A New Generalization of the Inverse Distributions: Properties and Applications

In this paper the generalized inverse distribution is defined. Some properties and applications of the generalized inverse distribution are studied in some detail. Characterization theorems generalizing the new family in terms of the hazard function are obtained. Recommendation for further study concludes the paper.


Introduction
Suppose random variable X has CDF F (x), the CDF of the inverse distribution associated with the random variable X is defined as The PDF is given by Two methods for creating probability distributions appeared in [1] and [2]. The work in [2] extends that of [1]. The CDF of the distribution generalizing that of [1] is given by

Some Sub-Models of the New Family
In this section we introduce three sub-models according to the support of the new family. The PDF, CDF, survival function (SF), and hazard function (HF) are visualized.
Case #1: T has support [0, ∞) The proposed sub-model is called the Inverse Standard Exponential-Dagum {Standard Log-Logistic} distribution. The PDF is given by The CDF is given by The SF is given by a and the HF is given by where x, a, b, c > 0. We write if J is an Inverse Standard Exponential-Dagum{Standard Log-Logistic} random variable. Case #2: T has support (−∞, ∞) The proposed sub-model is called the Inverse Standard Extreme Value-Dagum {Standard Logistic} distribution. The PDF is given by The CDF is given by The SF function is given by The HF is given by , where x, a, b, c > 0. We write if JJ is an Inverse Standard Extreme Value-Dagum{Standard Logistic} random variable.
Case #3: T has support [0, 1] or bounded support in general The proposed sub-model is called the Inverse Beta(2,2)-Dagum{Standard Uniform} distribution. The PDF is given by The CDF is given by The SF is given by The HF is given by if JK is an Inverse Beta(2,2)-Dagum{Standard Uniform} random variable.

Quantile Function
Theorem 5.1. The quantile function of the generalized inverse type distribution is given by where 0 < p < 1.
Proof. With 0 < p < 1, we solve the following equation for Q(p)

Random Number Generation
If U is uniform on (0, 1), then random numbers from the generalized inverse type distribution can be obtained using

rth Non-Central Moments
Assuming the random variable R follows the standard log-logistic distribution, then the random variable follows the generalized inverse type standard log-logistic distribution. Thus we have the following Theorem 5.2. The ordinary moments of the inverse type standard log-logistic distribution for r ∈ N, can be expressed as Proof. By the binomial series we can write Hence the result

Renyi Entropy
By definition the Renyi entropy is defined as where δ > 0 and δ = 1.
Since the PDF of the inverse type standard log-logistic distribution is given by The definition immediately above implies the following Theorem 5.3. The Renyi entropy of the generalized inverse type log-logistic distribution can be expressed as where δ > 0 and δ = 1.

Parameter Estimation
The method of maximum likelihood is used in this paper to estimate model parameters. Here we discuss this method for the generalized inverse type distribution. Suppose x 1 , x 2 , · · · , x n is a random sample of size n from the generalized inverse type distribution. It can be shown that the total log-likelihood function is given by where ξ is a vector of parameters associated with the distribution of the random variable R. Partial differentiation of the total log-likelihood function with respect to model parameters gives the following as the score function Equating the score function to zero and numerically solving the equation using techniques such as the quasi Newton-Raphson method, gives the maximum likelihood estimates for the model parameters. Let = (ξ), for the purposes of constructing confidence intervals for the parameters in the hyperbolic tan-X family of distributions, the observed information matrix, call it J( ), can be used due to the complex nature of the expected information matrix. The observed information matrix is given by ∂ξ∂ξ . When the usual regularity conditions are satisfied and that the parameters are within the interior of the parameter space, but not on the boundary, the distribution of √ n( − ) converges to the multivariate normal distribution N p (0, I −1 ( )), where I( ) is the expected information matrix, and it is assumed that ξ = (ξ 1 , · · · , ξ p ). The asymptotic behavior remains valid when I( ) is replaced by the observed information matrix evaluated at J( ). The asymptotic multivariate normal distribution N p (0, J −1 ( )) is a very useful tool for constructing an approximate 100(1 − ψ)% two-sided confidence intervals for the model parameters, where ψ is the significance level.

Simulation Study
In this section we show that the method of maximum likelihood is adequate in estimating the parameters in the generalized inverse type distribution. For this, a Monte Carlo simulation study is carried out to assess the performance of the estimation method in the ISEVDSL sub-model. Samples of sizes 200, 400, 500, and 700, are drawn from the ISEVDSL distribution, and the samples have been drawn for the following set of parameters The maximum likelihood estimators for the parameters a, b and c are obtained. The procedure has been repeated 400 times, and the mean and standard deviation for the estimates are computed, and the results are summarized in Tables 1-3 below for each of sets I, II and III, respectively, considered above From the table above, we find that the simulated estimates are close to the true values of the parameters and hence the estimation method is adequate. We have also observed that the estimated standard deviation consistently decrease with increasing sample size as can been seen by plotting the standard deviation against the sample size. From the table above, we find that the simulated estimates are close to the true values of the parameters and hence the estimation method is adequate. We have also observed that the estimated standard deviation consistently decrease with increasing sample size as can been seen by plotting the standard deviation against the sample size. From the table above, we find that the simulated estimates are close to the true values of the parameters and hence the estimation method is adequate. We have also observed that the estimated standard deviation consistently decrease with increasing sample size as can been seen by plotting the standard deviation against the sample size.
Overall the simulation study conducted, indicated that using the method of maximum likelihood in estimating model parameters is adequate.

Applications
We study some new generalizations of the inverse Dagum Distribution.

