2. Relative-$\chi_{\alpha}^2$ divergence measure and connection between $\chi_{\alpha}^2$ divergence measure and q-Fisher information

In this section, we first show that the $\chi_{\alpha}^2$ divergence measure in Eq. (1.2) has a close connection to $q-$Fisher information of mixing parameter of a given arithmetic mixture distribution. Next, we introduce a relative-$\chi_{\alpha}^2$ divergence measure and show that it includes some of the well-known chi-square-type divergence measures as special cases.

2.1. Connection between $\chi_{\alpha}^2$ divergence measure and q-Fisher information

The q-Fisher information of a density function f_θ about parameter θ, defined by [Reference Lutwak, Lv, Yang and Zhang14], is given by

(2.1)\begin{eqnarray} {\cal I}_q(\theta)=\int \left(\frac{\partial \,\log_q f_{\theta}(x)}{\partial \theta}\right)^2 f_{\theta}(x)\,{\rm d}x, \end{eqnarray}

where $\log_q (x)$ is the q-logarithmic function defined as

(2.2)\begin{eqnarray} \log_q(x)=\frac{x^q-1}{q}\quad \big( x\in \Re,\ q\neq 0\big) \end{eqnarray}

for more details, see [Reference Furuichi9, Reference Masi15, Reference Yamano23]. Then, we have the following result.

Theorem 2.1. Let f ₁ and f ₂ be two density functions. Then, the q-information measure of mixing parameter p in the two-component mixture model

(2.3)\begin{eqnarray} f_{p}(x)=pf_{1}(x)+(1-p)f_{2}(x), \quad p\in(0,1), \end{eqnarray}

is given by

\begin{eqnarray*} {{\cal I}_q}(p) = \frac{8}{2q+1} \mathcal{M}_{\frac{1}{2}}\left({\chi_{2q}^{2}(\,f_{p}, f_{1})},{\chi_{2q}^{2}(\,f_{p}, f_{2})} \right), \end{eqnarray*}

where $M_{\frac{1}{2}}(.,.)$ is the power mean with exponent $\frac{1}{2}$, defined as $\mathcal{M}_{\frac{1}{2}}(x,y)=\left(\frac{x^{\frac{1}{2}}}{2}+\frac{y^{\frac{1}{2}}}{2}\right)^{2}$ for positive x and y.

Proof. From the mixture model in Eq. (2.3), we readily see that

(2.4)\begin{eqnarray} f_1(x)-f_2(x)=\frac{f_1(x)-f_{p}(x)}{1-p}=\frac{f_p(x)-f_2(x)}{p}. \end{eqnarray}

Now, from the definition of q-Fisher information measure in Eq. (2.1), we find

(2.5)\begin{equation} {{\cal I}_q}(p)=\int \frac{\big(\,f_{1}(x)-f_{2}(x)\big)^2}{f_{p}^{1-2q}(x)}\,{\rm d}x =\left\{ \begin{array}{ll} \frac{2}{{(1+2q)(1-p)^2}}\chi_{2q}^{2}(\,f_p: f), & f=f_{1}, \\ \\ \frac{2}{{(1+2q)p^2}}\chi_{2q}^{2}(\,f_p: f), & f=f_{2}, \end{array} \right. \end{equation}

which readily yields

\begin{eqnarray*} {{\cal I}_q}(p)&=&\frac{2}{1+2q}\left(\sqrt{\chi_{2q}^{2}(\,f_{p}:f_{1})}+\sqrt{\chi_{2q}^{2}(\,f_{p}: f_{2})}\right)^{2}\\ &=&\frac{8}{1+2q}\left(\frac{1}{2}\sqrt{\chi_{2q}^{2}(\,f_{p}:{f_{1}})}+\frac{1}{2}\sqrt{\chi_{2q}^{2}(\,f_{p}: f_{2})}\right)^{2}\\ &=&\frac{8}{1+2q} \mathcal{M}_{\frac{1}{2}}\left({\chi_{2q}^{2}(\,f_{p}:f_{1})},{\chi_{2q}^{2}(\,f_{p}:f_{2})} \right), \end{eqnarray*}

as required.

2.2. Relative-$\chi_{\alpha}^2$ divergence measure

In this subsection, we introduce a relative-$\chi_{\alpha}^2$ divergence measure and show that it includes some of the well-known chi-square-type divergence measures as special cases. Further, we show that the special case of the proposed measure, when α = 0, is connected to the variance of density ratios.

Definition 2.2. Let f and g be two density functions on support ${\cal X}$. Then, a relative version of $\chi_{\alpha}^2$ divergence measure between f and g with respect to density function ψ on support ${\cal X}$, denoted by R-$\chi_{\alpha}^2$, for $\alpha\geq 0$, is defined as

(2.6)\begin{eqnarray} D_{{\alpha}}^{\psi}(\,f:g)=\frac{1+\alpha}{2}\int_{{\cal X}} \frac{\big({f(x)-g(x)}\big)^2}{\psi^{1-\alpha}(x)}\,{\rm d}x, \end{eqnarray}

provided the involved integral exists. In addition, the special case of R-$\chi_{\alpha}^2$ divergence measure, when α = 0, is of the form

(2.7)\begin{eqnarray} D_{{\alpha=0}}^{\psi}(\,f:g)=\frac{1}{2}\int_{{\cal X}} \frac{\big({f(x)-g(x)}\big)^2}{\psi(x)}\,{\rm d}x. \end{eqnarray}

Moreover, it is useful to note that $D_{{\alpha}}^{\psi}(\,f:g)$ reduces to $\chi_{\alpha}^2(\,f:g)$ when $\psi=f.$ It is easily seen from Eq. (2.6) that R-$\chi_{\alpha}^2$ divergence measure can be expressed based on two expectations under densities f and g to be

\begin{eqnarray*} D_{{\alpha}}^{\psi}(\,f:g)&=&\frac{1+\alpha}{2}\int_{{\cal X}} \frac{\big({f(x)-g(x)}\big)^2}{\psi^{1-\alpha}(x)}\,{\rm d}x\\ &=&\frac{1+\alpha}{2}E_f\left(\frac{f(X)-g(X)}{\psi^{1-\alpha}(X)}\right)+\frac{1+\alpha}{2}E_g\left(\frac{g(X)-f(X)}{\psi^{1-\alpha}(X)}\right). \end{eqnarray*}

From the definition of $D_{{\alpha}}^{\psi}(\,f:g)$, the weight density function, $\psi(x)$, can be utilized to assign varying degrees of importance to different regions of the dataset. For instance, a weight function that places less emphasis on extreme values can be employed to make the divergence measure more robust to outliers. On the other hand, a weight function that highlights a specific region of the data can be used to detect dissimilarities within that region of the data.

In general, $D_{{\alpha}}^{\psi}(\,f:g)$ divergence provides a flexible and powerful framework for assessing the differences between probability distributions in a wide range of applications. The parameters α and $\psi(x)$ can be adjusted to suit the specific characteristics and features of the data for the two models that are being compared, offering greater sensitivity and flexibility in the comparison process.

Remark 2.3.

1. (i) If α = 1, then $D_{\alpha=1}^{\psi}(\,f:g)=L_2(\,f:g)=\int \big(\,f(x)-g(x)\big)^2{\rm d} x$.

2. (ii) If $\psi(x)=f(x)$, then $D_{\alpha=0}^{\psi}(\,f:g)={\chi_{0}^2(\,f,g)}=\frac{\chi^2(\,f,g)}{2}$.

3. (iii) If $\psi(x)=g(x)$, then $D_{\alpha=0}^{\psi}(\,f:g)={\chi_{0}^2(g,f)}=\frac{\chi^2(g,\,f)}{2}$.

4. (iv) If $\psi(x)=p f(x)+(1-p) g(x)$, then $D_{\alpha=0}^{\psi}(\,f:g)=\frac{1}{2(1-p)^2}\chi^2\big(\psi:f\big)=\frac{1}{2p^2}\chi^2(\psi:g)$.

5. (v) If $\psi(x)= \frac{f(x)+g(x)}{2}$, then $D_{\alpha=0}^{\psi}(\,f:g)=\chi_{T}^2(\,f:g)$, where $\chi_{T}^2(\,f:g)$ is the triangular divergence defined in Eq. (1.4).

6. (vi) If $\psi(x)= \frac{f(x)+g(x)}{2}$, then $D_{\alpha=0}^{\psi}(\,f:g)= \chi_{\rm BS}^2(\,f:g)+\chi_{\rm BS}^2(g:f)$, where $D_{\rm BS}(\,f:g)$ is the Balakrishnan–Sanghvi divergence measure defined in Eq. (1.3).

