A Univariate, Bivariate and Multivariate Extensions for the Inverse Rayleigh Model with Properties and Applications to the Univariate Version

A new univariate extension of the Inverse Rayleigh distribution is proposed and studied. Some of its fundamental statistical properties such as some stochastic properties, ordinary and incomplete moments, moments generating functions, residual life and reversed residual life functions, order statistics, quantile spread ordering, Rényi, Shannon and q-entropies are derived. A simple type Copula based construction via Morgenstern family and via Clayton copula is employed to derive many bivariate and multivariate extensions of the new model. We assessed the performance of the maximum likelihood estimators using a simulation study. The importance of the new model is shown via two applications to real data sets.


1.Introduction and motivation
The cumulative distribution function (CDF) and probability density function (PDF) of the Generalized Odd Generalized Exponential G (GOGE-G) family are given, respectively, by where ( ) is the baseline CDF depending on a parameter vector and ℎ ( ) = ( ) is its corresponding PDF and , > 0 are two additional shape parameters.
Using (3) and (2) the CDF of the GOGEIR can be derived as the corresponding PDF of (5) can be expressed as Henceforth, ∼ GOGEIR( , , ) denotes a random variable having density function (6). The hazard rate function (HRF) of can be derived using the well-known relationship has CDF (5). Now, we provide a useful representation for (6). Using the series expansion which holds for | | < 1 and > 0 real non-integer and using the power series, the PDF of the GOGEIR in (6) can be expressed as Using the series expansion again we arrive at where ℎ (1+ 1 + 2 ) ( ) is the IR density with scale parameter [ (1 + 1 + 2 )]   The rest of the paper is outlined as follows. In Section 2, we derive some mathematical properties for the new model including some stochastic properties, ordinary and incomplete moments, moments generating functions, residual life and reversed residual life functions, order statistics, quantile spread ordering, Rényi, Shannon and q-entropies. A simple type Copula based construction is derived in Section 3. Maximum likelihood estimation of the model parameters is addressed in Section 4. In Section 5, we assess the performance of the maximum likelihood estimators using a Monte Carlo simulation study. In Section 6, we provide two applications to real data to illustrate the importance of the GOGEIR model. Finally, some concluding remarks are presented in Section 7.

2.Mathematical properties 2.1Moments, cumulants and numerical results
The th ordinary moment of is given by Then we obtain where is the complete gamma function and The last integration can be computed numerically for most parent distributions. The skewness and kurtosis measures can be calculated from the ordinary moments using wellknown relationships. The th central moment of , say , follows as The cumulants ( ) of follow recursively from The mean of , variance (V ( ) ) skewness (S ( ) ) and kurtosis (K ( ) ) measures also can be calculated from the ordinary moments using well-known relationships. The mean, variance, skewness and kurtosis of the GOGEIR distribution are computed numerically for some selected values using the R software. The numerical values displayed in Table 1 indicate that the skewness of the GOGEIR distribution is always positive and can range in the interval (0.59,1.016) . The spread for its kurtosis is ranging from 1.05 to 3.6 . With fixing the values of and , we note that the (S ( ) ) and (K ( ) ) has no changes.

2.2Incomplete moment
The r ℎ incomplete moment, say ( ) , of can be expressed, from (8), as where ( , ) is the incomplete gamma function.

2.3The moment generating function (MGF)
The MGF ( ) = ( ) of can be derived from equation (4) as Another alternative method for deriving the MGF can be introduced by the Wright generalized hypergeometric function (WHGF) which is defined by Then, the MGF of (1) can be defined as Combining expressions (8) and (10), we obtain the MGF of the GOGEIR as Equations (9) and (11) can be easily evaluated by scripts of the Maple, Matlab and Mathematica platforms.

Entropies
The Rényi entropy of a random variable represents a measure of variation of the uncertainty. The Rényi entropy is defined by Using the PDF (6), we can write Then, the Rényi entropy of the GOGEIR model is given by The q-entropy, say ( ) , can be obtained as and The Shannon entropy of a random variable , say , is defined by It is the special case of the Rényi entropy, ( )| ( >0 and ϑ≠1) , when ↑ 1.

