2. A new family of pseudometric spaces

This section explains a class of pseudometrics on $\mathbb{R}^n$ which preserves majorization ordering. Let $\tilde{\mathbf{x}}=(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n$ ; in particular, $\tilde{\mathbf{x}}_i=(x_{i1},x_{i2},\ldots,x_{in})$ , where $i\in \mathbb{N}$ , and $\mathbf{a}^*=(a,a,\ldots,a)$ , $\mathbf{a}^{**}=(a,0,,0,\ldots,0)$ for $a\in \mathbb{R}$ . We start with the concept of majorization in $\mathbb{R}^n$ (see [Reference Marshall, Olkin and Arnold26]).

Definition 2.1. Let $\tilde{\mathbf{x}},\tilde{\mathbf{y}}\in\mathbb{R}^n$ ; then $\tilde{\mathbf{x}}$ is majorized by $\tilde{\mathbf{y}}$ (denoted by $\tilde{\mathbf{x}}\prec\tilde{\mathbf{y}}$ ) if

\begin{equation*}\begin{cases}\sum_{i=1}^{k}x_{[i]}\leq \sum_{i=1}^{k}y_{[i]} & \text{for all $k=1,2,\ldots,n-1$,}\\[3pt] \sum_{i=1}^{n}x_{[i]} = \sum_{i=1}^{n}y_{[i]} , &\end{cases}\end{equation*}

where $(x_{[1]}, x_{[2]},\ldots,x_{[n]})$ represents decreasing rearrangements of $(x_1,x_2,\ldots,x_n)$ .

The following definition defines a general class of pseudometrics that preserves an essential property of majorization.

Note that from the relation between majorization and the Lorenz curve (see [Reference Arnold and Sarabia4]), axiom 4 is equivalent to:

1. 4^′ If $ \tilde{\mathbf{x}}_{1},\tilde{\mathbf{x}}_{2} \in\mathbb{R}^n$ with $\sum_{j=1}^{n}x_{1j}=\sum_{j=1}^{n}x_{2j}$ and $ L_{\tilde{\mathbf{x}}_{1}}(p)\geq L_{\tilde{\mathbf{x}}_{2}}(p)$ , $0 \lt p \lt 1$ and for some p, $ L_{\tilde{\mathbf{x}}_{1}}(p) \gt L_{\tilde{\mathbf{x}}_{2}}(p) \Rightarrow d(\tilde{\mathbf{x}}_{1},\mathbf{x}_{\mathbf{1.}}^{*}) \lt d(\tilde{\mathbf{x}}_{2},\mathbf{x}_{\mathbf{1.}}^{*}),$ where $L_{\tilde{\mathbf{x}}}(p)$ denotes the Lorenz curve of $\tilde{\mathbf{x}}$ .

Example 2.1. Let d be the Euclidean metric in $\mathbb{R}^n$ , that is,

\[d(\tilde{\mathbf{u}},\tilde{\mathbf{v}})=\Biggl[\sum_{i=1}^n(u_i-v_i)^2\Biggr]^{{{1}/{2}}},\]

where $\tilde{\mathbf{u}}=(u_1,u_2,\ldots,u_n)$ and $\tilde{\mathbf{v}}=(v_1,v_2,\ldots,v_n)$ . Then d is an m-pseudometric on $\Re^n$ .

Example 2.2. Let d be the $L^r$ -metric in $\mathbb{R}^n$ with $1 \lt r \lt \infty$ , that is,

\[d(\tilde{\mathbf{u}},\tilde{\mathbf{v}})=\Biggl[\sum_{i=1}^n(u_i-v_i)^r\Biggr]^{{{1}/{r}}}.\]

Assume $\tilde{\mathbf{u}}\prec\tilde{\mathbf{v}}$ and $\tilde{\mathbf{v}}\nprec \tilde{\mathbf{u}}$ . Since the function $|x-u_.|^r$ , where $r \gt 1$ , $x\in \mathbb{R}$ and $u_. = \frac{1}{n} \sum_{j=1}^{n} u_{j}$ , is a convex function, we have $d(\tilde{\mathbf{u}},\mathbf{u}_{\mathbf{.}}^*) \lt d(\tilde{\mathbf{v}},\mathbf{v}_{\mathbf{.}}^*)$ (see [Reference Berge7, page 184]). So d is an m-pseudometric on $\mathbb{R}^n$ .

The following metric property is still valid for an m-pseudometric.