Theorem 2.4. Let ψ be a density function. Then, $D_{\alpha=0}^{\psi}(\,f:g)$ divergence measure in Eq. (2.7) can be expressed as

(2.8)\begin{eqnarray} D_{\alpha=0}^{\psi}(\,f:g)&=&\frac{Var_\psi\left(\frac{f(X)}{\psi(X)}\right)+Var_\psi\left(\frac{g(X)}{\psi(X)}\right)}{2}-E_\psi\left(\frac{f(X)g(X)}{\psi^2(X)}\right)+1. \end{eqnarray}

Proof. From the definition of $D_{\alpha=0}^{\psi}(\,f:g)$, we have

\begin{eqnarray*} 2 D_{\alpha=0}^{\psi}(\,f:g)&=& \int \frac{f^2(x)}{\psi(x)}\,{\rm d}x-\left(\int{f(x)}{\rm d}x\right)^2\\ &&+\int \frac{g^2(x)}{\psi(x)}\,{\rm d}x-\left(\int{g(x)}{\rm d}x\right)^2 -2\int\frac{f(x)g(x)}{\psi(x)}\,{\rm d}x+2\\ &=&Var_\psi\left(\frac{f(X)}{\psi(X)}\right)+Var_\psi\left(\frac{g(X)}{\psi(X)}\right)-2E_\psi\left(\frac{f(X)g(X)}{\psi^2(X)}\right)+2, \end{eqnarray*}

as required.

3. Jensen-$\chi_{\alpha}^2$ divergence measure

In this section, we first introduce Jensen-$\chi_{\alpha}^2$ divergence measure and then establish some of its properties.

In fact, the Jensen-$\chi_{\alpha}^2$ divergence measure is an expansion of $D_{{\alpha}}^{\psi}(\,f:g)$ that is established based on a convex combination. This extension allows for the incorporation of additional divergence measures into the framework, further increasing the flexibility and applicability of the method.

Definition 3.1. Let $X_{1}, X_{2}$ and Y be random variables with density functions ${f}_{1},{f}_{2}$ and ψ, respectively, Then, the Jensen-$\chi_{\alpha}^2$ (J-$\chi_{\alpha}^2$) divergence measure, for $p\in (0,1)$, is defined as

(3.1)\begin{eqnarray} {\cal {J}}_{\alpha}^{\psi}\big({f}_{1},{f}_{2}; {\textbf{P}}\big)&=& p\chi_{\alpha}^{2}(\psi:f_1)+(1-p)\chi_{\alpha}^{2}(\psi:f_2)-\chi_{\alpha}^{2}\big(\psi:pf_1+(1-p)f_2\big). \end{eqnarray}

Proof. As $\phi(x)=x^2$ is a convex function, by using Jensen’s inequality, we readily find

\begin{eqnarray*} \frac{2}{1+\alpha}{\cal {J}}_{\alpha}^{\psi}\big({f}_{1},{f}_{2}; {\bf{P}}\big)&=&p\int \frac{\big(\,f_1(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)} \,{\rm d}x+(1-p)\int \frac{\big(\,f_2(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)}\, {\rm d}x\\ &&- \int \frac{\big(pf_1(x)+(1-p)f_2(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)} \,{\rm d}x\\ &=& p\int \frac{f_{1}^{2}(x)}{\psi^{1-\alpha}(x)}\,{\rm d}x+(1-p)\int \frac{f_{2}^{2}(x)}{\psi^{1-\alpha}(x)}\,{\rm d}x-\int \frac{\big(pf_1(x)+(1-p)f_2(x)\big)^2}{\psi^{1-\alpha}(x)}\,{\rm d}x,\\ &\geq& 0, \end{eqnarray*}

where the last expression follows from the fact that

\begin{eqnarray*} \big(pf_1(x)+(1-p)f_2(x)\big)^2\leq pf_{1}^2(x)+(1-p)f_{2}^2(x). \end{eqnarray*}

Theorem 3.3. A representation for ${\cal {J}}_{\alpha=0}^{\psi}\big({f}_{1},{f}_{2}; {\bf{P}}\big)$, based on variance of the ratio of densities, is given by

\begin{eqnarray*} {\cal {J}}_{\alpha=0}^{\psi}\big({f}_{1},{f}_{2}; {\bf{P}}\big)=\frac{1}{2}\left\{p\, Var_\psi\left(\frac{f_1(X)}{\psi(X)}\right)+(1-p)Var_\psi\left(\frac{f_2(X)}{\psi(X)}\right)-Var_\psi\left(\frac{pf_1(X)+(1-p)f_2(X)}{\psi(X)}\right)\right\}. \end{eqnarray*}

Proof. From the definition of ${\cal {J}}_{\alpha=0}^{\psi}\big({f}_{1},{f}_{2}; {\bf{P}}\big)$, we have

\begin{eqnarray*} 2{\cal {J}}_{\alpha=0}^{\psi}\big({f}_{1},{f}_{2}; {\bf{P}}\big)&=&p\int \frac{\big(\,f_1(x)-\psi(x)\big)^2}{\psi(x)} {\rm d}x+(1-p)\int \frac{\big(\,f_2(x)-\psi(x)\big)^2}{\psi(x)} {\rm d}x\\ &&- \int \frac{\big(pf_1(x)+(1-p)f_2(x)-\psi(x)\big)^2}{\psi(x)} {\rm d}x\\ &=& p\int \frac{f_{1}^{2}(x)}{\psi(x)}{\rm d}x+(1-p)\int \frac{f_{2}^{2}(x)}{\psi(x)}{\rm d}x-\int \frac{\big(pf_1(x)+(1-p)f_2(x)\big)^2}{\psi(x)}{\rm d}x\\ &=&p\left\{\int \frac{f_{1}^{2}(x)}{\psi(x)}{\rm d}x-1\right\}+(1-p)\left\{\int \frac{f_{2}^{2}(x)}{\psi(x)}{\rm d}x-1\right\}\\ &&-\left\{\int \frac{\big(pf_1(x)+(1-p)f_2(x)\big)^2}{\psi(x)}{\rm d}x-1\right\}\\ &=&p\,Var_\psi\bigg(\frac{f_1(X)}{\psi(X)}\bigg)+(1-p)Var_\psi\bigg(\frac{f_2(X)}{\psi(X)}\bigg)-Var_\psi\left(\frac{pf_1(X)+(1-p)f_2(X)}{\psi(X)}\right), \end{eqnarray*}

as required.

Theorem 3.4. Let the random variables X ₁ and X ₂ have density functions f ₁ and f ₂, respectively. Then, ${\cal {J}}_{\alpha}^{\psi}\big({f}_{1},{f}_{2}; {\bf{P}}\big)$ measure is a mixture of R-$\chi_{\alpha}^{2}$ divergence measures of the form

\begin{eqnarray*} {\cal {J}}_{\alpha}^{\psi}\big({f}_{1},{f}_{2}; {\bf{P}}\big)=p D_{\alpha}^{\psi}(\,f_1:f_T)+(1-p)D_{\alpha}^{\psi}(\,f_2:f_T), \end{eqnarray*}

where $D_{\alpha}^{\psi}(\,f_i:f_T)$ is the divergence measure in Eq. (2.6), with $f_{T}=pf_1+(1-p)f_2$ being the two-component mixture density.

Proof. With $f_{T}=pf_1+(1-p)f_2$, we first find

\begin{eqnarray*} \frac{2}{1+\alpha}{\cal {J}}_{\alpha}^{\psi}\big({f}_{1},{f}_{2}; {\bf{P}}\big) &=&p\int \frac{\big(\,f_1(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x+(1-p)\int \frac{\big(\,f_2(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &&- \int \frac{\big(pf_1(x)+(1-p)f_2(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &=& p\int \frac{f_{1}^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x+(1-p)\int \frac{f_{2}^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x-\int \frac{\big(pf_1(x)+(1-p)f_2(x)\big)^2}{\psi^{1-\alpha}(x)}{\rm d}x.\\ \end{eqnarray*}

On the other hand, with $k=pD_{\alpha}^{\psi}(\,f_1:f_T)+(1-p)D_{\alpha}^{\psi}(\,f_2:f_T)$, we also have

\begin{eqnarray*} \frac{2}{1+\alpha}k&=& p\int \frac{\big(\,f_1(x)-f_T(x)\big)^2}{{\psi^{1-\alpha}(x)}} {\rm d}x+(1-p)\int \frac{\big(\,f_2(x)-f_T(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &=&p\int \frac{f_{1}^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x-2p\int \frac{f_1(x)f_T(x)}{\psi^{1-\alpha}(x)}{\rm d}x+ p\int \frac{f_{T}^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x\\ &&+(1-p)\int \frac{f_{2}^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x-2(1-p)\int \frac{f_2(x)f_T(x)}{\psi^{1-\alpha}(x)}{\rm d}x+ (1-p)\int \frac{f_{T}^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x\\ &=&p\int \frac{f_{1}^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x+ (1-p)\int \frac{f_{2}^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x-2\int \frac{f_{T}^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x+\int \frac{f_{T}^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x\\ &=&p\int \frac{f_{1}^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x+ (1-p)\int \frac{f_{2}^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x-\int \frac{f_{T}^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x, \end{eqnarray*}

which establishes the required result.

