A New One-parameter G Family of Compound Distributions:Copulas, Statistical Properties and Applications

This work introduces a new one-parameter compound G family. Relevant statistical properties are derived. The new density can be “asymmetric right skewed with one peak and a heavy tail”, “symmetric” and “left skewed with one peak”. The new hazard function can be “upside-down”, “upside-down-constant”, “increasing”, “decreasing” and “decreasing-constant”. Many bivariate types have been also derived via different common copulas. The estimation of the model parameters is performed by maximum likelihood method. The usefulness and ﬂexibility of the new family is illustrated by means of two real data sets.


Introduction and motivation
In the last few years, huge efforts have been paid to derive many new G families using the well knowm methods. These new G families have been used for modeling non-censored and censored real data sets in many applied studies such as finance, econometrics, value at risk applications, insurance, biology, engineering, forecasting, medicine and environmental sciences see , for example, Marshall and Olkin (1997) ( where ζ • = 1 − exp (−ζ). Note that for ZTP R.V, the expected value E(N |ζ) and variance V ar(N |ζ) are, respectively, given by E(N |ζ) = ζζ −1 • , and Var(N |ζ) Suppose that the failure time of each subsystem has the reciprocal Rayleigh G ("RR-G(P)" for short) family defined by the cumulative distribution function (CDF) and probability density function(PDF) given by and h P (x) = 2g P (x) respectively, where Therefore, the CDF of the Poisson reciprocal Rayleigh (PRR-G) family can be expressed as where Ω =(ζ, P). The corresponding PDF as In this work, a special attention is paid to two special members called the Poisson reciprocal Rayleigh exponential and the Poisson reciprocal Rayleigh Fréchet distributions. The new density of the Poisson reciprocal Rayleigh exponential model can be "asymmetric right skewed with one peak and a heavy tail", "symmetric" and "left skewed with one peak". The new hazard rate function of the can be Poisson reciprocal Rayleigh exponential model "upside-down-constant", "increasing" and "decreasing-constant". The new density of the Poisson reciprocal Rayleigh Fréchet model can be "asymmetric right skewed with one peak and a heavy tail", "symmetric" and "left skewed with one peak". The new hazard rate function of the Poisson reciprocal Rayleigh Fréchet model can be "upside-down", "increasing" and "decreasing". So, the family may be useful in modeling the "asymmetric right skewed with one peak and a heavy tail" real data sets, "symmetric" real data setsand "left skewed with one peak" 944 A NEW ONE-PARAMETER G FAMILY OF COMPOUND DISTRIBUTIONS real data sets. Also, the family may be useful in modeling the real data sets which have "upside-down", "upsidedown-constant", "increasing", "decreasing" and "decreasing-constant" hazard rate functions. The new family is better than the odd Lindley family, Marshall-Olkin family, the Burr-Hatke family, generalized Marshall-Olkin family, Beta family, Marshall-Olkin Kumaraswamy family, Kumaraswamy family, the Burr X family and Kumaraswamy Marshall-Olkin family in modeling the bimodal right skewed relief times data set so the new family could be considered as a good alternative to these families. The new family is better than the Marshal-Olkin family, Generalized Marshal-Olkin family, Kumaraswamy family, beta family, Kumaraswamy Marshal-Olkin family and Marshal-Olkin Kumaraswamy family in modeling the gauge lengths data set so the new family could be considered as a good alternative to these families.

Useful expansions
Using the power series the PDF in (6) can be written as again applying (7) to (8) we get If υ1 υ2 < 1 and υ 3 > 0 is a real non-integer, the following power series holds Applying (10) to (9) we have where and π c * (x; P) = c * g P (x) G P (x) c * −1 is the PDF of the exponentiated-G (exp-G) family with power parameter c * . Equation (12) reveals that the density of X can be expressed as a linear mixture of exp-G densities. So, several mathematical properties of the new family can be obtained from those of the exp-G distribution. Similarly, the CDF 945 of the PRR-G family can also be expressed as a mixture of exp-G CDFs given by where Π c * (x; P) = G P (x) c * is the CDF of the exp-G family with power parameter c * .

Quantile function (QF) and random number generation
The QF of the R.V X, where X ∼PRR-G(ζ, P), is obtained by inverting its CDF in (5) as Simulating the PRR-G R.V is straightforward. If U is a uniform variate on the unit interval (0, 1), then the R.V X = Q (U ) follows (6).

Moments
Let Y c * be a R.V having density π c * (x; P). The rth ordinary moment of X, say µ r,X , follows from (12) as where can be evaluated numerically in terms of the baseline qf Q G (u) = G −1 (u) as Setting r = 1 in (14) gives the mean of X.