Proof. Since d is an m-pseudometric, the result follows from the elementary result that if $x \gt 0$ , $y \gt 0$ , and $x \lt y$ , then

\begin{equation*} \frac{x}{1+x} \lt \frac{y}{1+y}.\end{equation*}

3. A new family of randomness measures

This section introduces a new methodology to measure the randomness of a discrete random variable using an m-pseudometric. So this is a metric-based measure of randomness and can be considered an alternative measure of entropy for measuring randomness.

The following defines a special subset of $\mathbb{R}^n$ :

(3.1) \begin{equation}\Gamma=\Biggl\{\tilde{\mathbf{p}}=(p_1,p_2,\dots,p_n)\in\mathbb{R}^n \mid\sum_{i=1}^np_i=1, p_i\geq 0, i=1,2,\ldots,n\Biggr\}.\end{equation}

Let X be a random variable with support $\{x_1,x_2,\ldots,x_n\}$ . Let $\tilde{\mathbf{p}}=(p_1,p_2,\ldots,p_n)$ be the probability distribution of X. That is, $\mathbb{P}(X=x_i)=p_i,i\in\{1,2,\ldots,n\}$ . Then clearly $\tilde{\mathbf{p}} \in \Gamma$ .

For any $\tilde{\mathbf{p}} \in \Gamma$ , $\tfrac{\mathbf{1}}{\mathbf{n}}^*\prec \tilde{\mathbf{p}}\prec \mathbf{1}^{**}$ . That is, a random variable with equally likely probabilities represents the most randomness, and a random variable degenerate at one point represents non-randomness, and all other PMFs are between these two extreme cases. Let Y, for example, follow a uniform distribution with parameter n and let Z be a random variable which degenerates at a point (i.e. $\mathbb{P}(Z=a)=1$ , $ a\in \mathbb{R}$ ). Then, for any X with support $\{x_1,x_2,\ldots,x_n\}$ , where each $x_i^{\prime}$ , $i=1,2,\ldots,n$ , is distinct, the randomness of Y is greater than the randomness of X, which is greater than the randomness of Z. If $\tilde{\mathbf{p}}=(p_1,p_2,\ldots,p_n)$ and $\tilde{\mathbf{q}}=(p_{(1)},p_{(2)},\ldots,p_{(n)})$ , where $((1),(2),\ldots,(n))$ is a permutation of $(1,2,\ldots,n)$ , then $\tilde{\mathbf{p}}\prec \tilde{\mathbf{q}}$ and $\tilde{\mathbf{q}}\prec \tilde{\mathbf{p}}$ , and therefore the randomness of $\tilde{\mathbf{p}}$ and $\tilde{\mathbf{q}}$ is identical [Reference Hickey15].

We call the m-pseudometric defined on $\Gamma$ a random pseudometric. Since, for any $\tilde{\mathbf{p}} \in \Gamma $ , we have $\tilde{\mathbf{p}}\prec \mathbf{1}^{**}$ and $\mathbf{1}^{**} \nprec \tilde{\mathbf{p}}$ , this implies $d(\tilde{\mathbf{p}},\tfrac{\mathbf{1}}{\mathbf{n}}^*)\lt d(\mathbf{1}^{**},\tfrac{\mathbf{1}}{\mathbf{n}}^*)$ . So, the random pseudometric is bounded. The following definition introduces a new class of randomness measures (we call it the N-randomness measure) of a discrete random variable X having finite support.

Definition 3.2. Let $\Gamma$ be a class of probability distributions defined as

\[\Gamma=\Biggl\{\tilde{\mathbf{p}}=(p_1,p_2,\dots,p_n)\in\mathbb{R}^n \mid \sum_{i=1}^np_i=1, p_i\geq 0, i=1,2,\ldots,n \Biggr\},\]

and let d be a random pseudometric on $\Gamma$ . Let X be a random variable with PMF $\tilde{\mathbf{p}}=(p_1,p_2,\ldots,p_n)$ . Then the N-randomness measure of X is defined as

\[\zeta(X)=1-k_{n}d\biggl(\tilde{\mathbf{p}},\dfrac{\mathbf{1}}{\mathbf{n}}^*\biggr),\]

where

\[k_{n}=\dfrac{1}{\max d\bigl(\tilde{\mathbf{p}},\tfrac{\mathbf{1}}{\mathbf{n}}^*\bigr)}=\dfrac{1}{d\bigl(\mathbf{1}^{**},\tfrac{\mathbf{1}}{ \mathbf{n}}^*\bigr)}.\]

The following properties are essential for randomness measures, location invariance, invariance under injective transformation and symmetry (invariance under permutation). Since $\zeta$ depends only on probabilities, the results follow,

The following example illustrates the above theorems in binomial distributions.