Order statistics
Let 1 , 2 , … , be a random sample (RS) from the GOGEIR distribution and let where B (⋅,⋅) is the beta function. Substituting (1) and (2) in equation (12) and using a power series expansion, we have Then, the PDF of : can be expressed as The density function of the GOGEIR order statistics is a mixture of IR densities. Based on the last equation, the moments of : can be expressed as

Quantile spread order
The quantile spread ( ϒ ⋅ ( ) ) of the rv ∼ GOGEIR ( , , ) having the CDF (5) is given by is the survival function. The ϒ ⋅ ( ) of a distribution describes how the probability mass is placed symmetrically about its median and hence can be used to formalize concepts such as peakedness and tail weight traditionally associated with kurtosis. So, it allows us to separate concepts of kurtosis and peakedness for asymmetric models.
Let 1 and 2 be two random variables following the GOGEIR model with quantile spreads ϒ 1 and ϒ 2 , respectively. Then 1 is called smaller than 2 in quantile spread order, denoted as • The order ≤ [ϒ] implies ordering of the mean absolute deviation around the median, say (⋅) ,

3.Simple type Copula based construction
In this Section, we consider several approaches to construct the bivariate and the multivariate GOGEIR type distributions via copula (or with straightforward bivariate CDFs form, in which we just need to consider two different GOGEIR CDFs).

Via Morgenstern family
First, we start with CDF for Morgenstern family of two random variables ( , ) which has the following form then we have a seven-dimension parameter model. The estimation will be a big issue here. Estimation via Bayesian paradigm may be done, but again, the choice of appropriate priors will be challenging.

Via Clayton copula The bivariate extension
The bivariate extension via Clayton copula can be considered as a weighted version of the Clayton copula, which is of the form the associated CDF bivariate GOGEIR type distribution will be Note: Depending on the specific baseline CDF, one may construct various bivariate GOGEIR type model in which ( 1 + 2 ) ≥ 0.

The Multivariate extension
A straightforward -dimensional extension from the above will be .
Further future works could be be allocated for studying the bivariate and the multivariate extensions of the GOGEIR model.

Some stochastic properties
Suppose 1 ∼ GOGEIR ( 1 , 1 , ) and 2 ∼ GOGEIR ( 2 , 2 , ). Then 1 is stochastically smaller than 2 if 1 > 2 and 1 > 2 . Note that for any 1 > 2 , . This is true for both integer and fractional values of 1 and 2 . After some algebra, we get the following: Since for 1 > 2 we have Rest of the proof follows immediately from here This completes the proof. equations, it is usually more convenient to use nonlinear optimization methods such as the quasi-Newton algorithm to numerically maximize ℓ( ) . Further works could be devoted using other different methods to estimate the GOGEIR parameters such as least squares, moments, weighted least squares, Anderson-Darling, Jackknife, bootstrap, Cramér-von-Mises, Bayesian analysis, among others, and compare the estimators based on these methods.

Simulations
Using (7), we simulate the GOGEIR model via taking =20, 50, 150, 500 and 1000. We evaluate the MLEs of the parameters for each sample size. Then, repeating this process = 1000 times and calculate the averages of the estimates (AEs), mean squared errors (MSEs). Table 2 gives all simulation results. The numerical results in Table 2  increases. These results supports that the asymptotic normal model provides an adequate approximation to the finite sample distribution of the MLEs. Table 2 below gives the AEs and MSEs based on = 1000 simulations of the GOGEIR model for some values of , and .  The 1 data set consists of 100 observations of breaking stress of carbon fibers given by Nichols and Padgett (2006)  In order to compare the distributions, we consider the following criteria: the  2  ▪ (Maximized Log-Likelihood), AI (Akaike Information Criterion), CAI (Consistent Akaike Information Criterion), B (Bayesian information criterion) and HQI (Hannan-Quinn information Criterion). The model with minimum values for these statistics could be chosen as the best model to fit the data. Total time test (TTT ) plot (see Aarset (1987) anf Figure 1) is an important graphical approach to verify whether our data can be applied to a specific model or not. The TTT plots the two real data sets is presented in Figure 1. This plot indicates that the empirical HRFs of the the two data sets are increasing.          The GOGEIR model gives the lowest values for the AI , BI , HQI and CAI statistics (in bold values) among all ftted models to these data. So, it may be considered as the best model among them. Figures 2-6, respectively, display the plots of estimated CDFs, estimated PDFs, estimated HRFs, P-P plots and Kaplan-Meier survival plots for the 1 and 2 data. These plots reveal that the proposed distribution yields a sufficient fit for both data sets.

7.Conclusions
A new extension of the Inverse Rayleigh model is proposed and studied. Some of its fundamental statistical properties such as some stochastic properties, ordinary and incomplete moments, moments generating functions, residual life and reversed residual life functions, order statistics, quantile spread ordering, Rényi, Shannon and q-entropies are derived. A simple type Copula based construction via Morgenstern family and via Clayton copula is employed to derive many bivariate and multivariate extensions of the new model. We assessed the performance of the maximum likelihood estimators using a simulation study. The importance of the new model is shown via two applications to real data sets.