On the extension of inverse Lindley distribution

In this paper, we proposed another extension of inverse Lindley distribution, called extended inverse Lindley and studied its fundamental properties such as moments, inverse moments, mean deviation, stochastic ordering and entropy. The ﬂexibility of the proposed distribution is shown by studying monotonicity properties of density and hazard functions. It is shown that the distribution belongs to the family of upside-down bathtub shaped distributions. Maximum likelihood estimators along with asymptotic conﬁdence intervals are constructed for estimating the unknown parameters. An algorithm is presented for random number generation form the distribution. The property of consistency of MLEs has been veriﬁed on the basis of simulated samples. The applicability of the extended inverse Lindley distribution is illustrated by means of real data analysis.


Introduction
In 1958, Prof. D.V. Lindley (Lindley (1958)) investigated a probability distribution in con-text of fiducial statistic as a counter example of Bayesian theory. Later, this distribution is called as the Lindley distribution (LD). Ghitany et al. (2008) discussed the fundamental properties of the LD with application to waiting time data. Mazucheli and Achcar (2011) worked on the Lind-ley distribution applied to competing risks lifetime data. Krishna and Kumar (2011) estimated the parameter of Lindley distribution with progressive Type-II censoring scheme. They also showed that it may be better lifetime model than exponential, lognormal and gamma distribu-tions in some real life situations. Since then the distribution has been widely discussed in various context. Singh  It may be mentioned here that the Lindley distribution is useful when the data show increas-ing failure rate. This is the property that encourage the use of Lindley distribution in lifetime data analysis over exponential distribution. Although the family of Lindley distributions posses very nice properties and gained great applicability in various disciplines, its applicability may be restricted to non-monotone hazard rate data (bathtub and upside down bathtub (UBT) see Sharma et al. (2014a)). Therefore, the LD has been extended to various ageing classes and introduced various generalized class of lifetime distribution based on Lindley distribution. Zakerzadeh and Dolati (2009) Mahmoudi and Zakerzadeh (2010) and Oluyede and Yang (2015).
In the references cited above, authors mainly focused on the estimation of increasing, decreasing and bathtub shaped failure rates data. Nobody has paid attention to the modelling of the upside down bathtub data. Recently, Sharma et al. (2015) investigated inverted version of the Lindley distribution that has UBT shaped failure rate. The inverse Lindley distribution (ILD) is defined by the following probability density function (pdf) The above density (1) can be rewrite as where, p = θ 1+θ , f 1 (x) = θx −2 e −θ/x , x > 0, θ > 0 and f 2 (x) = θ 2 Γ2 x −3 e −θ/x , x > 0, θ > 0. Thus, inverse Lindley distribution is a two component mixture of inverse exponential distribution and special case of inverse gamma distribution. Sharma et al. (2015) discussed the properties of inverse Lindley distribution with application to stress strength reliability analysis. Sharma et al. (2014b) introduced two parameter extension of inverse Lindley distribution using power transformation to inverse Lindley random variable. Recently, Alkarni (2015) proposed three parameter inverse Lindley distribution with application to maximum flood level data.
We call the distribution in (3) as the extended inverse Lindley distribution and denote the density as EILD(α, θ). Note that if X follows EILD(α, θ), then Y = X −1 follows the generalized Lindley distribution by Abouammoha et al. (2015).
Rest of the article is organized in the following sections. The distributional properties such as moments, skewness, kurtosis, mean deviation, stochastic ordering and Reyni entropy are investigated in section 1. The maximum likelihood estimation along with asymptotic distribution is discussed in section 2. Section 3 consists algorithm for random sample generation form the EILD. Simulation study has also been carried out study the performances of the MLEs. A set of real data is used for illustration purposes in section 4. The paper is concluded in section 5.

Distributional properties
In this section, we discuss the fundamental properties of the EILD. Mainly the properties to be discussed include measures of central tendency, dispersion and shapes of the frequency distribution from EILD. First, we study the shapes of the pdf and hazard function in the following section.