Example 3.1. Let $X\sim B(3,\tfrac{1}{2})$ and $Y=X+2$ , where B(n,p) represents a binomial distribution with parameters n and p. Then, from Theorem 3.1, $\zeta(Y)=\zeta(X)$ . Now assume $Z=X^2$ . Since the square function is injective on the positive real line and from Theorem 3.2, $\zeta(Z)=\zeta(X)$ . Let W be a random variable with PMF

\[f_W(w)=\begin{cases}\frac{3}{8} & \text{if $w = 0,1$,}\\\frac{1}{8} & \text{if $w = 2,3$.}\end{cases}\]

Then, from Theorem 3.3, $\zeta(W)=\zeta(X)$ .

4. Gini randomness measure for discrete distributions

This section discusses a randomness measure based on a new pseudometric, which we call the Gini pseudometric.

Definition 4.1. Let $\Gamma$ be the class of discrete probability distributions as defined in (3.1). Then the Gini pseudometric $d_g$ is defined as $d_g\,\colon\, \Gamma \times\Gamma\rightarrow [0,\infty)$ such that

(4.1) \begin{equation}d_g(\tilde{\mathbf{u}},\tilde{\mathbf{v}})=\Biggl\lvert\sum_{i=1}^n\sum_{j=1}^n\lvert u_i-u_j\rvert-\sum_{i=1}^n\sum_{j=1}^n\lvert v_i-v_j\rvert \Biggr\rvert,\end{equation}

where $\tilde{\mathbf{u}},\tilde{\mathbf{v}}\in \Gamma$ .

The Gini pseudometric is a random pseudometric. The first three axioms in Definition 2.2 are direct. Now $d_g(\tilde{\mathbf{u}},\mathbf{u}_{\mathbf{.}}^*)=\sum_{i=1}^n\sum_{j=1}^n|u_i-u_j|$ is a constant multiple of the Gini coefficient of $\tilde{\mathbf{u}}$ (see [Reference Kendall, Stuart and Ord19]). Hence $\tilde{\mathbf{u}}\prec\tilde{\mathbf{v}}$ and $\tilde{\mathbf{v}}\nprec \tilde{\mathbf{u}}$ implies $d_g(\tilde{\mathbf{u}},\mathbf{u}_{\mathbf{.}}^*) \lt d_g(\tilde{\mathbf{v}},\mathbf{v}_{\mathbf{.}}^*)$ (see [Reference Allison2], [Reference Marshall, Olkin and Arnold26]). That is, the function $d_g$ satisfies axiom 4 of Definition 2.2.

Using the Gini pseudometric, a direct calculation gives $d_g(\mathbf{1}^{**},\tfrac{\mathbf{1}}{\mathbf{n}}^*)=2(n-1)$ . Hence the following definition introduces a randomness measure which we call the Gini randomness measure; a similar formula is used for measuring evenness in ecology (see [Reference Chao and Ricotta8], [Reference Kvålseth24]).

Definition 4.2. Let X be a discrete random variable with PMF

\[f_X(x)=\begin{cases}p_i & \text{if $x = x_i, i\in\{1,2,\ldots,n\}$,}\\0 & \text{otherwise.}\end{cases}\]

The Gini randomness measure of X ( $\zeta_g(X)$ ) is defined as

\[\zeta_g(X)=1-\dfrac{1}{2(n-1)}\sum_{i=1}^n\sum_{j=1}^n|p_i-p_j|.\]

Different formulas can be used to calculate the Gini randomness measure; see [Reference Sen31] and [Reference Yitzhaki33]. The major advantage of this randomness measure is the powerful theory available for the Gini coefficient. That can be useful for further study of randomness (see [Reference Yitzhaki and Schechtman34]). To illustrate this point, from [Reference Yitzhaki and Schechtman34], there are different ways to represent the Gini coefficient, which again means that the Gini randomness measure has different representations. The Gini coefficient, for instance, can be defined using the Lorenz curve (see [Reference Gastwirth11]), and hence the Gini randomness measure of a random variable X can be defined using the Lorenz curve; see Definition 7.2.