Theorem 3.5. A connection between ${\cal {J}}_{\alpha=0}^{\psi}\big({f}_{1},{f}_{2}; {\bf{P}}\big)$ with $\psi=\frac{f_1+f_2}{2}$ and Balakrishnan–Sanghvi divergence measure is given by

\begin{eqnarray*} {\cal {J}}_{\alpha=0}^{\psi}\big({f}_{1},{f}_{2}; {\bf{P}}\big)=p\left\{\chi_{\rm BS}^2(\,f_1:f_T)+\chi_{\rm BS}^2(\,f_T:f_1)\right\}+(1-p)\left\{\chi_{\rm BS}^2(\,f_2:f_T)+\chi_{\rm BS}^2(\,f_T:f_2)\right\}, \end{eqnarray*}

where $f_{T}=pf_1+(1-p)f_2$ is the two-component mixture density.

Proof. With $f_{T}=pf_1+(1-p)f_2$ and from Part (vi) of Remark 2.3 and Theorem 3.4, we have

\begin{eqnarray*} {\cal {J}}_{\alpha=0}^{\psi}\big({f}_{1},{f}_{2}; {\bf{P}}\big)&=&p D_{\alpha=0}^{\psi}(\,f_1:f_T)+(1-p)D_{\alpha=0}^{\psi}(\,f_2:f_T)\\ &=&p\left\{\chi_{\rm BS}^2(\,f_1:f_T)+\chi_{\rm BS}^2(\,f_T:f_1)\right\}+(1-p)\left\{\chi_{\rm BS}^2(\,f_2:f_T)+\chi_{\rm BS}^2(\,f_T:f_2)\right\}, \end{eqnarray*}

as required.

Theorem 3.6. We have

(3.2)\begin{eqnarray} \frac{-1}{2}\,\frac{\partial^2}{\partial p^2}{\cal {J}}_{\alpha}^{\psi}\big({f}_{1},{f}_{2}; {\bf{P}}\big)= D_{\alpha}^{\psi}(\,f_1:f_2). \end{eqnarray}

Proof. From the definition ${\cal {J}}_{\alpha}^{\psi}\big({f}_{1},{f}_{2}; {\textbf{P}}\big)$ in Eq. (3.1) and making use of the dominated convergence theorem, we have

\begin{eqnarray*} \frac{-1}{2}\,\frac{\partial^2}{\partial p^2}{\cal {J}}_{\alpha}^{\psi}\big({f}_{1},{f}_{2}; {\bf{P}}\big)&=&-\frac{1+\alpha}{4}\,\frac{\partial}{\partial p}\left(\int\frac{\big(\,f_1(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x+\int \frac{\big(\,f_2(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x\right)\\ &&+\frac{1+\alpha}{2}\,\frac{\partial}{\partial p}\left(\int\big(\,f_1(x)-f_2(x)\big) \frac{pf_1(x)+(1-p)f_2(x)-\psi(x)}{\psi^{1-\alpha}(x)} {\rm d}x \right)\\ &=&\frac{1+\alpha}{2}\int\frac{\big(\,f_1(x)-f_2(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &=&D_{\alpha}^{\psi}(\,f_1:f_2), \end{eqnarray*}

as required.

We now extend the definition of Jensen-$\chi_{\alpha}^{2}$ divergence measure in Eq. (3.1) to the case of n + 1 random variables. Let $X_{1},\ldots,X_{n}$ and Y be random variables with density functions ${f}_{1},\ldots,{f}_{n}$ and ψ, respectively, and $p_{1},\ldots,p_{n}$ be non-negative real numbers such that $\sum_{i=1}^{n}p_{i}=1$. Then, the Jensen-$\chi_{\alpha}^{2}$ measure is defined as

(3.3)\begin{eqnarray} {\cal {J}}_{\alpha}^{\psi}\big({f}_{1},\ldots,{f}_{n}; {\textbf{P}}\big)&=& \sum_{i=1}^{n}p_i\chi_{\alpha}^{2}(\psi:f_i)-\chi_{\alpha}^{2} \left(\psi:\sum_{i=1}^{n}p_i f_i\right). \end{eqnarray}

The special case of Jensen-$\chi_{\alpha}^2$ divergence measure, when α = 0, has the representation

\begin{eqnarray*} {\cal {J}}_{\alpha=0}^{\psi}\big({f}_{1},\ldots,{f}_{n}; {\textbf{P}}\big)&=&{\frac{1}{2}} \sum_{i=1}^{n}p_i\chi^{2}(\psi:f_i)-{\frac{1}{2}}\chi^{2}\left(\psi:\sum_{i=1}^{n}p_i f_i\right)\\ &=&{\frac{1}{2}}\sum_{i=1}^{n}p_i Var_\psi\left(\frac{f_i(X)}{\psi(X)}\right)-{\frac{1}{2}} Var_\psi\left(\frac{\sum_{i=1}^{n}p_if_i(X)}{\psi(X)}\right). \end{eqnarray*}

Corollary 3.7. The ${\cal {J}}_{\alpha}^{\psi}\big({f}_{1},\ldots,{f}_{n}; {\textbf{P}}\big)$ measure in Eq. (3.3) is a mixture of $D_{\alpha}^{\psi}$ measures in Eq. (2.6) of the form

\begin{eqnarray*} {\cal {J}}_{\alpha}^{\psi}\big({f}_{1},\ldots,{f}_{n}; {\bf{P}}\big)&=&\sum_{i=1}^{n}p_i D_{\alpha}^{\psi}(\,f_i:f_T). \end{eqnarray*}

Theorem 3.8. The ${\cal {J}}_{\alpha}^{\psi}\big({f}_{1},\ldots,{f}_{n}; {\textbf{P}}\big)$ measure in Eq. (3.3) is a mixture of $D_{\alpha}^{\psi}$ measures in Eq. (2.6) of the form

\begin{eqnarray*} {\cal {J}}_{\alpha}^{\psi}\big({f}_{1},\ldots,{f}_{n}; {\bf{P}}\big)&=&2\sum_{i=1}^{n}\sum_{j=1}^{n}p_ip_j D_{\alpha}^{\psi}(\,f_i:f_j). \end{eqnarray*}

Proof. From Corollary 3.7 and making use of the identity ([Reference Steele21], pp. 95–96)

\begin{equation*}\sum_{i=1}^{n}w_i\big(x_i-\bar{x}_w\big)^2={\frac{1}{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}w_i w_j\big(x_i-x_j\big)^2,\qquad\bar{x}_w=\sum_{i=1}^{n}w_ix_i,\ \sum_{i=1}^{n}w_i=1,\end{equation*}

we obtain

\begin{eqnarray*} {\cal {J}}_{\alpha}^{\psi}\big({f}_{1},\ldots,{f}_{n}; {\bf{P}}\big)&=&\sum_{i=1}^{n}p_i D_{\alpha}^{\psi}(\,f_i:f_T)\\ &=&\frac{1+\alpha}{2}\sum_{i=1}^{n}p_i\int \frac{\big(\,f_i (x) -\sum_{j=1}^{n}p_j f_j(x)\big)^2}{\psi^{1-\alpha}(x)}{\rm d}x\\ &=&\frac{({1+\alpha})}{4}\sum_{i=1}^{n}\sum_{j=1}^{n}p_ip_j \int \frac{\big(\,f_i(x)-f_j(x)\big)^2}{\psi^{1-\alpha}(x)}{\rm d}x\\ &=&\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}p_ip_j D_{\alpha}^{\psi}(\,f_i:f_j), \end{eqnarray*}

as required.

