A Member of the rT  X Family of Distributions Induced by a New V with Application to a Power Series Class of Distributions

Introduction

The cumulative distribution function (CDF) and the probability density function (PDF) of the $r_T - X$ family induced by $V$ is given by Ampadu CB [1]

$$J(x) = R(V(F(x)))$$

where the random variable $T \in [a,b]$, for $-\infty \leq a < b \leq \infty$, has CDF $R$, $V$ is an appropriate weight, and $F(x)$ is the CDF of any random variable $X$.

Theorem 3.2. The PDF of the $r_T - X$ family induced by $V$ is given by Ampadu CB [1]

$$j(x) = r(V(F(x)))V'(F(x))f(x)$$

where the random variable $T \in [a,b]$, for $-\infty \leq a < b \leq \infty$, has PDF $r$, $V$ is an appropriate weight $F(x)$ and $f(x)$ are the CDF and PDF, respectively, of any random variable $X$.

Remark 3.3. The choice of $V$ depends on the support of $T$. In particular, when the support of $T$ is $[a, \infty)$, where $a \geq 0$, we can take $V$ as follows.

(a) $$V(x) = 1 - e^{-x}$$

(b) $$V(x) = \frac{x}{1+x}$$

(c) $$V(x) = \left[1 - e^{-x}\right]^{\frac{1}{\alpha}}$$, where $\alpha > 0$

(d) $$V(x) = \left[\frac{x}{1+x}\right]^{\frac{1}{\alpha}}$$, where $\alpha > 0$

On the other hand, when the support of $T$ is $(-\infty, \infty)$, we can take $V$ as follows $$ V (x) = 1 - e ^ {- e ^ {x}} $$

(a) ( ) 1 $$ V (x) = \frac {e ^ {x}}{1 + e ^ {x}} $$

(b) ( ) +

1

1 $$ 0 = \left[ 1 - e ^ {- e ^ {x}} \right] ^ {\frac {1}{\alpha}}, $$

, where α> 0 x e e V x (c)

1 ( )

1 $$ ) = \left[ \frac {e ^ {x}}{1 + e ^ {x}} \right] ^ {\frac {1}{\alpha}}, $$

, where α> 0 x e (d)

1 ( ) x e V x Moreover, a so-called new Exponential-Weibull distribution arising from this broad class of distributions was shown to be a good fit to the breast cancer patients data [2], showing its practical significance.

The power series class of distributions was proposed and studied in Noack A [3]. This class of distributions includes binomial, geometric, logarithmic and Poisson distributions as special cases. However, these distributions may not be useful when a random variable takes the value of zero with high probability, that is, zero- inflated. In such situations, it is more appropriate to consider the distribution which is truncated at zero. More details on these distributions can be found in the context of univariate discrete distributions contained in Johnson NL, Kemp AW, Kotz S [4].

Power series distributions are usually motivated by the stochastic representations $$ Z = \min \left(X _ {1}, \dots , X _ {N}\right) $$ if the components are in series, or $$ Z = \max \left(X _ {1}, \dots , X _ {N}\right) $$ if the components are in parallel, and many lifetime data admit such representations, and the related distributions are very useful in modeling such data. We suppose X1,…..XN are independent and identically distributed random variables from a parent distribution with PDF f (x) and CDF F (x), and consider N to be a discrete random variable from a power series distribution (truncated at zero) and whose PDF is given by n an P N n λ $$ (= n) = \frac {a _ {n} \lambda^ {\prime \prime}}{C (\lambda)}, n = 1, 2, 3, \dots \dots $$ ( ) ( ) C $$ c (\lambda) = \sum_ {n = 1} ^ {\infty} a _ {n} \lambda^ {n}, a _ {n} $$ where ( ) , 1 depends on n, and λ > 0. C(λ) is = finite, and its first, second, and third derivatives with respect to λ are defined and given by C’(λ), C’’(λ), and C’’’(λ), respectively. The Table 1 below represents some useful quantities including an C(λ), C−1(λ), C’(λ), C’’(λ), and C’’’(λ), respectively, for the Poisson, geometric, logarithmic and binomial (with m being the number of replicas) distributions which belong to the power series family of distributions.