In addition to Theorems 3.1, 3.2, and 3.3, the Gini randomness measure is continuous with respect to each $p_1,p_2,\ldots,p_n$ . The entropy measure also satisfies these properties (see [Reference Csiszár9]). So, both the Gini randomness measure and the entropy measure satisfy many of the essential properties of a randomness measure. The following discussion shows the advantages of the Gini randomness measure over entropy measures in applied situations. Example 4.2 illustrates a comparative study of the Gini randomness measure and Shannon entropy on sensitivity. The example shows some situations in which the Gini randomness measure is more sensitive than the entropy measure.

Example 4.2. (Comparison of Shannon entropy and Gini randomness measure.) Let $X\sim B(1,p)$ . The Shannon entropy of X (denoted by H(X)) is

\[H(X)=-((1-p)\log_{2}(1-p)+p\log_{2}p), \quad 0\leq p \leq 1.\]

The Gini randomness measure of X is

\begin{equation*}\zeta_g(X)=\begin{cases}2p & \text{if $0 \leq p \leq \tfrac{1}{2} $,}\\2- 2p & \text{if $\tfrac{1}{2} \leq p \leq 1 $.}\end{cases}\end{equation*}

Figure 1 illustrates the difference between the two measures. The blue line represents the Shannon entropy for p, and the red line represents the Gini random measure for p. Note that both measures take the maximum when p is $\tfrac{1}{2}$ and the minimum when p is 0 and 1.

Figure 1. The figure illustrates the Shannon entropy and Gini randomness measure of a random variable following B(1,p) distribution.

Also, the sensitivity of the two measures is different. The Gini randomness measure is more sensitive when more probabilities are scattered, whereas Shannon entropy is sensitive to the ratio of probabilities (see [Reference Allison2]). To illustrate this point, let X and Y have probability distributions $(p_1,p_2,\ldots,p_n)$ and $(q_1,q_2,\ldots,q_n)$ respectively. The probabilities are in descending order with $p_k=q_k$ for $k=1,2,\ldots,i-1,i+1,\ldots,j-1,j+1,\ldots,n$ and $q_i=p_i+a$ , $q_j=p_j-a$ , where a is a constant. From [Reference Allison2], $\zeta_g(X)-\zeta_g(Y)=c_1 a(j-i)$ , whereas $H(X)-H(Y)=c_2 a \log({{p_i}/{p_j}})$ , where $c_1$ and $c_2$ are constants. Note that the sensitivity of the Gini randomness measure depends on the ranks of i and j in $1,2,\ldots,n$ , but the Shannon entropy depends on the numerical values of $p_i$ and $p_j$ . So $\zeta_g(X)-\zeta_g(Y)$ is high when more probability values lie between $p_i$ and $p_j$ , and $H(X)-H(Y)$ is high when $p_i$ is near to one and $p_j$ is near to zero.

This can be seen in Figure 1, since for each p, the Bernoulli distribution has two probabilities $(p_1,p_2)$ , and from the previous paragraph, $i=2$ and $j=1$ , and the difference in $\zeta_g(X)-\zeta_g(Y)$ depends only on the transfer probability a whereas $H(X)-H(Y)$ depends on both a and the values of probabilities $p_i$ and $p_j$ . So the Gini randomness curve has a constant slope and the Shannon entropy curve takes a higher slope when p is near zero or one (so $p_1$ is near to one and $p_2$ is near to zero).

For example, let p change from $0.52$ to $0.51$ ; then the corresponding change in entropy is $0.0009$ , which is almost zero, and the variation in the Gini randomness measure is $0.02$ . So, in Bernoulli distribution, when p changes from $0.52$ to $0.51$ , Gini ( $0.02$ ) exhibits more change in randomness than entropy ( $0.0009$ ). Now if p changes from $0.02$ to $0.01$ , then the variation in the Gini randomness measure shows the same value $0.02$ (since here both a and $j-i$ are the same as in the previous case). However, the corresponding change in entropy is 0.06. So, in this case, the entropy ( $0.06$ ) shows more change in randomness than Gini ( $0.02$ ) since here p is small. This implies that $p_1$ is near to 1 and $p_2$ is near to zero. The same thing can be seen when the number of probabilities in a distribution is more than two. So, to study changes in the randomness of probability distributions that are more scattered, the Gini randomness measure provides better results since it captures the changes more effectively than the Shannon entropy.

The following property of the Gini randomness measure is another advantage that popular entropy measures do not satisfy.