Incomplete moments and mean deviations
The rth incomplete moment of X is defined by ω r, We can write from (12) The integral m r,ζ (y) can be determined analytically for any special model with closed-form expressions for the Q G (u) or computed at least numerically for most baseline distributions. Two important applications of the first incomplete moment are related to the mean deviations about the mean and median and to the Bonferroni and Lorenz curves.

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A NEW ONE-PARAMETER G FAMILY OF COMPOUND DISTRIBUTIONS

Moment generating function (MGF)
The MGF of X, say M X (t) = E(exp (t X) ), is obtained from (12) as where M c * (t) is the generating function of Y c * given by The two integrals I +∞ −∞ (x) and I +1 0 (u) can be computed numerically for most base line distributions.

Copulas
A copula is a multivariate CDF for which the marginal distribution of each R.V is uniform on the interval [0, 1].
Copulas are used to describe the dependence between R.Vs. In this Section, we derive some new bivariate PRR

Modified FGM copula
The modified FGM copula is defined as 3.2.1. Modified FGM type-I Consider the following functional form for both A (υ) and C (a). Then, the B-PRR-FGM (Type-I) can be derived from

Modified FGM type-II Let A (υ) and C (a) be two functional form satisfying all the conditions stated earlier where
which satisfy all the conditions stated earlier. In this case, one can also derive a closed form expression for the associated CDF of the B-PRR-FGM (Type-

Clayton copula
The Clayton copula can be considered as C(a 1 , . Setting a 1 = F (υ) and a 2 = F (x), the B-PRR type can be derived from C(a 1 , a 2 ) = C(F (a 1 ) , F (a 2 )). Then Similarly, the M-PRR can be derived from

Renyi's entropy copula
Using the theorem of Pougaza and Djafari (2011) where C(υ, a) = z 2 υ + z 1 a − z 1 z 2 , the associated B-PRR can be derived from

Ali-Mikhail-Haq copula
Under the stronger Lipschitz condition, the Archimedean Ali-Mikhail-Haq copula can expressed as .

Special PRR-G submodels
In this Section we will provide many new distributions based on some common base line models namely  Table 1). A special attention is given to the Poisson reciprocal Rayleigh exponential (PRRE) and the Poisson reciprocal Rayleigh Fréchet (PRRF) distributions. Figure 1 gives some plots of the PDF of the PRRE model. Figure 2 provides different plots of the HRF of the PRRE model. Figure 3 gives some plots of the PDF of the PRRF model. Figure 4 provides different plots of the HRF of the PRRF model. Based on Figure 1, the new density of the PRRE model can be "asymmetric right skewed with one peak and a heavy tail", "symmetric" and "left skewed with one peak". Based on Figure 2, the new hazard rate function (HRF) of the can be PRRE model "upside-down-constant", "increasing" and "decreasing-constant". Based on Figure 3, the new density of the PRRF model can be "asymmetric right skewed with one peak and a heavy tail", "symmetric" and "left skewed with one peak". Based on Figure 4, the new hazard rate function (HRF) of the can be PRRF model "upside-down", "increasing" and "decreasing". Based on Figure 1 and Figure 3, the family may be useful in modeling the "asymmetric right skewed with one peak and a heavy tail" real data sets, "symmetric" real data setsand "left skewed with one peak" real data sets.Based on Figure 3 and Figure 4, the family may be useful in modeling the real data sets which have "upside-down HRF", "upside-down-constant HRF", "increasing HRF", "decreasing HRF" and "decreasing-constant HRF".

Parameter Estimation
Here, we will consider the estimation of the unknown parameters (ζ, P) of the new G family from complete samples by maximum likelihood method. Let z 1 , · · · , x n be a random sample (rs) from the PRR-G models parameter vector Ω =(ζ, P ) . The log-likelihood function for Ω is given by The above log-likelihood function can be maximized numerically by using R (optim), SAS (PROC NLMIXED) or Ox program (sub-routine MaxBFGS), among others. For confidence interval (C.I.) estimation of the parameters, the elements of the observed information matrix J(Ω) can be evaluated numerically, where

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A NEW ONE-PARAMETER G FAMILY OF COMPOUND DISTRIBUTIONS and where P is the number of parameters of the base line model, Setting the nonlinear system of equations U ζ = U β = U P P = 0 and solving them simultaneously yields the maximum likelihood estimations (MLEs) of Ω = (ζ, P ) . These equations can be solved numerically using convenient iterative method such as the Newton-Raphson type algorithms. For interval estimation of these parameters, we can obtain the observed information matrix J Ω = ∂ 2 n (Ω) ∂m∂n (for m, n = ζ, V) which can be computed numerically. Under standard regularity conditions when n → +∞, the distribution of Ω can be approximated by a multivariate normal N (P+1) 0, J Ω −1 distribution to construct approximate confidence intervals for the parameters. Here, J Ω is the total observed information matrix evaluated at Ω . Large sample theory for these estimators delivers simple approximations that work well in finite samples. The normal approximation for the MLEs is easily handled numerically. Likelihood ratio tests can be performed for the proposed family in the usual way.

