Unit Power Distribution: New Order Statistics Approach

• Unit power distribution is a probability model defined on (0,1) by transforming the inverse Weibull, enabling flexible modeling of bounded data.
• It naturally incorporates additional shape parameters through order statistics, leading to extensions like the Kumaraswamy and Fatima families.
• Explicit moment formulas and quantile functions facilitate practical applications, often providing better fit than classical models for skewed or heavy-tailed data.

The unit power distribution is a probability distribution defined on the unit interval (0, 1), systematically derived by transforming a classical distribution on the positive real line—specifically, the inverse Weibull—and using this transformation as the parent law for constructing additional families via order statistics. This construction yields a flexible class of distributions parameterized naturally through the machinery of order statistics, resulting in both the unit power distribution itself and generalizations such as the Kumaraswamy and novel "Fatima" families. The methodology departs from traditional power transformation or T–X (transformed–transformer) family techniques by introducing new shape parameters directly through the order statistics without requiring ad-hoc functional forms.

1. Construction of the Unit Power Distribution via Transformation

Starting from the inverse Weibull (IW) distribution, which possesses the PDF and CDF

$$f_X(x) = \beta a x^{-(a+1)} e^{-B x^{-a}}, \quad F_X(x) = e^{-B x^{-a}}, \qquad x > 0,\, a,B > 0,$$

the variable transformation

$$y = e^{-x^{-a}}, \qquad \text{with inverse} \quad x = (-\ln y)^{-1/a}$$

maps the support from the positive real line to the unit interval $(0,1)$. The corresponding Jacobian is

$$\left|\frac{dx}{dy}\right| = \frac{(-\ln y)^{1/a - 1}}{a y}.$$

Applying this transformation produces the unit power distribution with density

$$f_Y(y) = a \beta [-\ln y]^{1-a} \exp\big(-B (-\ln y)^a\big), \qquad 0 < y < 1,$$

and cumulative distribution function

$F_Y(y) = 1 - e^{-B (-\ln y)^a}, \qquad 0 < y < 1.$

Here, the parameter $B$ determines the shape and concentration of the distribution on $(0,1)$.

2. Generalization via Order Statistics

Order statistics provide a canonical mechanism for generalization by incorporating additional (integer or continuous) shape parameters reflecting sample size and order. For a sample $Y_1, \ldots, Y_n$ from $F_Y$, the PDF of the $i$th order statistic is

$$f_{i:n}(y) = \frac{n!}{(i-1)!(n-i)!} [F_Y(y)]^{i-1} [1-F_Y(y)]^{n-i} f_Y(y), \qquad 0 < y < 1,$$

where $f_Y$ and $F_Y$ derive from the unit power parent distribution.

Specializations yield notable families:

• Smallest order ($i=1$): $f_{1:n}(y) = n [1 - F_Y(y)]^{n-1} f_Y(y)$—this yields the Kumaraswamy distribution for suitable parameterizations, illustrating its derivation as the minimum of $n$ i.i.d. unit power variables.
• Largest order ($i=n$): $f_{n:n}(y) = n [F_Y(y)]^{n-1} f_Y(y)$—labeled Fatima 1 in the paper, constituting a separate three-parameter generalization.
• $i$th order: Further expansion as Fatima 2 and analog families for the transformed unit Rayleigh (see below), each increasing flexibility and control over the density's shape and tail behavior.

3. Extension to Other Parent Distributions (Unit Rayleigh Example)

Analogous treatment applies to the Rayleigh distribution. The Rayleigh parent,

$$f_W(w) = 2w e^{-w^2}, \quad F_W(w) = 1 - e^{-w^2}, \quad w>0,$$

is mapped to $(0,1)$ by $y = e^{-w^2} \implies w = [-\ln y]^{1/2}$, with Jacobian $|dw/dy| = \frac{1}{2y (-\ln y)^{1/2}}$. This gives the unit Rayleigh distribution: $$f_Y(y) = \frac{1}{2(-\ln y)^{1/2}} y^{-1}, \quad 0<y<1.$$ Order statistics generalize this parent as Fatima 3 (minimum), Fatima 4 (maximum), and so on, following the same formulas as above.

4. Comparison with Power Transformation and T-X Family Methods

The order statistics approach differs fundamentally from:

• Power transformation: Simple exponentiation of the parent CDF, producing distributions like $F^{\theta}(y)$, e.g., Johnson SB.
• T–X family method: Applying an invertible function $T$ to the parent CDF as $T(F_X(x))$ for generalization.

In contrast, the order statistics method embeds the original parent density within the combinatorial structure of order statistics, with the additional parameter $n$ (sample size) or $i$ (order index) introduced naturally. This results in richer modeling flexibility, as these parameters control the shape (e.g., skewness, kurtosis) and are especially effective in cases where hazard functions or tail behaviors need tuning.

5. Properties and Functional Forms

For these unit distributions, the paper provides explicit formuli for:

• Raw moments: For example, for Fatima 1 (largest order statistic of unit power), the $r$th raw moment is

$$E(Y^r) = \frac{1}{a\beta} (\text{expression involving Gamma functions}),$$

with closed-form solutions available for special parameter values.

• Quantile functions: For distributions like Fatima 1 and Fatima 3 with invertible CDF, the quantile funtion $Q(u)=F_Y^{-1}(u)$ is derived analytically, allowing for straightforward random variate generation and percentile computations.

6. Real Data Applications and Model Comparison

An application is presented using a water quality satisfaction dataset (proportion data). Competing models—Beta, Kumaraswamy, and order-statistics–generalized families (Fatima 1–7)—are evaluated using standard goodness-of-fit measures (AIC, CAIC, BIC, HQIC) and test statistics (KS, CVM, AD). The findings indicate that while the classical Beta often yields the lowest AIC, order-statistics–based generalizations provide competitive fit and greater shape flexibility. For skewed or heavy-tailed empirical data, Fatima distributions may better capture the observed sample behavior due to their adaptable hazard rates and cumulative function forms.

Family Name	Construction Base	Added Parameter(s)
Unit Power	IW transform	$a,B$
Kumaraswamy	Min order statistic	$a, b$
Fatima 1-7	Order statistics	$n, i$, derived from parent

7. Summary and Significance

The unit power distribution, as derived from the inverse Weibull and extended via order statistics, constitutes a flexible and analytically tractable family of distributions on $(0,1)$. The generalization via order statistics systematically introduces additional shape parameters, distinguishing this approach from power transformation and transformer-based families. Analytically-expressible moments, quantile functions, and explicit functional forms support practical applications such as distribution fitting for fractional data, with empirical evidence showing competitive or superior fit to classical alternatives. This construction provides a principled and unifying framework for modeling bounded data with varying shapes and tail behaviors, and the methodology is extendable to other parent distributions on $(0,1)$, as demonstrated with the unit Rayleigh and related Fatima families (Attia, 14 Sep 2025).