To check that the Gini randomness measure satisfies the value validity property, we use the following procedure from Kvålseth [Reference Kvålseth23, page 4859]. Let U and V be uniform and degenerate random variables, respectively. Let X be a random variable with PMF $(1-\lambda+{{\lambda}/{n}},{{\lambda}/{n}}, \ldots, {{\lambda}/{n}})$ , $\lambda \gt 0$ (denoted by $\widetilde{P_X}$ ). Then a measure $\gamma$ satisfies the value validity property if

(4.2) \begin{equation}\dfrac{\gamma(U)-\gamma(X)}{\gamma(U)-\gamma(V)}=\dfrac{d(U,X)}{d(U,V)}=1-\lambda,\end{equation}

where d denotes the metric, and Kvålseth suggests that the Euclidean distance and the Minkowski class of distance metrics satisfy (4.2). Since

\[d_g(\tfrac{\mathbf{1}}{\mathbf{n}}^*,\widetilde{P_X})=2(n-1)(1-\lambda),\quad d_g\big(\tfrac{\mathbf{1}}{\mathbf{n}}^*, \mathbf{1}^{**}\big)=2(n-1)\quad\text{and}\quad \zeta_g(X)=\lambda,\]

the Gini pseudometric satisfies (4.2) and the Gini randomness measure satisfies the value validity property.

Kvålseth [Reference Kvålseth23, Reference Kvålseth24] argues that the value validity property is crucial for a realistic representation of evenness characteristic of a species distribution and in many other applied situations. The popular entropy measures do not satisfy this property, so in such situations, the Gini randomness measure is more appropriate than the entropy measures. Another crucial advantage of the Gini randomness measure compared to entropy measures is its nice continuous extension (see Section 7).

The theory discussed for discrete random variables in the previous sections can be extended to continuous random variables without losing many of its crucial properties, which are the topic of the following sections.

5. Random pseudometric on the space of bounded random variables

This section extends the methodology defined on $\Gamma$ , a subset of $\mathbb{R}^n$ , to the space of bounded random variables (here ‘bounded random variable’ means either bounded discrete random variables or continuous random variables with bounded interval support). So, the results could be extended to continuous cases. Let $(\Omega,\texttt{A},\mu)$ be a measure space where $\Omega$ is a non-empty finite set of real numbers or a bounded interval subset of $\mathbb{R}$ , and $\mu$ is a counting or Lebesgue measure, respectively. Let $\Lambda$ be a class of all bounded non-negative random variables such that its integral with respect to $\mu$ is unity. That is, if a random variable $X\in \Lambda$ , then $\int_{0}^{\infty}X \,{\mathrm{d}} \mu =1$ .

Proof. Let $\Omega$ be a sample space with finite measure and suppose $X\equiv a$ . Then

\begin{equation*} \int_{\Omega }X(w) \,{\mathrm{d}} \mu(w) =1 \end{equation*}

implies

\begin{equation*} a =\dfrac{1}{\int_{\Omega } \,{\mathrm{d}} \mu(w)}.\end{equation*}

Let $\nu$ denote the degenerate random variable at ${{1}/{(\mu(\Omega))}}$ , that is, if $\Omega$ is a finite set with n elements or a bounded interval (a,b), then $\nu$ is a random variable degenerate at ${{1}/{n}}$ or ${{1}/{(b-a)}}$ respectively. Let

\[ L_X(p)=\frac{1}{\mu_X}\int_0^pQ_X(q)\,{\mathrm{d}} q,\quad 0\leq p \leq1 , \]

where $\mu_X=\mathbb{E}(X)$ and $Q_X$ is the quantile function of X (for the Lorenz curve definition using the quantile function, see [Reference Gastwirth11]). The following class of metrics is a generalization of the random pseudometrics in the class of random variables $\Lambda$ .

Then d is a random pseudometric.

Example 5.1. Let $X,Y \in \Lambda$ and

\[ M_g(X)=g^{-1}\biggl(\int_{J}g(x)\,{\mathrm{d}} F(x)\biggr) \]

be the quasi-arithmetic mean (see [Reference Porcu, Mateu and Christakos28]), where g is a continuous, strictly increasing, and concave function, J is the support of X, and $\mu_X$ represents E(X). Then the function

\[d(X,Y)=\biggl\lvert\dfrac{M_g(X)}{\mu_X}-\dfrac{M_g(Y)}{\mu_Y}\biggr\rvert\]

is a random pseudometric. The first three axioms in Definition 5.1 are direct, and since

\[d(X,\nu)=1-\dfrac{M_g(X)}{\mu_X},\]

which is the Atkinson index, then axiom 4 follows (see [Reference Allison2], [Reference Atkinson5]).