Theorem 3.9. Let $f_i\geq \frac{\psi^{1-\alpha}}{2}, i=1,\ldots,n$. Then, a lower bound for ${\cal {J}}_{\alpha}^{\psi}\big({f}_{1},\ldots,{f}_{n}; {\textbf{P}}\big)$ is given by

\begin{eqnarray*} {\cal {J}}_{\alpha}^{\psi}\big({f}_{1},\ldots,{f}_{n}; {\bf{P}}\big)\geq \frac{1+\alpha}{4} JS_{{\bf{P}}}(\,f_1,\ldots,f_n), \end{eqnarray*}

where $JS_{{\textbf{P}}}(\,f_1,\ldots,f_n)$ is the Jensen–Shannon entropy; see [Reference Lin13].

Proof. From the assumption, Theorem 3.8 and by making use of the identity

\begin{equation*}\sum_{i=1}^{n}w_i\big(x_i-\bar{x}_w\big)^2={\frac{1}{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}w_i w_j\big(x_i-x_j\big)^2,\end{equation*}

and then setting $w_i=p_i$, $w_j=p_j$, $x_i=f_i(x)$, $x_j=f_j(x)$ and $\bar{x}_w=\sum_{i=1}^{n}p_i f_i(x)$ , we find

\begin{eqnarray*} \frac{2}{1+\alpha}{\cal {J}}_{\alpha}^{\psi}\big({f}_{1},\ldots,{f}_{n}; {\bf{P}}\big) &=&\sum_{i=1}^{n}p_i\int \frac{\big(\,f_i (x) -\sum_{j=1}^{n}p_j f_j(x)\big)^2}{\psi^{1-\alpha}(x)}{\rm d}x\\ &=&{\frac{1}{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}p_ip_j \int \frac{\big(\,f_i(x)-f_j(x)\big)^2}{\psi^{1-\alpha}(x)}{\rm d}x\\ &=&{\frac{1}{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}p_ip_j \int \frac{\big(\,f_i(x)-f_j(x)\big)^2}{f_i(x)}\frac{f_i(x)}{\psi^{1-\alpha}(x)}{\rm d}x\\ &\geq& {\frac{1}{2}} \sum_{i=1}^{n}\sum_{j=1}^{n}p_ip_j \int \frac{\big(\,f_i(x)-f_j(x)\big)^2}{f_i(x)} {\rm d}x\\ &\geq& {\frac{1}{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}p_ip_j \int f_i(x)\log\bigg(\frac{f_i(x)}{f_j(x)}\bigg) {\rm d}x\\ &=&{\frac{1}{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}p_ip_j KL(\,f_i:f_j)\\ &\geq&{\frac{1}{2}} JS_{{\bf{P}}}(\,f_1,\ldots,f_n), \end{eqnarray*}

where the second inequality follows from the fact that $\log(x) \lt x-1, x \gt 0$, and the last inequality follows from [Reference Asadi, Ebrahimi, Soofi and Zohrevand3].

4. ( p, w)-Jensen-$\chi_{\alpha}^2$ divergence measure

In this section, we first review the definition of (p, w)-Jensen–Shannon divergence measure. Then, we introduce (p, w)-Jensen-$\chi_{\alpha}^2$ divergence measure in a way similar to (p, w)-Jensen–Shannon divergence. Furthermore, we establish some results for this extended divergence measure. Let f and g be two density functions. Then, the Kullback–Leibler divergence between f and g is defined as

\begin{eqnarray*} KL(\,f,g)=\int f(x)\,\log\left(\frac{f(x)}{g(x)}\right){\rm d}x, \end{eqnarray*}

where log denotes the natural logarithm. The (p, w)-Jensen–Shannon divergence between two density functions f ₁ and f ₂, for α and p $\in (0,1)$, is defined as

\begin{eqnarray*} JS_{(p,w)}(\,f_1,f_2)&=& H\big((1-{\bar{s}})f_1+{\bar{s}}f_2\big)-w H\big((1-p)f_1+p f_2\big)-(1-w)H\big(p f_1+(1-p) f_2\big)\\ &=& wKL\left((1-p)f_1+p f_2: (1-\bar{s})f_1+\bar{s}f_2\right)\\ &&+(1-w)KL\left(p f_1+(1-p) f_2: (1-{\bar{s}})f_1+{\bar{s}}f_2\right), \end{eqnarray*}

where ${\bar{s}}=wp+(1-w)(1-p)$. For more details, one may refer to [Reference Melbourne, Talukdar, Bhaban, Madiman and Salapaka16, Reference Nielsen17].

Definition 4.1. Let $X_{1}, X_{2}$ and Y be random variables with density functions ${f}_{1},{f}_{2}$ and ψ, respectively, Then, the (p, w)-Jensen-$\chi_{\alpha}^2$ divergence measure, for w and p $\in (0,1)$, is defined as

\begin{eqnarray*} {\cal {J}}_{\alpha}^\psi\big({f}_{1},{f}_{2};{\textbf{P}},\textbf{w}\big)&=& w\chi_{\alpha}^{2}\big(\psi:(1-p)f_1+p f_2\big)+(1-w)\chi_{\alpha}^{2}\big(\psi:p f_1+(1-p) f_2\big)\\ &&-\chi_{\alpha}^{2}\big(\psi:(1-{\bar{s}})f_1+{\bar{s}}f_2\big), \end{eqnarray*}

where ${\bar{s}}=wp+(1-w)(1-p)$.

Theorem 4.2. Let the random variables X ₁ and X ₂ have density functions f ₁ and f ₂, respectively. Then, ${\cal {J}}_{\alpha}^\psi\big({f}_{1},{f}_{2};{\textbf{P}},\textbf{w}\big)$ is a mixture of relative measures in Eq. (2.6) of the form

\begin{align} {\cal {J}}_{\alpha}^\psi\big(\,{f}_{1},{f}_{2};{\textbf{P}},\textbf{w}\big)=wD_{\alpha}^{\psi}\left((1-p)f_1+p f_2:f_{{\bar{s}}}^{T}\right)+(1-w)D_{\alpha}^{\psi}\left(p f_1+(1-p) f_2:f_{{\bar{s}}}^{T}\right), \end{align}

with $f_{{\bar{s}}}^{T}=(1-{\bar{s}})f_1+{\bar{s}}f_2$ is the two-component mixture density.

Proof. With $f_{{\bar{s}}}^{T}=(1-{\bar{s}})f_1+{\bar{s}}f_2$, we find

\begin{eqnarray*} &&{\frac{2}{1+\alpha}}{\cal {J}}_{\alpha}^\psi\big({f}_{1},{f}_{2};{\textbf{P}},\textbf{w}\big)\\ &=&w\int \frac{\big((1-p)f_1(x)+p f_2(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x+(1-w)\int \frac{\big(p f_1(x)+(1-p) f_2(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &&- \int \frac{\big((1-{\bar{s}})f_1(x)+{\bar{s}}f_2(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &=& w\int \frac{\big((1-p)f_1(x)+p f_2(x)\big)^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x+(1-w)\int \frac{\big(pf_1(x)+(1-p)f_2(x)\big)^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x\\ &&-\int \frac{\big((1-{\bar{s}})f_1(x)+{\bar{s}}f_2(x)\big)^2}{\psi^{1-\alpha}(x)}{\rm d}x.\\ \end{eqnarray*}