Shapes of Density and Hazard functions
The pdf of extended inverse Lindley distribution is given by The first derivative of the pdf with respect to x is readily obtained as By putting f (x) = 0, we get the mode of EILD which is given by For various shapes of the pdf of EILD, see Figure 1a. The survival function of the density is given by S (x) = 1 − F (x). The survival function of EILD have been plotted in Figure 1b for various combination of α and θ. The hazard function of extended inverse Lindley distribution is given by The hazard function of EILD for various choices of α and θ is shown in Figure 2 indicates that the hazard function is upside down bathtub shaped.

Moments, inverse moments and associated Measures
The r th moment can be defined by, On putting the density function, we get Using the definition of inverse gamma, we obtain The r th inverse moment E 1 X r can be obtained by After substituting the pdf of the ILD, we have x α+1 e −θ/x dx.
On some simplifications, we get The harmonic mean of the EILD can be computed using the first inverse moment as 1/E The first four moments of EILD are given by , α > 4, , α > 5.
The variance of the EILD can be computed using the following relation, σ 2 = µ 2 − µ 1 2 .   The variance of the EILD random variable is given by The coefficients of the skewness and kurtosis measures can now be computed from the following expression, where, The skewness for EILD is shown in Figure 3a for various choices of α and θ. From the figure it can be observed that the γ 1 > 0 so the EILD is positively skewed. The kurtosis is shown in Figure 3b for various values of α and θ. It can be observed from the figure that γ 2 > 0. EILD is more peaked than normal curve so it is leptokurtic.

Mean Deviation and Median Deviation
Proof Consider the given integral, x α e −θ/x dx.
The mean deviation about mean and median are defined by,

Stochastic ordering
A random variable X is said to be stochastically greater than Y if F X (x) <= F Y (x) for all x. In the similar way, X is said to be stochastically greater than Y in the • hazard rate order (Y ≤ hr X) if h X (x) ≤ h Y (x) for all x.
• mean residual life order (Y ≤ mrl X) if m X (x) ≤ m Y (x) for all x.
f (x) is an increasing function of x. Shaked and Shanthikumar (1994) have stated the results for stochastic ordering of distribution as follows: Following the above relations, to show the above ordering in hr, mlr and lr, it is suffix to show the ordering in likelihood only. Proposition 1: Let X and Y are two independent random variables follow the extended inverse Lindley distribution with shape parameters α 1 and α 2 and scale parameters θ 1 and θ 2 respectively. If θ 2 ≥ θ 1 , then (Y ≤ X) and if α 1 ≥ α 2 , then (Y ≤ X) for all x, while other parameters kept fixed and same.
Proof It is given that Y ∼ EILD (α 1 , θ 1 ) and X ∼ EILD (α 2 , θ 2 ). Then the likelihood ratio is given by, On differentiating, we get We can easily find that the function above is increasing in x for θ 2 ≥ θ 1 . Which means that X is stochastically greater than Y with respect to likelihood ratio if θ 2 ≥ θ 1 and α 1 = α 2 = α.

Renyi Entropy
An entropy is a measure of variation of the uncertainty in the distribution of a random variable X. For a probability distribution, the expression of the Renyi entropy (Renyi (1961)) is defined by, where, γ > 0 and γ = 1. Substituting (4) in (7) and after some simplifications, we have

Approximate confidence intervals
Obtaining the exact confidence intervals for α and θ is not an easy task since the MLE's of α and θ are not in closed forms. We, therefore, can use the asymptotic behaviour of the maximum likelihood estimator to obtained the asymptotic confidence intervals for the del parameters. The diagonal elements var (α) and var (θ) of this matrix are the asymptotic variances of the estimators of α and θ, respectively. Thus, the asymptotic 100(1 − )% confidence intervals for α and θ are given by where, Z ψ/2 is the upper 100 × (ψ/2) th percentile of a standard normal distribution.

