Generalization of Two-Sided Length Biased Inverse Gaussian Distributions and Applications

: The notion of length-biased distribution can be used to develop adequate models. Length-biased distribution was known as a special case of weighted distribution. In this work, a new class of length-biased distribution, namely the two-sided length-biased inverse Gaussian distribution (TS-LBIG), was introduced. The physical phenomenon of this scenario was described in a case of cracks developing from two sides. Since the probability density function of the original TS-LBIG distribution cannot be written in a closed-form expression, its generalization form was further introduced. Important properties such as the moment-generating function and survival function cannot be provided. We offered a different approach to solving this problem. Some distributional properties were investigated. The parameters were estimated by the method of the moment. Monte Carlo simulation studies were carried out to appraise the performance of the suggested estimators using bias, variance, and mean square error. An application of a real dataset was presented for illustration. The results showed that the suggested estimators performed better than the original study. The proposed distribution provided a more appropriate model than other candidate distributions for ﬁtting based on Akaike information criterion.


Introduction
Recorded observations may not have original distributions when practitioners collect natural observations according to certain stochastic models. Each observation is taken with unequal probabilities of recording. Weighted distributions can be adopted in this situation for selecting appropriate models [1]. One of the most widely known for special cases of weighted distributions is length-biased distributions. Precisely, let X denote a non-negative random variable with a probability density function shortly called PDF or f X (x). The weighted version of X denoted by X w has a PDF defined as where w(x) is the weighted function and 0 < E[w(x)] < ∞. Different weighted models are formulated depending on choices of the weight function w(x). In cases of w(x) = x, the resulting distribution is called length-biased whose PDF is defined by

TS-LBIG Random Variable
In this section, the TS-LBIG random variable (τ) introduced by Simmachan et al. [20] is described. Let X be a non-negative continuous random variable and let F(x) = F LBIG (x, λ, θ) denote the distribution function of the breakdown time moment τ for one-sided loading. The parameters λ and θ were previously defined. Let Y = k/τ be the random variable denoted as a crack speed. Under the object consideration, a crack expands from two sides with the same distribution function of the time to approach the length k. The random variables from both sides, τ 1 and τ 2 , are supposed to be independent and identically distributed. The crack speed for the two-sided situation is definded as The breakdown moment of the interested object is defined as the following random variable
Proof of Proposition 1. For the reciprocal random variable 1/τ, the distribution function is given by and applying the chain rule, the density function is Proof of Proposition 2. By Proposition 1, Proposition 3. If the random variable τ > 0 has IG(λ, θ) distribution, then the reciprocal random variable 1/τ is LBIG[λ, 1/(λ 2 θ)] distributed.
Proof of Theorem 1. We know that a moment-generating function (MGF) of IG(λ, θ) of a random variable τ is defined as Now, we have two independent LBIG random variables τ 1 and τ 2 . That is, Initially, we find the MGF of the random variable X = τ −1 1 + τ −1 2 . According to Proposition 2, if the random variable τ > 0 has LBIG(λ, θ) distribution, then the reciprocal random variable 1/τ is IG[λ, 1/(λ 2 θ)] distributed. Therefore, Theorem 2. If a random variable Y ∼ TS-LBIG(λ, θ), the moment generating function of Y is given as Proof of Theorem 2. By Theorem 1, it is known that X ∼ IG[2λ, 1/(λ 2 θ)]. To find the MGF of the TS-LBIG random variable Y, the reciprocal of X is considered.

The Probability Density Function of TS-LBIG Distribution
Theorem 3. Define X as a random variable of the TS-LBIG distribution. Then, the corresponding probability density function (PDF) of X is given by . (12) Several shapes of the PDF for the TS-LBIG distribution are illustrated in Figures 1 and 2 for various parameter values. The different shapes indicate that the TS-LBIG distribution is right-skewed and unimodal. Moreover, this distribution is a family of asymmetric distributions which are useful for skewed data analysis.

The Cumulative Density Function of TS-LBIG Distribution
Theorem 4. Let X be a random variable of the TS-LBIG distribution. The cumulative density function (CDF) of X is given by where Φ(x) is the standard normal distribution function.

Proof of Theorem 4. The CDF of LBIG distribution is
Hence, the CDF of TS-LBIG is where Φ(x) is the standard normal distribution function.

The Survival Function of TS-LBIG Distribution
Theorem 5. Let X be a random variable of the TS-LBIG distribution with parameters λ and θ.
The survival function of X is obtained as: where Φ(x) is the standard normal distribution function.
Proof of Theorem 5. Let X be a continuous random variable with a cumulative density function F(x) on the interval [0, ∞). The survival function of X can be written in this form: Inserting Equation (13) into Equation (17) leads to the survival function of TS-LBIG distribution in equation: where Φ(x) is the standard normal distribution function.