On the other hand, letting

\begin{equation*}\frac{1+\alpha}{2}k=wD_{\alpha}^{\psi}\left((1-p)f_1+p f_2:f_{{\bar{s}}}^{T}\right)+(1-w)D_{\alpha}^{\psi}\left(p f_1+(1-p) f_2:f_{{\bar{s}}}^{T}\right)\end{equation*}

and using the fact that

\begin{eqnarray*} f_{{\bar{s}}}^{T}(x)&=&(1-{\bar{s}})f_1(x)+{\bar{s}}f_2(x)\\ &=&\big(1-(wp+(1-w)(1-p))\big)f_1(x)+(wp+(1-w)(1-p))f_2(x)\\ &=&w\big((1-p)f_1(x)+p f_2(x)\big)+(1-w)\big(p f_1(x)+(1-p)f_2(x)\big), \end{eqnarray*}

we find

\begin{eqnarray*} k&=& w\int \frac{\big((1-p)f_1(x)+p f_2(x)-f_{{\bar{s}}}^{T}(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x+(1-w)\int \frac{\big(p f_1(x)+(1-p) f_2(x)-f_{{\bar{s}}}^{T}(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &=&w\int \frac{\big((1-p)f_1(x)+p f_2(x)\big)^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x-2w\int \frac{\big((1-p) f_1(x)+p f_2(x)\big)f_{{\bar{s}}}^{T}(x)}{\psi^{1-\alpha}(x)}{\rm d}x+ \int \frac{\big(\,f_{{\bar{s}}}^{T}(x)\big)^2}{\psi^{1-\alpha}(x)}{\rm d}x\\ &&+(1-w)\int \frac{\big((pf_1(x)+(1-p)f_2(x)\big)^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x-2(1-w)\int \frac{\big(pf_1(x)+(1-p)f_2(x)\big)f_{{\bar{s}}}^{T}(x)}{\psi^{1-\alpha}(x)}{\rm d}x\\ &=&w\int \frac{\big((1-p)f_1(x)+p f_2(x)\big)^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x+ (1-w)\int \frac{\big((pf_1(x)+(1-p)f_2(x)\big)^{2}(x)}{\psi^{1-\alpha}(x)}{\rm d}x\\ &&+\int \frac{\big(\,f_{{\bar{s}}}^{T}(x)\big)^2}{\psi^{1-\alpha}(x)}{\rm d}x. \end{eqnarray*}

Now, from the above results, we have

\begin{eqnarray*} {{\cal {J}}_{\alpha}^\psi\big({f}_{1},{f}_{2};{\textbf{P}},\textbf{w}\big)=\frac{1+\alpha}{2}k=wD_{\alpha}^{\psi}\left((1-p)f_1+p f_2:f_{{\bar{s}}}^{T}\right)+(1-w)D_{\alpha}^{\psi}\left(p f_1+(1-p) f_2:f_{{\bar{s}}}^{T}\right),} \end{eqnarray*}

which establishes the required result.

From Definitions 3.1 and 4.1, we readily have the following Corollary.

Corollary 4.3. A connection between ${\cal {J}}_{\alpha}^\psi\big({f}_{1},{f}_{2};{\textbf{P}},\textbf{w}\big)$ and ${\cal {J}}_{\alpha}^\psi\big({f}_{1},{f}_{2};\textbf{w}\big)$ measures is given by

\begin{equation*}{\cal {J}}_{\alpha}^\psi\big({f}_{1},{f}_{2};{\textbf{P}},\textbf{w}\big)={\cal {J}}_{\alpha}^{\psi}\bigg((1-p){f}_{1}+p{f}_{2}: p{f}_{1}+(1-p){f}_{2}; \textbf{w}\bigg).\end{equation*}

Proof. From Theorem 3.6 and Corollary 4.3, we have

\begin{eqnarray*} \frac{-1}{2}\,\frac{\partial^2}{\partial w^2}{\cal {J}}_{\alpha}^\psi\big({f}_{1},{f}_{2};{\bf{P}},\bf{w}\big)&=&\frac{-1}{2}\,\frac{\partial^2}{\partial w^2}{\cal {J}}_{\alpha}^\psi\left((1-p){f}_{1}+pf_2,p{f}_{2}+(1-p)f_1,{\bf{w}}\right)\\ &=& D_{\alpha}^{\psi}\left((1-p){f}_{1}+pf_2:p{f}_{1}+(1-p)f_2\right), \end{eqnarray*}

which proves Part (i). From Corollary 4.3 and using the facts that

\begin{eqnarray*} f_{{\bar{s}}}^{T}(x)&=& w\big((1-p)f_1(x)+p f_2(x)\big)+(1-w)\big(p f_1(x)+(1-p)f_2(x)\big) \end{eqnarray*}

and

\begin{eqnarray*} (1-w){f}_{1}(x)+wf_2(x) -\big(w{f}_{1}(x)+(1-w)f_2(x)\big)=w(\,f_2(x)-f_1(x))+(1-w)(\,f_1(x)-f_2(x)), \end{eqnarray*}

we find

\begin{eqnarray*} \frac{-1}{2}\,\frac{\partial^2}{\partial p^2}{\cal {J}}_{\alpha}^\psi\big({f}_{1},{f}_{2};{\bf{P}},\bf{w}\big)&=&\frac{-1}{2}\,\frac{\partial^2}{\partial p^2}{\cal {J}}_{\alpha}^{\psi}\left((1-p){f}_{1}+pf_2,p{f}_{1}+(1-p)f_2; \bf{w}\right)\\ &=&-w\frac{\alpha+1}{4}\,\frac{\partial^2}{\partial p^2}\int \frac{\big((1-p)f_1(x)+p f_2(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &&-(1-w)\frac{\alpha+1}{4}\,\frac{\partial^2}{\partial p^2}\int \frac{\big(p f_1(x)+(1-p) f_2(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &&+\frac{\alpha+1}{4}\,\frac{\partial^2}{\partial p^2}\int \frac{\big(\,f_{\bar{s}}^{T}(x)-\psi(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &=&\frac{\alpha+1}{2}\int \frac{\left(w(\,f_2(x)-f_1(x))+(1-w)(\,f_1(x)-f_2(x))\right)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &&-\frac{\alpha+1}{2}\int \frac{\big(\,f_2(x)-f_1(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &=&\frac{\alpha+1}{2}\int \frac{\left((1-w){f}_{1}(x)+wf_2(x) -\big(w{f}_{1}(x)+(1-w)f_2(x)\big)\right)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &&-\frac{\alpha+1}{2}\int \frac{\big(\,f_2(x)-f_1(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &=&D_{\alpha}^{\psi}\left((1-w){f}_{1}+wf_2:w{f}_{1} +(1-w)f_2\right)-D_{\alpha}^{\psi}\big(\,f_1:f_2\big) , \end{eqnarray*}

which proves Part (ii). Hence, the theorem.

5. $D_{\alpha}^{\psi}$ divergence measure of escort and arithmetic mixture densities

In this section, we examine $D_{\alpha}^{\psi}$ divergence measure of escort and arithmetic mixture densities.

5.1. $D_{\alpha}^{\psi}$ divergence measure of escort and generalized escort densities

The escort distribution is a key concept in nonextensive statistical mechanics and coding theory and is closely associated with Tsallis and Rényi entropy measures. Bercher [Reference Bercher6] studied some connections between coding theory and the measure of complexity in nonextensive statistical mechanics in terms of escort distributions.

Let f be a density function. Then, the escort density with order η > 0, associated with f, is defined as

(5.1)\begin{eqnarray} f_{\eta}(x)=\frac{f^\eta(x)}{\int f^\eta(x) {\rm d}x}. \end{eqnarray}

Theorem 5.1. Let f and g be two density functions and f_α be the escort density corresponding to f. Then, for $0\leq\eta\leq1$ and $\psi(x)=f_{\eta}(x)$, we have

(5.2)\begin{eqnarray} D_{\alpha}^{\psi}(\,f:g)=\frac{1+\alpha}{1+\beta}G_{\eta}^{1-\alpha}(\,f)\chi_{\beta}^{2}(\,f:g), \end{eqnarray}

where $\beta=1-\eta(1-\alpha)$ and $G_{\eta}(\,f)$ is the information generating function of density f with order η defined as

(5.3)\begin{eqnarray} G_{\eta}(\,f)=\int f^{\eta}(x) {\rm d}x. \end{eqnarray}

Proof. From the definition of $D_{\alpha}^{\psi}(\,f:g)$ and the assumption that $\psi(x)=f_{\eta}(x)$, we have

\begin{eqnarray*} D_{\alpha}^{\psi}(\,f:g)&=&\frac{1+\alpha}{2}\int \frac{\big(\,f(x)-g(x)\big)^2}{f_{\eta}^{1-\alpha}(x)} {\rm d}x\\ &=&\frac{1+\alpha}{2}\left(\int f^{\eta}(x){\rm d}x\right)^{1-\alpha}\int \frac{\big(\,f(x)-g(x)\big)^2}{{f^{\eta(1-\alpha)}(x)}}{\rm d}x\\ &=&\frac{1+\alpha}{2} G_{\eta}^{1-\alpha}(\,f)\int \frac{\big(\,f(x)-g(x)\big)^2}{{f^{\eta(1-\alpha)}(x)}}{\rm d}x\\ &=&\frac{1+\alpha}{1+\beta}G_{\eta}^{1-\alpha}(\,f)\chi_{\beta}^{2}(\,f:g), \end{eqnarray*}

where $\beta=1-\eta(1-\alpha),$ as desired.

