The T ‒ R{Y{U}} Family of Distributions of Type I: Some Properties and Applications

CDF-quantile distributions appeared in [1]. In the present paper, we show it can be used to generalize the T ‒ R{Y} class of distributions [2] to a new family which we call T ‒ R{Y{U}} family of distributions. Some properties and applications associated with the T ‒ R{Y{U}} family of distributions are obtained.

family of distributions. Some properties and applications associated with family of distributions are obtained.

Y R T − Family of Distributions
This family of distributions was proposed in [2]. In particular, let T, R, Y be random variables with CDF's ( ) ( ),

The CDF-quantile Family of Distributions
Let ( ) σ µ, , x G denote a CDF for random variable X with support ( ), is a location parameter, and 0 > σ is a scale parameter. Then, where F is a CDF with support which is denoted ,

The New Class of Distributions
To motivate the new class of distributions, we first make the following observation for Case I of the previous section. Note that similar observations hold for the remaining cases. For Case I, the CDF can be written as CDF is given by the following , , 1 1 By differentiating the CDF in the previous definition, we have the following 1 ln where R F is the CDF of the random variable R. Thus, the above implies the following Proof. Using the fact that , The result follows from Definition 3.1 by solving the following equation for ( ) class of distributions has the following representation as a power series for its CDF Proof. From Theorem 4.1, we know that the CDF of the standard Logistic- By the negative binomial series, we can write By the power series representation for the exponential function, we can write By the binomial theorem, we can write class of distributions has the following representation as a power series for its PDF By the negative binomial series, we can write By the power series representation for the exponential function, we can write By the binomial theorem, we can write Proof. From Theorem 4.1, the following random variable below follows the standard is a quantile function. According to [4], we can write where the coefficients are suitably chosen real numbers that depend on the parameters of the ( ) By the power series representation for the exponential function, we can write By the Binomial theorem we can write Again by the Binomial theorem we have It now follows that we have the following By the transformation technique, the CDF of Y for 1 0 ≤ ≤ y is given by ( ) Consequently, the PDF is given by thus the result follows by noting that we have the following

Practical Illustration and Numerical Comparison
In this section, we show a member of the family of distributions of type I is a good fit to the coupons data, We assume the random variable T with support ( ) We also assume that the random variable Y with support ( ) is (standard) Cauchy distributed, so that the quantile function is given by Finally, we have the added assumption that the random variable R is framework we deduce the following if Q is a random variable with the CDF given by the previous proposition. In the rest of this section we compare the Normal-Standard Cauchy{Pareto} distribution, and the Normal-Standard Cauchy{Pareto{U}} distribution of type I in fitting the coupons data [5]. Table 1. Estimated parameters for the coupons data. In order to compare the two distribution models, we used the following criteria: -2(Loglikelihood), AIC (Akaike information criterion), AICC (corrected Akaike information criterion), and BIC (Bayesian information criterion) for the data set. The better distribution corresponds to the smaller -2(Log-likelihood) AIC, AICC, and BIC values: where k is the number of parameters in the statistical model, n is the sample size, and l is the maximized value of the log-likelihood function under the considered model. From distribution to the coupons data.

Simulation Study
In this section a Monte Carlo simulation study is carried out to assess the performance of the maximum likelihood estimation method in the distribution  Table 3 and Table 5 for each of sets I and II, respectively, considered above. From Table 3, we observe that the estimated standard deviation and variance consistently decrease with increasing sample size as seen in Table 4, hence the estimation method is adequate. From Table 5, we observe that the standard deviation and variance consistently decrease with increasing sample size as seen in Table 6, hence the estimation method is adequate. Table 6. Decreasing variance (VAR) and standard deviation (SD) for increasing sample size.

A Characterization Theorem
The characterization of statistical distributions plays a major role in stochastic modeling. In this section we present a characterization of the of type I. Our characterization theorem is based on a simple relationship between two truncated moments, and for related works in this direction, the reader is referred to [6]- [11].

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At first, we recall the following which will be useful later is defined with some real function .
η Assume that where the function s is a solution of the differential equation Remark 7.2. The characterization based on the ratio of two truncated moments is stable in the sense of weak convergence, and for more details see [12].
The main result of this section is as follows be a continuous random variable, and let ( ) , we deduce the following Now in view of Theorem 7.1, X has PDF q is given by the previous proposition, then we have the following The PDF of X is there exists functions 1 q and η defined in Theorem 7.1 satisfying the following differential equation Remark 7.5. The general solution of the differential equation in the above corollary is given by where D is a constant.