6. Randomness measure of general random variable

Let $\Phi$ be a set of all random variables, either discrete or continuous, with bounded support. Let $f_X$ be the PMF or PDF of X. Let U be a uniform random variable that is either discrete or continuous.

For a given $X\in \Phi$ , let $f_X(U)$ be a random variable created by transforming U using the PMF or PDF of X; U and X have the same support (a similar transform can be seen in the literature; see [Reference Ahmad and Kochar1], [Reference Di Crescenzo, Paolillo and Suárez-Llorens10], and [Reference Rommel, Bonnans, Gregorutti and Martinon30]). If X is a discrete random variable having outcomes $\{x_1,x_2,\ldots,x_n\}$ and the PMF is $\mathbb{P}(X=x_i)=p_i$ , $i\in\{1,2,\ldots,n\}$ , then $L_{\tilde{\mathbf{p}}}(r)=L_{f_X(U)}(r)$ , for all $r\in(0,1)$ . For example, if $X\sim B(4,\tfrac{1}{2})$ , then to calculate $f_X(U)$ , consider $\mathbb{P}(U=i)=\tfrac{1}{5}$ , $i=0,1,2,3,4$ . Then use the binomial PMF, that is,

\[f_X(x)=\begin{cases}\tfrac{1}{16} & \text{if $x= 0 , 4$,}\\[3pt] \tfrac{4}{16} & \text{if $x= 1 , 3$,}\\[3pt] \tfrac{6}{16} & \text{if $x= 2 $.}\end{cases}\]

This implies that $f_X(U)$ is a random variable with PMF

\[ f_{f_X(U)}(x)= \begin{cases} \tfrac{2}{5} & \text{if $x=\tfrac{ 1}{16} $,}\\[3pt] \tfrac{2}{5} & \text{if $x=\tfrac{ 4}{16} $,}\\[3pt] \tfrac{1}{5} & \text{if $x= \tfrac{ 6}{16} $.} \end{cases} \]

Then its Lorenz curve is

\[L_{\tilde{\mathbf{p}}}(r)=L_{f_X(U)}(r)=\begin{cases}\tfrac{5}{16}r & \text{if $0\leq r \lt \tfrac{2}{5}$,}\\[3pt] \tfrac{5}{4}r- \tfrac{3}{8} & \text{if $\tfrac{2}{5}\leq r \lt \tfrac{4}{5}$,}\\[3pt] \tfrac{15}{8}r-\tfrac{7}{8} & \text{if $\tfrac{4}{5}\leq r \leq 1$,}\end{cases}\]

where $\tilde{\mathbf{p}}=(\tfrac{1}{16},\tfrac{4}{16},\tfrac{6}{16},\tfrac{4}{16},\tfrac{1}{16})$ . Also, if $\tilde{\mathbf{p}}$ and $\tilde{\mathbf{q}}$ correspond to $f_X(U)$ and $f_Y(U)$ , then $\tilde{\mathbf{p}}\prec \tilde{\mathbf{q}}$ if and only if $L_{f_X(U)}(r)\geq L_{f_Y(U)}(r)$ , for all $ r\in[0,1]$ (see [Reference Marshall, Olkin and Arnold26, page 718]). Theorem 6.1 extends this result to continuous random variables, that is, the relation between $f_X$ (the density function of X) and $f_X(U)$ (a random variable created by transforming U using the PDF of X; U and X have the same support). The theorem establishes that axiom 4 in Definition 5.1 is a valid extension of axiom 4 in Definition 2.2. To prove Theorem 6.1, the following results (from [Reference Hickey16, page 924] and [Reference Marshall, Olkin and Arnold26, page 719] respectively) are required.

Lemma 6.1. Let $f_X$ and $f_Y$ be two densities with the same support. Then $f_X$ is majorized by $f_Y$ ( $f_X\prec f_Y$ ) if and only if

\[\int_{-\infty}^{\infty}h(f_X(x))\,{\mathrm{d}} x\leq\int_{-\infty}^{\infty}h(f_Y(x))\,{\mathrm{d}} x,\]

for every continuous convex function h.