Next, let f and g be two density functions. Then, the generalized escort density, for $1 \gt \eta \gt 0$, is defined as

(5.4)\begin{eqnarray} h_{\eta}(x)=\frac{f^\eta(x) g^{1-\eta}(x)}{\int f^\eta(x) g^{1-\eta}(x) {\rm d}x}. \end{eqnarray}

Let $\psi(x)=h_{\eta}(x)$. We then have

(5.5)\begin{eqnarray} 2 D_{\alpha=0}^{\psi}(\,f:g)&=&\int \frac{\big(\,f(x)-g(x)\big)^2}{h_{\eta}(x)} {\rm d}x=\int \frac{\big(\,f(x)-g(x)\big)^2}{\frac{f^\eta(x) g^{1-\eta}(x)}{\int f^\eta(x) g^{1-\eta}(x) {\rm d}x}}{\rm d}x\\ &=&\left(\int f^{\eta}(x) g^{1-\eta}(x){\rm d}x\right) \int \frac{\big(\,f(x)-g(x)\big)^2}{{f^{\eta}(x)g^{1-\eta}(x)}}{\rm d}x\\ &=& R_{\eta}(\,f:g)\int \frac{\big(\,f(x)-g(x)\big)^2}{{f^{\eta}(x)g^{1-\eta}(x)}}{\rm d}x, \end{eqnarray}

where $R_{\eta}(\,f:g)$ is the relative information-generating function between density functions f and g defined as

(5.6)\begin{eqnarray} R_{\eta}(\,f:g)=\int f^{\eta}(x)g^{1-\eta}(x){\rm d}x. \end{eqnarray}

Theorem 5.2. A lower bound for $D_{\alpha=0}^{\psi}(\,f:g)$ in Eq. (5.5) is given by

\begin{eqnarray*} D_{\alpha=0}^{\psi}(\,f:g)\geq \frac{R_{\eta}(\,f, g)}{2(1-\eta)^2}\chi^2(\,f_\eta:f), \end{eqnarray*}

where $f_\eta=\eta f+(1-\eta)g$ is the two-component mixture density, $\chi^2(.:.)$ is the chi-square divergence, and $ R_{\eta}(\,f:g)$ is as defined in Eq. (5.6).

Proof. From the definition of $D_{\alpha}^{\psi}(\,f:g)$ and the assumption that

\begin{equation*}\psi(x)=h_\eta(x)=\frac{f^\eta(x) g^{1-\eta}(x)}{\int f^\eta(x) g^{1-\eta}(x) {\rm d}x},\end{equation*}

for $0\leq \eta \leq1$, and using the geometric mean-arithmetic mean inequality between densities f and g given by

\begin{equation*}{f^{\eta}(x)g^{1-\eta}(x)}\leq \eta f(x)+(1-\eta) g(x),\end{equation*}

and the fact that $g(x)-f(x)=\frac{1}{1-\eta}(\,f_\eta(x)-f(x))$, we obtain

\begin{eqnarray*} 2D_{\alpha=0}^{\psi}(\,f:g)&=&\int \frac{\big(\,f(x)-g(x)\big)^2}{h_{\eta}(x)} {\rm d}x\\ &=&\left(\int f^{\eta}(x) g^{1-\eta}(x){\rm d}x\right) \int \frac{\big(\,f(x)-g(x)\big)^2}{{f^{\eta}(x)g^{1-\eta}(x)}}{\rm d}x\\ &=& R_{\eta}(\,f:g)\int \frac{\big(\,f(x)-g(x)\big)^2}{{f^{\eta}(x)g^{1-\eta}(x)}}{\rm d}x\\ &\geq&R_{\eta}(\,f:g) \int \frac{{\big(\,f(x)-g(x)\big)^2}}{\eta f(x)+(1-\eta) g(x)}{\rm d}x\\ &=&{\frac{R_{\eta}(\,f:g)}{(1-\eta)^2} \int \frac{{\big(\,f_\eta(x)-f(x)\big)^2}}{f_\eta(x)}{\rm d}x}\\ &=&\frac{R_{\eta}(\,f:g)}{(1-\eta)^2}\chi^2(\,f_\eta:f), \end{eqnarray*}

as required.

5.2. $D_{\alpha}^{\psi}$ divergence measure between two arithmetic mixture densities

In this subsection, we study $D_{\alpha}^{\psi}$ divergence measure between two arithmetic mixture densities. Consider two mixture density functions $f_m(x)=\sum_{i=1}^{n} p_i f_i(x)$ and $g_m(x)=\sum_{i=1}^{n} p_i g_i(x)$. Then, we have

\begin{eqnarray*} D_{\alpha}^{\psi}(\,f_m:g_m)&=&\frac{1+\alpha}{2}\int \frac{\big(\,f_m(x)-g_m(x)\big)^2}{\psi^{1-\alpha}(x)} {\rm d}x=\frac{1+\alpha}{2}\int \frac{\left(\sum_{i=1}^{n} p_i f_i(x)-\sum_{i=1}^{n} p_i g_i(x)\right)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &=&\frac{1+\alpha}{2}\int \frac{\left(\sum_{i=1}^{n} p_i\big(\,f_i(x)- g_i(x)\big)\right)^2}{\psi^{1-\alpha}(x)} {\rm d}x\\ &=&\frac{1+\alpha}{2}\int\left\{\sum_{i=1}^{n} p_{i}^{2} \frac{\big(\,f_i(x)- g_i(x)\big)^2}{\psi^{1-\alpha}(x)} +2\underset{i \lt j}{\sum_{i=1}^{n}\sum_{j=1}^{n}} p_{i}p_{j}\frac{\big(\,f_i(x)- g_i(x)\big)\big(\,f_j(x)- g_j(x)\big)}{\psi^{1-\alpha}(x)} \right\} {\rm d}x\\ &=&\frac{1+\alpha}{2}\sum_{i=1}^{n}p_{i}^{2} \int \frac{\big(\,f_i(x)- g_i(x)\big)^2}{{\psi^{1-\alpha}(x)}}{\rm d}x \\ &\quad&+(1+\alpha) \underset{i \lt j}{\sum_{i=1}^{n}\sum_{j=1}^{n}} p_{i}p_{j}\int \frac{\big(\,f_i(x)- g_i(x)\big)\big(\,f_j(x)- g_j(x)\big)}{{\psi^{1-\alpha}(x)}}{\rm d}x\\ &=&\frac{1+\alpha}{2}\sum_{i=1}^{n}p_{i}^{2} D_{\alpha}^{\psi}(\,f_i:g_i)+(1+\alpha)\underset{i \lt j}{\sum_{i=1}^{n}\sum_{j=1}^{n}} p_{i}p_{j}\int \frac{\big(\,f_i(x)- g_i(x)\big)\big(\,f_j(x)- g_j(x)\big)}{{\psi^{1-\alpha}(x)}}{\rm d}x. \end{eqnarray*}

Theorem 5.3. Let $f_1,\ldots,f_n$ be n density functions. Now, consider the probability mixing vector ${\textbf{P}}=(p_1,\ldots,p_n)$ and its corresponding negation probability vector

\begin{equation*}\bar{\textbf{P}}=(\bar{p}_1,\ldots,\bar{p}_n)=\left(\frac{1-p_1}{n-1},\ldots,\frac{1-p_n}{n-1}\right).\end{equation*}

Then, we have the lower bound for $D_{\alpha=0}^{\psi}$ as

\begin{eqnarray*} D_{\alpha=0}^{\psi}\left(\sum_{i=1}^{n}p_i f_i:\sum_{i=1}^{n}\bar{p}_i f_i\right)\geq {\frac{1}{2} \sum_{i=1}^{n} \left(\frac{np_i-1}{n-1}\right)^2}\big(KL(\,f_i:\psi)+1\big)+L, \end{eqnarray*}

where $L=\underset{i \lt j}{\sum_{i=1}^{n}\sum_{j=1}^{n}}\frac{(np_i-1)(np_j-1)}{(n-1)^2}\int\frac{f_i(x)f_j(x)}{\psi^2(x)}{\rm d}x.$ For more details about negation probability, see [Reference Wu, Deng and Xiong22].