Data Set #1
The proposed sub-model is called the Inverse Standard Exponential-Dagum {Standard Log-Logistic} distribution. The first application is a real data set given by [17]. It consists of thirty successive values of March precipitation (in inches) in Minneapolis/St Paul as recorded in [18].

The PDF of the Proposed Sub-Model
where x, a, b, c > 0.

The CDF of the Proposed Sub-Model
where x, a, b, c > 0.

The Competitor
The competing model is called the Inverse Standard Extreme Value-Dagum {Standard Logistic} distribution. The PDF is given by where x, a, b, c > 0, and the CDF is given by where x, a, b, c > 0.
Using the R software, we report below in Table 4, the estimates for the parameters in each of the two distributions alongside their standard errors.   Table 5. Whilst it appears from the fits above, that all the distributions are competitive in fitting the March precipitation data, Table 5 reveals that the ISEVDSL distribution is most compatible with this data set, and hence can be considered the best in this instance.

Data Set #2
The proposed sub-model is called the Inverse Standard Extreme Value-Dagum {Standard Logistic} distribution. The second application is given by [19]. The data refers to the time between failures for repairable items as recorded in [18].

The PDF of the Proposed Sub-Model
where x, a, b, c > 0.

The CDF of the Proposed Sub-Model
where x, a, b, c > 0.

The Competitor
The competing model is called the Inverse Beta(2,2)-Dagum{Standard Uniform} distribution. The PDF is given by where x, a, b, c > 0, and the CDF is given by Using the R software, we report below in Table 6, the estimates for the parameters in each of the two distributions alongside their standard errors. The fitted CDF and PDF of ISEVDSL to the repairbale items data using the above table are shown below  Table 7. Whilst it appears from the fits above, that all the distributions are competitive in fitting the repairable items data, Table 7 reveals that the ISEVDSL distribution is most compatible with this data set, and hence can be considered the best in this instance.

Data Set #3
The proposed sub-model is called the Inverse Beta(2,2)-Dagum{Standard Uniform} distribution. The third application is the vinyl chloride data obtained from clean upgrading, monitoring wells in mg/L; this data set was used by [20] and is recorded in [18].

The PDF of the Proposed Sub-Model
where x, a, b, c > 0.

The CDF of the Proposed Sub-Model
, where x, a, b, c > 0.

The Competitor
The competing model is called the Inverse Standard Exponential-Dagum {Standard Log-Logistic} distribution. The PDF is given by where x, a, b, c > 0, and the CDF is given by where x, a, b, c > 0.
Using the R software, we report below in Table 8, the estimates for the parameters in each of the two distributions alongside their standard errors. The fitted CDF and PDF of IBDSU to the vinyl chloride data using the above table are shown below  Table 9. Whilst it appears from the fits above, that all the distributions are competitive in fitting the vinyl chloride data, Table 9 reveals that the IBDSU distribution is most compatible with this data set, and hence can be considered the best in this instance.

Characterization Theorems
In this section, we present two generalizations of the generalized inverse type distributions using the hazard rate function

Hazard Function Characterization I
It is well known that the hazard function, h F , of a twice differentiable function, F , satisfies the first order differential equation In this section we present a Kumaraswamy-generalized inverse type distribution. The result here is inspired by [21]. First let us introduce the following Definition 9.1. We say a random variable X follows a Kumaraswamy-G type distribution if its CDF is given by where G is some baseline distribution, x ∈ Supp(G), and ξ is a vector of parameters in the baseline distribution whose support depends on G.
Remark 9.2. Note that if we take λ = 1 and ϕ = 2 in equation (1) of [22], then we get the CDF in the above definition.
The PDF of the Kumaraswamy-G type distribution is given by where g is the PDF of the baseline distribution. Clearly the hazard rate function of the Kumaraswamy-G type distribution is given by G(x; ξ)) .
Theorem 9.3. Let X : Ω → R be a continuous random variable. The PDF of X is for some baseline distribution with PDF g and CDF G if and only if its hazard rate function h(x) satisfies the following differential equation with boundary condition h(0) = 2g(0).
Proof. If X has PDF as stated in the theorem, then the differential equation as stated holds. Now if the stated differential equation holds, then which is the hazard rate function of the Kumaraswamy-G type distribution.
Clearly, a characterization of the Kumaraswamy-generalized inverse type distribution. is obtained from the above theorem by letting the baseline PDF be given as in Section 3.2, and letting the baseline CDF be given as in Section 3.1.

Hazard Function Characterization II
It is well known that the hazard function, h F , of a twice differentiable function, F , satisfies the first order differential equation In this section we present a Weibull-generalized inverse type distribution. The result here is inspired by [23]. First let us introduce the following or h F (x; ξ) = αg(x; ξ) G(x; ξ) α−1 G(x; ξ) α+1 which is the hazard function of Weibull-G Clearly, a characterization of the Weibull-generalized inverse type distribution. is obtained from the above theorem by letting the baseline PDF and CDF be given as in Section 3.2, and Section 3.1, respectively.

Further Recommendation
In the sense of [24] and [25], the "CDF" of the quantile generated family of distributions is given by where Q T is a quantile function, V is an appropriate weight depending on the support of T , and F (x) is some baseline distribution. The truncated distribution in the sense of [26] has CDF Suppose the random variable X has CDF given as above, and consider the random variable Z = 1 X , the CDF of Z is given by We call the random variable Z with the above CDF the inverse truncated quantile generated random variable. Obviously the PDF can be obtained by differentiating the CDF. A future interesting problem is to obtain some properties and applications of the inverse truncated quantile generated family of probability distributions.