Real data applications
In this Section we analyze two real data sets. For data the first data set (the first application), we considered the standard one-parameter exponential distribution as our base line model and then we compare the fits of the PRRE distribution with other competitive exponential extensions such as For comparing models, we consider the Cramér-Von Mises (CM) and the Anderson-Darling (AD) and the Kolmogorov-Smirnov (KS) statistic (and its corresponding P-value), the Akaike Information Criterion (C 1 ), Bayesian Information Criterion (C 2 ), consistent Akaike Information Criterion (C 3 ) and Hannan-Quinn Information Criterion (C 4 ) where q l , where a (n) = 1 + 9 4 n −2 + 3 4 n −1 , where z l = F (y l ), the y l 's values are the ordered observations and n is the sample size. However, other potential goodness-of-fit statistic tests for validation such as Nikulin-Rao-Robson statistic test and the modified Nikulin-Rao-Robson statistic test may be used (see Goual

First application (failure times data)
The failure times data set is given in ). The data represents the lifetime data relating to relief times (in minutes) of patients receiving an analgesic. This data was recently analyzed by Al-babtain et al. (2020) and . Table 2 lists the MLEs, standard errors (SEs) and 95% C.I.s (LC.I., UC.I.). Table 3 lists the C 1 , C 2 , C 3 , C 4 , AD, CM, K.S. and its p-value. Figure 3 gives the total time in test (TTT) plot (see Aarset (1987)) for the relief times data along with the corresponding quantile-quantile (QQ) plot , box plot and the nonparametric Kernel density estimation (N-KDE) plot. Based on Figure 5, the HRF of the relief times is "monotonically increasing HRF" (top right plot) and this data has an extreme value ( see top right and bottom right plots) and its density is right skewed and bimodal. Figure 6 gives the estimated PDF (E-PDF), estimated CDF (E-CDF), estimated HRF (E-HRF) and P-P plot for relief times data. Based on results of Table 3 and Table 4, it is concluded that the PRRE model is much better than the exponential, odd Lindley exponential, Marshall-Olkin exponential, moment exponential, the logarithmic Burr-Hatke exponential, generalized Marshall-Olkin exponential, Beta exponential, Marshall-Olkin Kumaraswamy exponential, Kumaraswamy exponential, the Burr X exponential and Kumaraswamy Marshall-Olkin exponential models with C 1 =34.86, C 2 =36.85, C 3 =35. 56 Figure 6. E-PDF, E-CDF, P-P and E-HRF for relief times data.

Second application (gauge lengths data)
The fgauge lengths data set consists of 74 observations (see Kundu and Raqab (2009)). Table 5 lists the MLEs, SEs confidence intervals (C.I.s) for the gauge lengths data. Table 6 lists the C 1 , C 2 , C 3 , C 4 , AD, CM, K.S. and p-value. Figure 5 gives the total time test (TTT) plot (Aarset (1987)) for the relief times data along with the corresponding box plot, QQ plot and the N-KDE plot. Based on Figure 7, the HRF of the gauge lengths data is "monotonically increasing HRF" (see top right plot) and this data has no extreme observation (see top right and bottom right plots) and its density is semi-bimodal. Figure 6 gives the E-PDF, E-CDF, E-HRF and P-P plot for gauge lengths data. Based on results Table 5

Concluding remarks
In this work, a new one-parameter compound G family of continuous distributions is derived and studied. Relevant statistical properties such as moments, incomplete moments and moment generating function are derived. The density of the new family is re-expressed in terms of the exponentiated G family. The new density can be "asymmetric right skewed with one peak and a heavy tail", "symmetric" and "left skewed with one peak". The corresponding hazard function can be "upside-down", "upside-down-constant", "increasing", "decreasing" and "decreasing-constant". Many bivariate types have been also derived via different common copulas. The estimation of the model parameters is performed by the maximum likelihood method. The usefulness and flexibility of the new family is illustrated by means of two real data sets. The new family is better than the odd Lindley family, Marshall-Olkin family, the Burr-Hatke family, generalized Marshall-Olkin family, Beta family, Marshall-Olkin Kumaraswamy family, Kumaraswamy family, the Burr X family and Kumaraswamy Marshall-Olkin family in modeling the bimodal right skewed relief times data set with C 1 =34.86, C 2 =36.85, C 3 =35.56, C 4 =35.24, AD=0.146, CM=0.025, K.S=0.092 and p-value=0.9955 so the new family could be considered as a good alternative to these families.