maximized at the same point, then for simple calculation, take the derivative of the logarithm of the function f ILBMD ðx; aÞ as The derivative of Ç with respect to x yields @Ç @x x 3 : Equating the preceding derivative to 0 leads to À 5 But since a > 0, then the mode of the ILBMD is : It is clear that the distribution is a unimodal and the mode decreases with increases values of α.

Moments and Quantile Function
In this section, we derived the various moments of the suggested ILBMD as Let X $ f ILBMD ðx; aÞ, then the rth moment of X is If X $ f ILBMD ðx; aÞ, then the moment generating function of X is where MeijerG a 1 . . . a n a nþ1 . . .
From Eq. (11), the first and second moments of the ILBMD, respectively, are given as and E X 2 The variance of the ILBMD is The degree of long-tail is measured by skewness (Sk) and for the ILBMD it is given by The coefficient of variation of the ILBMD is which is a very small value.
Let X $ f ILBMD ðx; aÞ, then the rth order inverse moment about the origin of X is Based on Eq. (6), the harmonic mean of the ILBMD distribution can be obtained for r ¼ 1 as given by If QðkÞ is the quantile function of order k of the ILBM random variable, then it can be the solution of the equation The Moors and Bowley measures of kurtosis and skewness, respectively, are given by

Fishers Information
Theorem: Let X $ f ILBMD ðx; aÞ, then the Fisher's information of a is FI ILBMD ðaÞ ¼ 8 a 2 : Proof: To find the Fisher's information of the ILBMD, we have The first derivative of this function with respect to ayields Again differentiate the last equation with respect to a to get Now, take the expectation of @ 2 ln f ILBMD ðx; aÞ @a 2 as This information is very helpful in determining the variance of estimator or lower bound of an estimator.

Order Statistics
Let X ð1:nÞ ; X ð2:nÞ ; …; X ðn:nÞ be the order statistics of the random sample X 1 ; X 2 ; …; X n selected from a pdf and cdf f ILBMD ðx; aÞ and F ILBMD ðx; aÞ, respectively. The pdf of the ith order statistics say X ði:nÞ , is and the corresponding cdf is defined as

Stochastic Ordering
The stochastic ordering can be considered to compare the behavior of two random variables. Let X and Y be two random variables, then X is said to be smaller than in Based on these relations, we have X Proof: Based on the concept of the likelihood ratio order, we have where its logarithm is The first derivative of this equation with respect to x is

Mean and Median Deviations
This section, introduced the mean and median deviations of the ILBMD, È l and È M , respectively. It is a measure of the scatter in the population the mean deviation about the mean and the mean deviation about the median, where where l and M are the population mean and median, respectively.
Theorem: Let X $ f ILBMD ðx; aÞ;the mean and median deviations about the mean and median, respectively, are where Erfc is the complementary function of the error function erfc, and Since the median of the ILBMD is The qth quantile x q of the ILBMD can be found by solving the equation q ¼ F ILBMD x q ; a À Á , that is q and the quantile is the solution of the lnðqÞ ¼ ln

Gini Index and Some Curves
Let X be a non-negative random variable with a continuous twice differentiable cumulative distribution function FðxÞ. In this section, we want to obtain the Gini index, Bonferroni and Lorenz curves for the ILBMD. The Gini coefficient is developed by the Italian statistician Gin in (1912). The Gini index measures the inequality among values of a frequency distribution, for example levels of total income. The Gini index value running from zero to one. A Gini index of zero value indicates perfect equality (that is all values are the same, a group has the same monthly income), while most unequal group where a single person receives Gini index of 1 of the total.
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 q ¼ F À1 ðpÞ and p 2 ð0; 1.
The Lorenz curve for the ILBMD is defined as

Stress-Strength Reliability
Let X and Y be independent random variables observed from the pdf f ðxÞ. The stress-strength reliability clarify the life of a component that has a random strength Y which is subjected to a random stress X, where Theorem: Let the random variables X and Y be independent selected from the ILBMD. The stressstrength reliability is given by

Entropies
The Rényi entropy is defined as where g > 0 and g 6 ¼ 1.
Let X $ f ILBMD ðx; aÞ, then the and Rényi entropy of X is defined as The q-entropy, say Q E ðqÞ is given by f ðxÞ q dx 0 @ 1 A ; q > 0; q 6 ¼ 1. For the ILBMD we have :

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

Maximum Likelihood Method
Let X 1 ; X 2 ; …; X n be a random sample of size n selected from the ILBMD with parameter a > 0. The maximum likelihood estimator for the ILBMD parameter can be derived based on the likelihood function as: The log likelihood function is given by The derivative of É with respect to a is É 0 ¼ À 4n a þ

Method of Moments
The mean of the ILBMD random variable is E X 1

Cramèr-von-Mises Estimation
Let x 1 ; x 2 ; …; x n be the observed values of a random sample of size n selected from a the f ILBMD ðx; aÞ, and let x ð1:nÞ ; x ð2:nÞ ; …; x ðn:nÞ be the order statistics of the sample. The Cramèr-von Mises estimation method (Cv) is suggested by Swain et al. [18]. The Cv method is based on the minimum difference between the cumulative and empirical distribution functions. The Cv estimator of the parameter can be calculated by minimizing For the proposed ILBMD, the Cv estimator,â Cv of a, can be obtained by minimizing the equation with respect to a.

Ordinary and Weighted Least Squares Methods
Let X ð1:nÞ ; X ð2:nÞ ; …; X ðn:nÞ be the order statistics of the random sample X 1 ; X 2 ; …; X n selected from a the f ILBMD ðx; aÞ. The least square estimator (LSE) [19] can be obtained by minimizing the residual sum of the square, which is defined as the differences of theoretical cdf and empirical cdf as For the ILBMD, the LS estimator,â LS of a can be obtained by minimizing the equation with respect to a. Similarly, the weighted least squares (WLS) estimate of a denoted byâ WLS can be obtained by minimizing the function with respect to a. For the ILBMD we have with respect to a.

Method of Maximum Product of Spacing
The maximum product of spacing (MPS) method as suggested by Cheng et al. [20,21] is a powerful alternative to the MLE method for estimating the parameters of continuous distributions. Reference Shanker et al. [22] showed that the MPS method possess similar properties as the MLE method.

Simulation Study
In this section, we compared the various suggested estimators of the model parameters. We selected the values of the parameters a ¼ 1;2,3 with samples sizes n ¼ 10;20,40,60, 80, 100 and 200. The results are presented in Tabs. 1 and 2 foe the parameter estimates (Es) and the corresponding mean squared errors (MSE).  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.

Real Data Application
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. [23] with pdf given by 3) Rama distribution (Rm) suggested by Gross et al. [24] with pdf given by f RD ðx; aÞ ¼ a 4 a 3 þ 6 x 3 þ 1 À Á e Àax x > 0; a > 0: 4) Exponential distribution (Exp), f ðx; aÞ ¼ a e À ax ; x > 0; a > 0: 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 AIC ¼ À2 log L þ 2m; CAIC ¼ À2 log L þ 2m n n À m À 1 ; BIC ¼ À2 log L þ j logðnÞ; 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 Fig. 4.

Conclusions
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 [26][27][28][29][30][31][32].