Extended Reciprocal Rayleigh Distribution: Copula, Properties and Real Data Modeling

A new extension of the reciprocal Rayleigh distribution is introduced. Simple type copula-based construction is presented for deriving and many bivariate and multivariate type distributions of the reciprocal Rayleigh model. The new reciprocal Rayleigh model generalizes another three reciprocal Rayleigh distributions. The performance of the estimation method is assessed using a graphical simulation study.


Introduction A random variable (RV)
is said to have the reciprocal Rayleigh (RR) distribution if its survival function (SF) is given by where > 0 and ( ) refer to the cumulative function (CDF). The density function (PDF) corresponding (1) can be expressed as ( ) = 2 2 −3 (− 2 −2 ) | ≥0 (2) The RR model is a special case from the well-known Reciprocal Weibull (RW) model. Recently, Cordeiro et al. (2016) proposed and studied a new class called the generalized odd log-logistic-G (GOLL-G) with two extra shape parameters 1 > 0 and 2 > 0 . For an arbitrary baseline CDF ( ) , the SF of the GOLL-G family is given by The PDF corresponding to (3) is given by The PDF corresponding to (5) is given by The HRF for the new model can be get from 1 , 2 , ( )/[1 − 1 , 2 , ( )] . Table 1 provides some sub-models of the GOLL-RR model. Reduced model Reduced CDF Trayer (1964) Fig. 1 give some plots of the GOLL-RR PDF and HRF. From Fig. 1 (left panel) the new GOLL-RR PDF can be a unimodal-PDF with right skewed and symmetric shape. From Fig. 1 (left panel) we note that the new GOLL-RR HRF can be upside-down-HRF with different useful shapes. The quantile function (QF) of (by inverting (5)), say Q = (Q) = −1 (Q) , as

The Multivariate extension
The -dimensional type extension can be derived as .

Representations
Based on Cordeiro et al. (2016), the PDF in (6) can be expressed as where and (1+ ) ( ; ) is the PDF of the RR model with scale parameter √(1 + ) also the CDF of becomes where (1+ ) ( ; ) is the CDF of the RR distribution.

Moments
The ℎ ordinary moment of is given by then we obtain where The ℎ incomplete moment (IM) can be obtained as where ( 1 , 2 ) is the incomplete gamma function 2 1 + , and 1 1 [⋅,⋅,⋅] is a confluent hypergeometric function (CHF). The first IM given by (11) with = 1 as The MGF ( ) = ( ) of can be derived from equation (8) as The Lorenz (L) and Bonferroni (B) curves are defined by respectively. Then, for the new RR we have , and Extended Reciprocal Rayleigh Distribution: Copula, Properties and Real Data Modeling 40

Residual life and reversed residuals and their moments
The ℎ moment of the residuals . Then, the ℎ moment is given by Therefore, and Then, the ℎ moment becomes

Maximum likelihood estimation (MLE)
The log-likelihood function The components of the "score vector" is available if needed.

Simulation studies
We can perform the simulation experiments to assess of the finite sample behavior of the MLEs based on the following algorithm: 1.Use (7)  We repeated these steps for = 50,100, … ,200 with 1 = 1 , 2 = 1 , = 1 , so computing biases (Bias ℎ ( )) , mean squared errors ( ℎ ( )) for 1 , 2 , and = 50,100, … ,200. Fig.s 2, 3 and 4 give the biases and MSEs for 1 , 2 and ∀ = 50,100, … ,200 for the GOLL-RR model. Fig.s 2, 3 and 4 shows how the biases and MSEs vary with respect to . The broken line in Fig. 4 corresponds to the biases being 0 . From Fig. 2, 3 and 4, the biases decrease to zero as → ∞ , the MSEs for each parameter decrease to zero as → ∞ .  The 1 st data (see Nichols and Padgett (2006)). Fig. 6 gives the total time test (TTT) plot (see Aarset (1987)) for data set I. It indicates that the empirical HRFs of data sets I is increasing HRF (IHRF). The 2 nd data (see Smith and Naylor (1987)). Fig. 6 gives the TTT plot for data set II. It indicates that the empirical HRFs of data sets II is IHRF. The 3rd data set (wingo data). Fig. 7 gives the TTT plot for data set III. It indicates that the empirical HRFs of data sets III is increasing.
Many other useful real data sets can be found in Aryal    The statistics are presented in Tables 2, 4 and 6 for data sets I-III respectively. The MLEs and corresponding standard errors (SEs) are given in Tables 3, 5 and 7 for data sets I-III respectively. Fig. 5, 6 and 7 gives the estimated density (E-PDF), estimated CDF (E-CDF), P-P plot (P-P) and estimated HRF (E-HRF) for data set I-III respectively. The GOLL-RR distribution in Tables 2, 4 and 6 give the best results.

Concluding remarks
A new extension of the reciprocal Rayleigh distribution is introduced. Simple type copula-based construction is presented for deriving and many multivariate and bivariate type reciprocal Rayleigh models. The new PDF can is expressed as a "double linear mixture" of the reciprocal Rayleigh PDF. The new reciprocal Rayleigh model generalizes another three reciprocal Rayleigh distributions. The performance of the estimation method is assessed using a graphical simulation.