Since $f_X(U)\in \Lambda$ , the following definition generalizes Definition 3.2 to a bounded random variable.

Definition 6.1. Let $X\in \Phi$ be a random variable. Then the N-randomness measure of X is defined as

\[\zeta(X)=1-K_Xd(f_X(U),f_U(U)),\]

where d is a random pseudometric, U is a uniform random variable with X and U having the same support, and

\[K_X=\dfrac{1}{\sup_{f_X(U)\in \Lambda} d(f_X(U),f_U(U))}.\]

The following example illustrates the possible X which attain the supremum.

Example 6.1. Assume, without loss of generality, that [0,1] is the support of $f_X(U)$ . Then consider the sequence of random variables $\{f_{Z_n}(U)\}_{n\geq 1}$ with PDF

\[f_{f_{Z_n}(U)}(z)=\dfrac{1}{n}z^{{{(1-n)}/{n}}},\quad 0\leq z\leq 1.\]

Then $f_{Z_n}(U)\in \Lambda$ . Also, it is easy to verify that $\mathbb{E}(f_{Z_n}(U))={{1}/{(n+1)}}$ ,

\begin{align*}F_{f_{Z_n}(U)}(z)&=z^{{{1}/{n}}},\quad 0\leq z\leq 1,\\[3pt] Q_{f_{Z_n}(U)}(p)&=p^n,\quad 0\leq p\leq 1,\\[3pt] L_{f_{Z_n}(U)}(p)&=p^{n+1},\quad 0\leq p\leq 1.\end{align*}

Therefore,

\begin{equation*} \lim_{n\rightarrow\infty}L_{f_{Z_n}(U)}(p)=\begin{cases}0 & \text{if $0 \lt p \lt 1 $,}\\1 & \text{if $p = 1$.}\end{cases}\end{equation*}

So the random variable X which satisfies $\sup_{f_X(U)\in \Lambda} d(f_X(U),f_U(U))$ has the above Lorenz curve; see Theorem 7.3.

Note that Definitions 3.2 and 6.1 are the same when X is a discrete random variable. Since $\zeta$ depends only on the PDFs, the result follows.

The above theorem shows that the location change does not affect the randomness of a random variable.

7. Gini randomness measure for general distributions

This section introduces the Gini randomness measure of a random variable. The following definition gives the Gini pseudometric on $\Lambda$ .

Definition 7.1. (Gini pseudometric.) Let $X,Y\in \Lambda$ be two random variables. Then the Gini pseudometric d is defined as $d\,\colon\, \Lambda \times \Lambda\rightarrow [0,1]$ such that

\[d(X,Y)=\int_0^1|L_{X}(p)-L_{Y}(p)|\,{\mathrm{d}} p.\]

The Gini pseudometric is the usual $L_1$ -metric defined on $\Lambda$ , which implies properties 1, 2, and 3 of Definition 5.1. Since $L_{\nu}(p)=p$ , $0\leq p\leq 1$ , and for any random variable X, $L_{X}(p) \leq p$ for $0\leq p\leq 1$ ,

\begin{align*}d(X,\nu) &= \int_0^1|L_{X}(p)-L_{\nu}(p)|\,{\mathrm{d}} p\\&= \int_0^1 (p-L_{X}(p))\,{\mathrm{d}} p \\&= \dfrac{1}{2}-\int_0^1L_X(p)\,{\mathrm{d}} p .\end{align*}

So, $d(X,\nu)$ is the Gini coefficient (see [Reference Arnold and Sarabia4, page 51]), which satisfies the Lorenz ordering (see [Reference Wei, Allen and Liu32]), and this implies property 4.

The following defines the constant $K_X$ :

\begin{equation*}K_X=\begin{cases}\dfrac{n}{n-1} & \text{if} \, \textit{X} \text{is discrete having support $\{x_1,x_2,\ldots,x_n\}$,}\\[7pt]1 & \text{if} \, \textit{X} \, \text{is continuous with bounded support}.\end{cases}\end{equation*}

Using Definitions 6.1 and 7.1, the following definition introduces the Gini randomness measure for a bounded random variable.