$$ \text {I f} X _ {(1)} = \min \left(X _ {1}, \dots , X _ {N}\right), $$

, then the conditional CDF of X(1)|N = n is given by () 1 1 F x n − −     The CDF of the power series class of distributions associated with some parent distribution with PDF f (x) and CDF F (x) is the marginal CDF of X(1) which is given by λ − − ( (1 ( )) 1 C F x λ ( ) C In this paper we will adopt the following terminology Definition 3.4. A power series class of distributions associated with some parent distribution whose PDF is $f(x)$ and whose CDF is $F(x)$, and defined by the marginal CDF of $X_{(1)}$ which is given by

$$1 - \frac{C(\lambda(1 - F(x))}{C(\lambda)}$$

will be called a Minimal Family of Power Series Distributions.

The $\left(\frac{1}{e}\right)^\alpha PT - G$ Family of Distributions of Ampadu CB [5] was further explored in Anafo AY [6]. In particular, by modifying the parameter space for $\alpha$ in the $\left(\frac{1}{e}\right)^\alpha PT -$ Standard Uniform family and using it in Remark 3.3 [7], we introduced a new weight as follows

$$W^*(x, \alpha) = -\log \left(\frac{1 - e^\alpha - ax}{1 - e^\alpha}\right)$$

where $\alpha \in \mathbb{R}$ and $\alpha \neq 0$ and $x \in [0, 1]$.

By inverting $W^*(x, \alpha)$, we introduce a new weight in class of distributions proposed in Ampadu CB [1], and hence a new family in the next section. In Section 3, we consider a power series application of the new family. Section 4 illustrates applicability of the new families. In Section 5, we compare the new families via some goodness of fit measures. The last section is devoted to some further recommendations.

The New Family

By finding the inverse of $W^*(x, \alpha)$, we define the new weight as

$$V^*(x, \alpha) = \frac{\log \left(\frac{e^\alpha + x}{e^\alpha + e^x - 1}\right)}{\alpha}$$

where $\alpha \in \mathbb{R}$ and $\alpha \neq 0$ and $x \in [0, 1]$.

The new weight can be obtained by solving the following equation for $V^*(x, \alpha)$:

$$x = -\log \left(\frac{1 - e^\alpha - ax}{1 - e^\alpha}\right)$$

Now suppose a random variable $T$ has support $[0, \infty]$ with PDF $r_T$ and CDF $R_T$, and the random variable $X$ has CDF $F_X$ and PDF $f_X$, we define the CDF of the $r_T - X$ family induced by $V^*$ with the following integral

$$\log \left(\frac{e^\alpha + F_X(x; \xi)}{e^\alpha + e^X(x; \xi) - 1}\right) K(x; \alpha; \xi, \beta) = \int_0^1 r_T(t, \beta) dt$$

where $x, \alpha \in \mathbb{R}$ and $\alpha \neq 0$, $\xi$ is a vector of parameters in the distribution of $X$, and $\beta$ is a vector of parameters in the distribution of $T$. The parameter space for $\xi$ and $\beta$, depends on the chosen distribution of the random variables $X$ and $T$, respectively. By evaluating the above integral, we get the following

Proposition 4.1: The CDF of the $\gamma_T - X$ family induced by $V^*$ is given by

$$K(x; \alpha, \xi, \beta) = r_T \left(\frac{\log \left(\frac{e^\alpha + F_X(x; \xi)}{e^\alpha + e^X(x; \xi) - 1}\right)}{\alpha}; \beta\right)$$

where $T$ has support $[0, \infty)$ with PDF $\gamma_T$ and CDF $R_T$, and the random variable $X$ has CDF $F_X$ and PDF $f_X$, with $x, \alpha \in \mathbb{R}$ and $\alpha \neq 0$, $\xi$ is a vector of parameters in the distribution of $X$, and $\beta$ is a vector of parameters in the distribution of $T$. The parameter space for $\xi$ and $\beta$, depends on the chosen distribution of the random variables $X$ and $T$, respectively.