Proof. From the definition of $D_{\alpha=0}^{\psi}$ divergence measure between mixture densities $\sum_{i=1}^{n}p_i\,f_i$ and $\sum_{i=1}^{n}\bar{p}_i f_i$ and upon setting

\begin{equation*}L=\underset{i \lt j}{\sum_{i=1}^{n}\sum_{j=1}^{n}}\frac{(np_i-1)(np_j-1)}{(n-1)^2}\int\frac{f_i(x)f_j(x)}{\psi^2(x)}{\rm d}x,\end{equation*}

we find

\begin{eqnarray*}D_{\alpha=0}^{\psi}\bigg(\sum_{i=1}^{n}p_i f_i:\sum_{i=1}^{n}\bar{p}_i f_i\bigg)&=&\frac{1}{2}\int \frac{\big(\sum_{i=1}^{n}p_i f_i(x)-\sum_{i=1}^{n}\bar{p}_i f_i(x)\big)^2}{\psi(x)}{\rm d}x\\ &=&\frac{1}{2}\sum_{i=1}^{n}{\left(\frac{np_i-1}{n-1}\right)^2} \int \frac{f_{i}^2(x){\rm d}x}{\psi(x)}{\rm d}x+L\\ &=&\frac{1}{2}\sum_{i=1}^{n}{\left(\frac{np_i-1}{n-1}\right)^2}\left(\chi^{2}(\,f_i:\psi)+1\right)+L\\ &\geq& \frac{1}{2}\sum_{i=1}^{n}{\left(\frac{np_i-1}{n-1}\right)^2}\left(KL(\,f_i:\psi)+1\right)+L, \end{eqnarray*}

where the last inequality follows from the inequality between Kullback–Leibler and chi-square divergence measures. This proves the required result.

6. Optimal information under $\chi_{\alpha}^2$ divergence measure

In this section, we first introduce $(p,\eta)$-mixture density as a generalization of arithmetic and harmonic mixture densities. Then, we examine optimal information property of $(p,\eta)$-mixture density. To follow this, we consider optimization problem for $\chi_{\alpha}^2$ divergence under three types of constraints. For more details about optimal information properties of some mixture distributions (arithmetic, geometric and $\alpha-$mixture distributions), one may refer to [Reference Asadi, Ebrahimi, Kharazmi and Soofi2] and the references therein.

6.1. $(p,\eta)$-mixture density

Definition 6.1. Let f ₀ and f ₁ be two density functions. Then, a generalized mixture density, called the $(p,\eta)$-mixture density, is defined as

\begin{eqnarray*} f_{m}(x)= \frac{pf_{0}^{\eta}(x)+(1-p)f_{1}^{\eta}(x)}{pf_{0}^{\eta-1}(x)+(1-p)f_{1}^{\eta-1}(x)}\left(\int \frac{pf_{0}^{\eta}(x)+(1-p)f_{1}^{\eta}(x)}{pf_{0}^{\eta-1}(x)+(1-p)f_{1}^{\eta-1}(x)}{\rm d}x\right)^{-1}. \end{eqnarray*}

The $(p,\eta)$-mixture density provides arithmetic and harmonic mixture densities as special cases:

1. (i) If p = 0, then $f_m(x)=f_1(x)$.

2. (ii) If p = 1, then $f_m(x)=f_0(x)$.

3. (iii) If η = 1, then $f_m(x)=pf_0(x)+(1-p)f_1(x)$ is the arithmetic mixture density.

4. (iv) If η = 0, then $f_m(x)=\frac{{\big(\frac{p}{{f}_0(x)}+\frac{1-p}{{f}_1(x)}\big)^{-1}}}{\int \big(\frac{p}{{f}_0(x)}+\frac{1-p}{{f}_1(x)}\big)^{-1} {\rm d}x}$ is the harmonic mixture density.

6.2. Optimal information property of $(p,\eta)$-mixture density

Theorem 6.2. Let f, f ₀ and f ₁ be three density functions. Then, the solution to the optimization problem

(6.1)\begin{eqnarray} \min_{{f}} \chi_{\alpha}^2({f}_0:{f}) \ \ \text{subject to}\ \chi_{\alpha}^2({f}_1:{f})=\eta, \ \int {f}(x) {\rm d}x =1 \end{eqnarray}

is the $(p,\eta)$-mixture density with $\eta=\alpha$ and mixing parameter $p=\frac{1}{1+\lambda_0}$, and $\lambda_0 \gt 0$ is the Lagrangian multiplier.

Proof. We use the Lagrangian multiplier technique for finding the solution of the optimization problem in Eq. (6.1). Thus, we have

\begin{eqnarray*} L({f},\lambda_0, \lambda_1)=\frac{1+\alpha}{2}\int \frac{({f}(x)-{f}_0(x))^{2}}{{f}_{0}^{1-\alpha}(x)}{\rm d}x+\frac{1+\alpha}{2}\lambda_0 \int \frac{({f}(x)-{f}_1(x))^{2}}{{f}_{1}^{1-\alpha}(x)}{\rm d}x+\lambda_1 \int{f}(x){\rm d}x. \end{eqnarray*}

Now, differentiating with respect to f, we obtain

(6.2)\begin{eqnarray} \frac{\partial}{\partial{f}} L({f},\lambda_0, \lambda_1)=(1+\alpha)\frac{{f}(x)-{f}_0(x)}{{f}_{0}^{1-\alpha}(x)}+(1+\alpha)\lambda_0\frac{{f}(x)-{f}_1(x)}{{f}_{1}^{1-\alpha}(x)}+\lambda_1. \end{eqnarray}

Setting Eq. (6.2) to zero, we get the optimal density function to be

\begin{eqnarray*} f(x)= \frac{pf_{0}^{\alpha}(x)+(1-p)f_{1}^{\alpha}(x)}{pf_{0}^{\alpha-1}(x)+(1-p)f_{1}^{\alpha-1}(x)}\left(\int \frac{pf_{0}^{\alpha}(x)+(1-p)f_{1}^{\alpha}(x)}{pf_{0}^{\alpha-1}(x)+(1-p)f_{1}^{\alpha-1}(x)}{\rm d}x\right)^{-1}, \end{eqnarray*}

where $p=\frac{1}{1+\lambda_0}$, as required.

Theorem 6.3. Let f, f ₀ and f ₁ be three density functions. Then, the solution to the optimization problem,

(6.3)\begin{eqnarray} \min_{{f}} \{w \chi_{\alpha}^2({f}_0:{f})+(1-w)\chi_{\alpha}^2({f}_1:{f})\} \quad \text{subject to} \ \int{f}(x) {\rm d}x =1, \quad 0\leq w \leq 1 , \end{eqnarray}

is the $(p,\eta)$-mixture density with mixing parameter p = w.

Proof. Making use of the Lagrangian multiplier technique in the same way as in Theorem 6.1, the required result is obtained.

Theorem 6.4. Let f, f ₀ and f ₁ be three density functions and $T_\alpha(X)=\frac{{f}(X)}{{f}_{1}^{1-\alpha}(X)}$. Then, the solution to the optimization problem,

(6.4)\begin{eqnarray} \min_{{f}} \chi_{\alpha}^2({f}_0:{f}) \quad \text{subject to}\ E_{f}(T_\alpha(X))=\eta, \quad \int{f}(x) {\rm d}x =1, \end{eqnarray}

is the $(p,\eta)$-mixture density with mixing parameter $p=\frac{1}{1+\lambda_0}$ and $\lambda_0 \gt 0$ is the Lagrangian multiplier.

Proof. Making use of the Lagrangian multiplier technique in the same way as in Theorem 6.1, the required result is obtained.

Now, we extend Theorem 6.2 to the case of n + 2 density functions.

Theorem 6.5. Let f, ${f}_0,\ldots,f_n$ be n + 2 density functions. Then, the solution to the optimization problem,

(6.5)\begin{eqnarray} \min_{{f}} \chi_{\alpha}^2({f}_0:{f}) \quad \text{subject to}\ \chi_{\alpha}^2({f}_i:{f})=\eta_i, i=1,\ldots,n, \quad \int {f}(x) {\rm d}x =1, \end{eqnarray}

is the extended $(p,\eta)$-mixture density with $\eta=\alpha$ and mixing parameters $p_i=\frac{\lambda_i}{1+\sum_{i=0}^{n-1}\lambda_i}$ and $\lambda_i \gt 0$, $i=0,\ldots,n$, is the Lagrangian multiplier.