Definition 7.2. For any $X\in \Phi$ , the Gini randomness measure of X $(\zeta_g(X))$ is defined as

\begin{equation*}\zeta_g(X)=\begin{cases}\displaystyle 2\dfrac{n}{n-1}\int_0^1L_{f_X(U)}(p)\,{\mathrm{d}} p-\dfrac{1}{n-1} & \text{if X has support $\{x_1,x_2,\ldots,x_n\}$,}\\[11pt]\displaystyle 2\int_0^1L_{f_X(U)}(p)\,{\mathrm{d}} p & \text{if} \textit{X} \text{is continuous with bounded support.}\end{cases}\end{equation*}

Note that Definitions 4.2 and 7.2 are the same when X is a discrete random variable. The following theorems show that the Gini randomness measure attains the maximum 1 for a uniform distribution and the minimum 0 for a degenerate distribution.

Example 7.1. (Truncated exponential distribution.) Let X be a random variable which follows a truncated exponential distribution. Then its PDF is

\[f_X(x)=\dfrac{\theta}{1-{\mathrm{e}}^{-\theta \nu}}{\mathrm{e}}^{-\theta x}, \quad 0 \lt x \lt \nu,\ \nu \gt 0,\ \theta \gt 0.\]

Let U be a uniform random variable in the interval $(0,\nu)$ . Then

\[f_X(U)=\dfrac{\theta}{1-{\mathrm{e}}^{-\theta \nu}}{\mathrm{e}}^{-\theta U}.\]

The PDF of $f_X(U)$ is

\begin{equation*}f_{f_X(U)}(x)=\begin{cases}\dfrac{1}{\theta \nu x} & \text{if $\dfrac{\theta}{{\mathrm{e}}^{\theta \nu}-1} \lt x \lt \dfrac{\theta}{1-{\mathrm{e}}^{-\theta \nu}}$,}\\[7pt]0 & \text{otherwise,}\end{cases}\end{equation*}

and the distribution function of $f_X(U)$ is

\[F_{f_X(U)}(x)=\dfrac{1}{\theta \nu}\ln\biggl\{\dfrac{x({\mathrm{e}}^{\theta\nu}-1)}{\theta}\biggr\}, \quad\dfrac{\theta}{{\mathrm{e}}^{\theta \nu}-1} \lt x \lt \dfrac{\theta}{1-{\mathrm{e}}^{-\theta \nu}}.\]

Therefore the quantile function of $f_X(U)$ is

\[Q_{f_X(U)}(p)=\dfrac{\theta}{{\mathrm{e}}^{\theta \nu}-1}{\mathrm{e}}^{\theta\nu p},\quad 0 \lt p \lt 1.\]

A simple calculation gives $\mathbb{E}(f_X(U))={{1}/{\nu}}$ . Using the Lorenz curve equation

\[L_X(p)=\dfrac{1}{\mu_X}\int_0^pQ_X(q)\,{\mathrm{d}} q,\quad 0\leq p \leq1,\]

we get

\[L_{f_X(U)}(p)=\dfrac{{{\mathrm{e}}^{\theta \nu p}-1}}{{\mathrm{e}}^{\theta \nu}-1},\quad 0\leq p\leq 1.\]

So, by Definition 7.2, the randomness measure of X is

(7.1) \begin{equation}\zeta_g(X)=\dfrac{1}{\theta\nu}-\dfrac{1}{{\mathrm{e}}^{\theta\nu}-1}.\end{equation}

For a fixed $\nu$ , $\zeta_g(X)$ is a decreasing function of $\theta$ , which implies that $\theta$ is an uncertainty parameter in the sense of [Reference Hickey16]. That is, for a constant $\nu$ , if $X_{\theta}$ denotes X, which follows a truncated exponential distribution with parameter $\theta$ , then $X_{\theta_1}$ has more randomness than $X_{\theta_2}$ if $\theta_1 \lt \theta_2$ .

Example 7.2. In the previous example, if $Y=aX$ with $a \gt 0$ , then it is easy to verify that Y has the same randomness measure as in (7.1). That is, $\zeta_g(aX)=\zeta_g(X)$ for every $a \gt 0$ .

Also, if Y follows

\[f_Y(x)=\dfrac{\theta}{\tfrac{\alpha}{\nu}(1-{\mathrm{e}}^{-\theta\nu})}{\mathrm{e}}^{{{(-\theta \nu x)}/{\alpha}}}, \quad 0 \lt x \lt \alpha,\ \nu \gt 0,\ \theta \gt 0,\ \alpha \gt 0,\]

then Y has the same randomness for different values of $\alpha$ and the randomness is independent of the parameter $\alpha$ .

The following theorem shows that the Gini randomness measure is scale-invariant, an important property of a randomness measure.