Remark 4.2. The PDF can be obtained by differentiating the CDF

Power Series Application of the New Family

Using Proposition 4.1 in Definition 3.4, we have the following

Proposition 5.1: The $r_T - X(V^*)$ minimal family of power series distributions has CDF

                                                        α ξ + ( ; ) log ( ; ) 1 1 ;

F x X e

α ξ λ β α F x x e e C RT

+ − − −

1 ( ) λ C

where T has support [0, ∞) with PDF T γ and CDF RT,

and the random variable X has CDF FX and PDF fX, with x, α ∈R and α ≠ 0, ξ is a vector of parameters in the distribution of X, and β is a vector of parameters in the distribution of T, and C is some useful quantity for power series distributions. Remark 5.2: The PDF can be obtained by differentiating the CDF

Practical Illustrations

In this section we show applicability of a so-called Exponential-Normal(V*) family of distributions, and a so- called Exponential-Normal(V* ) Poisson family of distributions in fitting the breaking stress of carbon fibers data contained in Table 2 of Alzaatreh A, Lee C, Famoye F [8]. We make the following assumptions throughout this section. First, we assume T is an Exponential random variable with CDF RT (t; b) = 1 − e−bt where x, b > 0, and X is a Normal random variable with CDF $$ ) = \frac {1}{2} e r f c \left(\frac {c - x}{\sqrt {2 d}}\right) $$

1 ( ; , ) 2 2 c x F x c d erfc X d

2 2 1 0 $$ 1 - \frac {2}{\sqrt {\pi}} \int_ {0} ^ {z} e ^ {= t ^ {2}} d t, d > 0, x, c \in \mathrm {R} $$ t z e dt π where erfc(z) = 1 − erf(z) = Using Table 1 above, we assume C is Poisson, that is, C(λ) = eλ − 1, where λ > 0

Numerical Comparisons

In this section we compare the Exponential-Normal (V*), and the Exponential-Normal (V*) Poisson, family of distributions, respectively, in fitting the breaking stress of carbon fibers data contained in Table 2 of Alzaatreh A, Lee C, Famoye F [8].

In order to compare the two distribution models, we used the following criteria: -2(Log- likelihood) and 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:

2 2 AIC k l = −

( ) 2 1 $$ = A I C + \frac {2 k (k +}{n - k} $$

k k AICC AIC

− −

1 n k $$ B I C = k \log (n) - 2 l $$

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 Table 3 above, it is clear that the Exponential-Normal (V*) distribution has the smallest values across all criteria considered, hence we see the Exponential-Normal ($V^*$) distribution is a better fit than the Exponential-Normal ($V^*$) Poisson distribution to the breaking stress of carbons fibers data.

Concluding Remarks and Further Recommendations

In this paper we have introduced two new broad classes of statistical distributions, and special members of these two classes of distributions are shown to be a good fit to real life data.

Now supposing $X(n) = \max(X_1, \cdots, X_N)$, then the conditional CDF of $X(n)$ $N=n$ is given by

$$F(x)^n$$

The CDF of the power series class of distributions associated with some parent distribution with PDF $f(x)$ and CDF $F(x)$ is the marginal CDF of $X(n)$ which is given by

$$\frac{C(\lambda F(x))}{C(\lambda)}$$

Now we introduce the following

Definition 8.1. A power series class of distributions associated with some parent distribution whose PDF is $f(x)$ and whose CDF is $F(x)$, and defined by the marginal CDF of $X(n)$ which is given by

$$\frac{C(\lambda F(x))}{C(\lambda)}$$

will be called a Maximal Family of Power Series Distributions.

Using Proposition 4.1 in Definition 8.1, we introduce the following

Proposition 8.2. The $r_T - X(V^*)$ maximal family of power series distributions has CDF

$$C\left(\lambda R_T \left( \frac{\log \left( \frac{e^{\alpha + F_X(x,\xi)}}{e^{\alpha + e_F_X(x,\xi)} - 1} \right)}{\alpha}; \beta \right)\right)$$

where $T$ has support $[0, \infty)$ with PDF $r_T$ and CDF $R_T$, and the random variable $X$ has CDF $F_X$ and PDF $f_X$, with $x, \alpha \in R$ and $\alpha \neq 0$, $\xi$ is a vector of parameters in the distribution of $X$, and $\beta$ is a vector of parameters in the distribution of $T$, and $C$ is some useful quantity for power series distributions.

Remark 8.3. The PDF of the $r_T - X(V^*)$ maximal family of power series distributions can be obtained by differentiating the CDF.

The future interesting problem is to obtain some properties and applications of the $r_T - X(V^*)$ maximal family of power series distributions.

**References**

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