Modified Inverse Lomax Distribution: Model and properties

Here, a new distribution having three called Modified Inverse Lomax Distribution is proposed. Important statistical properties like the survival, hazard rate, the probability density function (PDF) is studied. Least Square, Cramer-Von Mises and Maximum Likelihood estimation methods are used for estimation of parameters using R programming software. A data set is discussed and performed the goodness-of-fit to assess the application of the proposed distribution. Various methods of model comparison and model validation are also used. The proposed model called Modified Inverse Lomax Distribution is more applicable as compared to some existing probability model.


INTRODUCTION
Probability distribution helps to simulate the real-life problems and to compute the real-life data precisely and effectively.During last decade, different new probability distributions are introduced by researchers using different techniques.
These techniques may me adding some extra parameters to distribution, merging the distribution or inverting the variables etc.These methods make new distribution more flexible and useful than the existing distributions.Many standard techniques to analyze the problem of real-life data are available; we still need models to solve the problem more effectively and precisely.That is in all the cases, classical techniques are not effective as the new distributions in literature, we can find numerous distributions.We can derive parametric distribution by changing the number of parameters to existing distribution such as Lomax and exponential family of distribution (Marshall and Olkin, 1997) We can find different new distributions derived by using the Lomax distribution.Power Lomax distribution is more flexible than existing Lomax distribution with decreasing and inverted bathtub hazard rate functions (Rady et al., 2016).Taking alpha as power, a new distribution named alpha power inverted exponential distribution was introduced using inverted exponential distribution (Ceren et al., 2018).Using Lomax random variable as a generator Ogunsanya et for obtaining Odd generalized exponentiated Inverse Lomax distribution (Maxwell et al., 2019).Lomax exponential distribution is obtained using Lomax distribution (Ijaz and Asim, 2019).The inverse Lomax-exponentiated G-family Falgore and Doguwa, (2020) is introduced using Inverse Lomax distribution.In real life, we can find many data that have bathtub-shaped hrf.In literature we can also find many modifications of Weibull distribution.Two parameter Weibull distributions is given as ( , , ) An extra parameter α is added to modify the Lomax distribution as Modified Inverse Lomax distribution.CDF and PDF proposed model is given as.
And the PDF of Modified Inverse Lomax Distribution is given as

The Modified Inverse Lomax (MILX) distribution
A three parameters Modified Inverse Lomax distribution with CDF and PDF are as follows And pdf of MILX is, ; 0, , , 0

Survival function
Survival function of MILX is Hazard rate and reversed hazard rate function MILX model has hrf as ( ) The various shapes of pdf and hazard rate function of MILX ( )    for at various values of parameters are displayed below in (Figure 1).

Figure 1 Graphs of PDF (left) and Hazard rate function (right) for some values of parameters
Here we have discussed the quantile function also.Quantile function can be defined here by expression ( 6) where X is non negative random variable.

Skewness and Kurtosis
Quartile based coefficient of skewness defined by Bowley's is, Moors, (1988), defined coefficient of kurtosis based on octiles can be defined as

Maximum Likelihood Estimation
Here, we have presented the ML estimators (MLE's) of the MILX model are estimated by using MLE method.Let a sample ( ) Differentiating equation ( 7) with respect to parameters as ( ) ( ) and solving simultaneously for all parameters, ML estimators of the MILX ( ) ,,

  
distribution can be obtained.Normally, it is not possible of solving non-linear equations above so with the aid of suitable computer software one can solve them easily.Let is worthless.Asymptotic variance can be approximated by plugging in the estimated value of parameters.We have used () O  as an estimate of ( ) I  and is given by Newton Raphson algorithm is used to optimize Likelihood function.Matrix below is the called Variance Covariance matrix, ( ) .

Least Square Estimation
We have also used the LSE to estimate the ,  β, & λ of MILX.For this we have to minimize function below Let us consider that as a sample having n items from a distribution function F (.).LSE of parameters are calculated by minimizing the function Differentiating (10), we get,


We can obtain weighted least square estimators by minimizing ( ) Thus, weighted LSE of α, β, and λ are given by minimizing the function below,

Cramer-Von Mises estimation
This method of estimation can be obtained by minimizing the function Differentiating (12), we get, ( ) CVM estimators can be obtained.

APPLICATION TO REAL DATASET
For applicability of the model, we have considered a real set data.Data is by Lee   Graph of P-P plot and Q-Q plots are shown in (Figure 3).It is found that MILX model has better fitting to the data taken in consideration (Figure 4).MLE, LSE and CVE methods are used for estimation and the estimated value of the parameters of MILX model with log-likelihood, AIC, BIC, CAIC, and HQIC criteria in (Table 2).KS, W and A2 statistic and calculated p-value are given in (Table 3).
The PDF of LE (Bhati et al., 2015) distribution is, Various information criteria values for the testing of the applicability of the MILX are tabulated below (Table 4).We have displayed the graph of goodness-of-fit of MILX and models defined as (Figure 5), We have plotted the empirical distribution function of the proposed model MILX and the fitted distribution function and is shown in (Figure 6) Marshall and Olkin family of distribution Ghitany et al., (2007) is used to extend Lomax distribution.
the Fisher's information matrix defined as,

Z
is upper percentile of SNV then using asymptotic normality of MLEs, approximated 100(1-b) % CI of ,

Figure 2
Figure 2 Profile Log-likelihood function of parameters.

Figure 3
Figure 3 The P-P plot (left) & Q-Q plot (right) of the MILX Model.

Figure 4
Figure 4 Histogram and density function of fitted distributions (left) & fitted quantile vs sample quantile (right) of different estimation methods of MILX.

Figure 5
Figure 5 Histogram & pdf (left) of fitted model and Empirical versus estimated distribution function (right).

Figure 6
Figure 6 Empirical distribution functions versus the fitted distribution function of the MILX Lai et al., (2003)is modified to generate several distributions that possess bathtub hrf.Modifications of Weibull distribution is exponentiated Weibull distribution (Mudhokar and Srivastav, 1993).As given byLai et al., (2003)to get new lifetime distributions

Table 1
(Mailund, 2017)tware of the optim () function Team, (2020), calculation of MLEs of MILX by maximizing the log likelihood function defined in equation (5.3.1)(Mailund,2017).Values determined are tabulated in (Table1).Figure2displays the graph of profile log-likelihood function of parameters showing that ML estimates is uniquely defined.Estimated parameters using MLE, LSE and CVME

Table 2
Estimated parameters with LL and values of information criteria

Table 5
Test statistics and p-values