The Hazard Rate Function of TS-LBIG Distribution
Theorem 6. Let X be a random variable of the TS-LBIG distribution with parameters λ and θ.
The hazard rate function of X is given by where Φ(x) is the standard normal distribution function.
Proof of Theorem 6. Let X be an absolutely continuous non-negative random variable with the probability density function f (x) and the survival function S(x); then, the hazard rate function of X can be defined as: where Φ(x) is the standard normal distribution function.

The Mean and the Variance of TS-LBIG Distribution
From [22], let Y ∼ IG(x; λ, θ) and Z ∼ LB(x; λ, θ); then, the rth moment of Z for r = 1, 2, 3, . . . is given by Hence, the first four raw moments of the LBIG distribution are Let X ∼ TS-LBIG. Therefore, the first four raw moments of TS-LBIG distribution are Therefore, the mean of TS-LBIG distribution is and the variance of TS-LBIG distribution is

Parameter Estimation by the Method of Moments for the TS-LBIG Distribution
Recall that the rth raw population moment is equal to the rth raw sample moment.

Simulation Study
In this section, the Monte Carlo simulation to test the performance of the suggested estimators of the TS-LBIG distribution parameters is presented. Different values of the true parameters are considered. All 60 scenarios are the combination of sample size (n) = 10, 50 and 100, λ = 1, 3, 5 and 10, and θ = 0.5, 1, 3, 5 and 10. The proposed estimators, λ andθ, are compared to the estimators presented by Simmachan et al. [20],λ andθ, via bias, MSE and variance. The random numbers of the TS-LBIG distribution are generated via the composition method using the "twoCrack" package [23] in R [24], and the replications are repeated 1000 times in each scenario. The parameter estimates with their bias, MSE and variance are reported in Tables 1-6. For easier consideration, bar charts are created and presented in Figures 3 and 4. The blue and yellow bars represent the proposed method and the method of Simmachan et al. [20], respectively. It reveals that the bias, MSE and variance become smaller as the sample size increases and the estimates become closer to the true value of parameters. For bias consideration, the estimators of λ give over-estimates for both methods. The bias of the proposed estimator is slightly smaller than that of the original estimator. On the other hand, the estimators of θ provide mostly under-estimates for both methods. The bias of the proposed estimator is much smaller than that of the original estimator. MSE and variance indicators have similar behavior. For parameter λ, the MSE and variance of the proposed estimator are slightly smaller than those of the original estimator. For parameter θ, however, the MSE and variance of the proposed estimator are much smaller than those of the original estimator. Interestingly, the bias, MSE and variance of the proposed estimators are superior to those of the estimators from the previous study.

Illustrative Examples
In this section, the suggested distribution is implemented via a real dataset. The following data are collected from the BackBlaze data center [25], and they present the lifetime of the hard drives (in days) containing only the model ST8000DM002 in December 2017: 490, 497, 521, 394 489, 323, 376, 319, 431, 484, 547, 383, 534 and 316. This dataset was analyzed by Chananet and Phaphan [26], and the result indicated that the lifetime of the hard drives follow the right skewed distribution. Consequently, five right skewed distributions-two-parameter crack [18], Birnbaum-Saunders [16], inverse Gaussian [17], length-biased inverse Gaussian [17], and the proposed two-sided length-biased inverse Gaussian-are selected for goodness of fit comparison. The parameters of the TS-LBIG distribution are estimated by the suggested estimators. The parameters of other candidate distributions are estimated via the maximum likelihood estimation. The "nlminb" function in R [24] is employed for maximizing their likelihood functions. The Akaike information criterion (AIC) is used as an assessment criterion; hence, the best model is the one that provides the minimum AIC. As the result in Table 7 indicates, the TS-LBIG distribution gives the minimum AIC. This indicates that the proposed distribution is the best of the candidate distributions by considering at the value of AIC. Hence, by Equations (24) and (25), the average and standard deviation of the lifetime of the hard drives are 436 and 80.15877 days, respectively.

Conclusions and Discussion
In this article, a new form of the TS-LBIG distribution is introduced, since the original version offered by Simmachan et al. [20] does not present a closed-form PDF. This distribution is a right-skewed distribution. Some distributional properties of this distribution were studied, and its two parameters were estimated using the method of moment. Sixty combination scenarios are used to construct the simulation study in assessing the performance of the proposed method. An application of the TS-LBIG distribution was implemented in the lifetime of the hard drives. Results show that the proposed estimators are more efficient than the Simmachan et al. [20] estimators. The original study dealing with the indirect method of parameter estimation affects the parameter estimates far from the true values, especially the parameter θ. This is different from the proposed estimators that dealt with the direct method. The TS-LBIG distribution gives a better fit than the other candidate distributions in terms of AIC. Our contribution provides an alternative right-skewed distribution that can be applied in other aspects such as survival analysis and forestry.
For the future directions of this work, other methods of parameter estimation could be considered. Confidence intervals of the parameters could be also examined. Additionally, the concept of the two-sided model could be extended to generate a new distribution. Moreover, other applications of the TS-LBIG distribution should be applied.