Simulation Study
In this section, we investigate the properties of the MLEs of α and θ with respect to sample size n. For this purpose, we need to simulate the random sample from the EILD. Since the EILD is two component mixture of inverse gamma distributions, we can use the definition of the mixture distribution for generating the random numbers from the distribution. Algorithm for sample generation from the EILD is given by Algorithm 1: Step 1. Set n, α, & θ.
On the basis of simulated samples, we study the behaviour of biases, absolute biases and mean squared error (MSE) of MLEs of α and θ with varying sample size n. We also investigated the coverage probability for the asymptotic confidence intervals. Simulation algorithm consists the following steps: Algorithm 2: Step 1. Generate 10,000 samples of size n from the EILD(2, 2) by using the Algorithm 1.
Step 3. Compute the standard errors of the MLEs for the 10,000 sample say sα i , sθ i . The standard errors were computed by inverting the observed information matrices.
Step 4. Compute average estimate (AE) as Step 5. Compute the biases, absolute biases (AB) and mean squared error (MSE) using the following formulae,  We repeat these steps for n = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 500, 1000 with α = 2 and θ = 2 , and computed Bias Θ (n), MSE Θ (n), CP Θ (n) and CW Θ (n). Trends for the results have also been shown graphically in Figures 4, 5, 6. From the results, it can be observed that the biases and MSEs of the MLEs of α and θ decrease as sample size n increases. That proves the consistency of the MLEs. In general, the CP for the CIs of α and θ decrease with increasing sample size. However, it stabilizes at approximately 0.95 for large sample size. The average width of CIs for α and θ decreases as sample size increase.

Real data analysis
In this section, we use a set of real data (given in table 1) of flood levels to demonstrate the applicability of the extended inverse Lindley distribution. The data were obtained in a civil engineering context and gives the maximum flood level (in millions of cubic feet per second) for the Susquehanna river at Harrisburg, Pennsylvania over 20 four-year periods from 1890 to 1969. These data have been widely discussed by many authors and were initially reported by Dumonceaux and Antle (1973). First, we use the likelihood ratio (LR) test statistic to check   whether the shape parameter associated with extended inverse Lindley distribution improves its applicability. The hypothesis can be stated as null hypothesis; H 0 : X ∼ ILD(θ) (i.e. α = 1) versus alternative hypothesis; H 1 : X ∼ EILD(α, θ) (i.e. α = 1).
In this case, the LR test statistic for testing H 0 versus H 1 is ζ (l 0 − l 1 ), where l 1 and l 0 are the log-likelihood functions under H 1 and H 0 , respectively. The statistic ζ is asymptotically (asn → ∞) distributed as χ 2 k with k degree of freedom, where k is the number of parameters. The LR test rejects H 0 if ζ ≥ χ 2 k (γ), where χ 2 k (γ) denotes the upper 100γ% quantile of the χ 2 k distribution. For given real data set, the log-likelihood under the ILD is l 0 = 0.5854 witĥ θ = 0.6344 and under H 1 , l 1 is 16.1422 with (α = 15.556,θ = 5.7875). Clearly, the shape parameter α can never be 1 for the data sinceα = 15.556 which is very far than unity. However, the LR test statistic is ζ = 31.11 which is greater than χ 2 1 (0.05) = 3.84. The results indicate that evidences do not support the null hypothesis. Therefore, the EILD is a better model than its special case, ILD.
The selection criterion is that the lowest AIC and BIC correspond to the best model fitted. The MLEs, AIC and BIC are shown in Table 2. From the Table, we can observed that the extended inverse Lindley distribution shows the smaller AIC and BIC than other competing distributions. Thus, the EILD fits well the data set. The plots of probability-probability and the fitted cumulative distribution of the EILD are shown in Figure 7 for maximum flood level data. Figures also indicate that the EILD is a good fitted model for the data. The fitted density of EILD is shown in Figure 8.    Figure 8: Plot of the fitted density of the EILD for real data.

Summary and conclusion
In this article, we introduced two parameter extension of inverse Lindley distribution, called extended inverse Lindley distribution (EILD). The distribution shows the upside-down bathtub shape for its hazard rate. The current extension provides a rather general and flexible framework for statistical analysis of positive data. The EILD can be expressed as a two component mixture of the inverse gamma distributions. That provides some expansions for the ordinary moments, mean deviations, stochastic ordering, inverse moments and Renyi entropy. The estimation of parameters is approached by the method of maximum likelihood. The confidence intervals for the parameters of EILD are also obtained by using asymptotic distribution of MLEs. Simulation study has been carried out to study the behaviours of MLEs of the parameters with respect to sample size. We considered the likelihood ratio test statistic to compare the model with its baseline model. An application of the EILD shows that it could provide a better fit than other alternative inverse distributions. Finally, we hope that our proposed model will attain great applicability in the real problems encountered in various disciplines such as medical, engineering and social sciences etc.