Proof. We use the Lagrangian multiplier technique for finding the solution to the optimization problem in Eq. (6.5). Thus, we have

\begin{eqnarray*} L({f},\lambda_0,\ldots, \lambda_n)&=&\frac{1+\alpha}{2}\int \frac{({f}(x)-{f}_0(x))^{2}}{{f}_{0}^{1-\alpha}(x)}{\rm d}x+\sum_{i=0}^{n-1}\frac{\lambda_i(1+\alpha)}{2}\int \frac{({f}(x)-{f}_{i+1}(x))^{2}}{{f}_{i+1}^{1-\alpha}(x)}{\rm d}x\\ &&+\lambda_{n} \int{f}(x){\rm d}x. \end{eqnarray*}

Now, differentiating with respect to f, we obtain

(6.6)\begin{eqnarray} \frac{\partial}{\partial{f}} L({f},\lambda_0,\ldots,\lambda_n)=(1+\alpha)\frac{{f}(x)-{f}_0}{{f}_{0}^{1-\alpha}}+(1+\alpha)\sum_{i=0}^{n-1}\lambda_i\frac{{f}(x)-{f}_{i+1}}{{f}_{i+1}^{1-\alpha}}+\lambda_n. \end{eqnarray}

Setting Eq. (6.6) to zero, we get the optimal density function to be

\begin{eqnarray*} f(x)=k\frac{(1-\sum_{i=1}^{n}p_i)f_{0}^{\alpha}(x)+\sum_{i=1}^{n}p_i f_{i}^{\alpha}(x)}{(1-\sum_{i=1}^{n}p_i)f_{0}^{\alpha-1}(x)+\sum_{i=1}^{n}p_i f_{i}^{\alpha-1}(x)}, \end{eqnarray*}

where

\begin{equation*}k=\int \left(\frac{(1-\sum_{i=1}^{n}p_i)f_{0}^{\alpha}(x)+\sum_{i=1}^{n}p_i f_{i}^{\alpha}(x)}{(1-\sum_{i=1}^{n}p_i)f_{0}^{\alpha-1}(x)+\sum_{i=1}^{n}p_i f_{i}^{\alpha-1}(x)}\right)^{-1}{\rm d}x\end{equation*}

and $p_i=\frac{\lambda_i}{1+\sum_{i=0}^{n-1}\lambda_i}$, as required.

7. Relative-$\chi_{\alpha}^2$ divergence measure of mixed reliability systems

Consider a system with component lifetimes $X_1,\ldots,X_n$, which are independent and identically distributed (i.i.d.) with a common lifetime cumulative distribution function (c.d.f.) F and a probability density function (p.d.f.) f. Then, the system lifetime $T =\phi(X_1,\ldots , X_n)$, where ϕ is referred to as the system’s structure function, is connected to signature vector $\textbf{s}=(s_1,\ldots,s_n)$ through

\begin{equation*}s_i=P(T=X_{i:n})=\frac{n_i}{n!},~i=1,\ldots,n,\end{equation*}

where $X_{1:n},\ldots,X_{n:n}$ are the order statistics of component lifetimes and n_i is the number of ways that component lifetimes can be arranged such that $T =\phi(X_1,\ldots, X_n)=X_{i:n}$; for more details, see [Reference Samaniego20]. Then, the reliability function of T can be expressed as a mixture of reliability functions of $X_{i:n}, i=1,\ldots,n$, as

\begin{eqnarray*} {\bar{F}_{T}(t)}=\sum_{i=1}^{n}s_{i}{\bar{F}_{i:n}(t)}. \end{eqnarray*}

Consequently, the corresponding p.d.f. of T is

(7.1)\begin{eqnarray} {f_{T}(t)}=\sum_{i=1}^{n}s_{i}{f_{i:n}(t)}, \end{eqnarray}

where $f_{i:n}$ is the p.d.f. of $X_{i:n}$, given by

\begin{equation*}f_{{i:n}}(x)=\frac{n!}{(i-1)!(n-i)!}f(x) F^{i-1}(x)(1-F(x))^{n-i};\end{equation*}

see [Reference Arnold, Balakrishnan and Nagaraja1].

7.1. $D_{\alpha}^{\psi}$ measure for order statistics

Suppose $X_1,\ldots,X_n$ are i.i.d. variables from an absolutely continuous c.d.f. F and p.d.f. f, and $X_{1:n},\ldots,X_{n:n}$ are the corresponding order statistics.

Theorem 7.1. The $ D_{\alpha}^{\psi}$ divergence measure between densities $f_{i:n}$ and f is given by

(7.2)\begin{eqnarray} D_{\alpha}^{\psi}(\,f_{i:n}:f)=\frac{1+\alpha}{2}\int_{0}^{1} \frac{f(F^{-1}(u))}{\psi^{1-\alpha}\big(F^{-1}(u)\big)}\big(\,f_U(u)- f_{U_{i:n}(u)}\big)^2{\rm d}u, \end{eqnarray}

where the random variables U and $U_{i:n}$ are uniform and $Beta(i, n-i+1)$ random variables on $(0,1)$ with density functions f_U and $f_{U_{i:n}}$, respectively.

Proof. By using the definition of $D_{\alpha}^{\psi}$ divergence measure and the transformation $u=F(x)$, we obtain

\begin{eqnarray*} D_{\alpha}^{\psi}(\,f_{i:n}:f)&=&\frac{1+\alpha}{2}\int_{0}^{\infty}\frac{\big(\,f_{i:n}(x)-f(x)\big)^2}{\psi^{1-\alpha}(x)}{\rm d}x\\ &=&\frac{1+\alpha}{2}\int_{0}^{1} \frac{f(F^{-1}(u))}{\psi^{1-\alpha}\big(F^{-1}(u)\big)}\left(1- \frac{n!}{(i-1)!(n-i)!} u^{i-1}(1-u)^{n-i} \right)^2{\rm d}u\\ &=&\frac{1+\alpha}{2}\int_{0}^{1} \frac{f(F^{-1}(u))}{\psi^{1-\alpha}\big(F^{-1}(u)\big)}\big(\,f_U(u)- {f_{U_{i:n}}(u)}\big)^2{\rm d}u, \end{eqnarray*}

as required.

Corollary 7.2. From Theorem 7.1, we readily deduce the following:

1. (i) If $\psi(x)=f(x)$, then

   \begin{eqnarray*} D_{\alpha}^{\psi}(\,f_{i:n}:f)=\chi_{\alpha}^2(\,f_U:f_{U_{i:n}}). \end{eqnarray*}

2. (ii) If $\psi(x)=f_{i:n}(x)$, then

   \begin{eqnarray*} D_{\alpha}^{\psi}(\,f_{i:n}:f)=\chi_{\alpha}^2(\,f_{U_{i:n}}:f_U). \end{eqnarray*}

From Corollary 7.2, it is immediately seen that under the imposed assumptions, $D_{\alpha}^{\psi}(\,f_{i:n}:f)$ divergence is free of the baseline distribution.

Theorem 7.3. The $D_{\alpha}^{\psi}$ divergence measure between two density functions $f_{i:n}$ and $f_{j:n}$ is given by

(7.3)\begin{eqnarray} D_{\alpha}^{\psi}(\,f_{i:n}:f_{j:n})=\frac{1+\alpha}{2}\int_{0}^{1} \frac{f(\,f^{-1}(u))}{\psi^{1-\alpha}\big(\,f^{-1}(u)\big)}\big(\,f_{U_{i:n}}(u)-f_{U_{j:n}}(u)\big)^2{\rm d}u. \end{eqnarray}

Proof. By using the definition of $D_{\alpha}^{\psi}$ divergence measure and the transformation $u=F(x)$ in the same way as in the proof of Theorem 7.1, the required result is obtained.

In the special case when $\psi(x)=f_{i:n}(x)$, we find that

(7.4)\begin{eqnarray} D_{\alpha}^{\psi}(\,f_{i:n}:f_{{j:n}})=\chi_{\alpha}^{2}(\,f_{U_{i:n}}:f_{U_{j:n}}). \end{eqnarray}

7.2. $D_{\alpha}^{\psi}$ measure for mixed systems

In this section, we examine the $D_{\alpha}^{\psi}$ divergence measure associated with mixed reliability systems.

Theorem 7.4. If $\psi(x)=f_{i:n}(x)$, then the $D_{\alpha}^{\psi}(\,f_T:f_{i:n})$ divergence measure is given by

(7.5)\begin{eqnarray} D_{\alpha}^{\psi}(\,f_T:f_{i:n})={\chi_{\alpha}^2}\left(\,f_{U_{i:n}}:\sum_{i=1}^{n}s_{i}f_{U_{i:n}}\right). \end{eqnarray}

Proof. From the assumption that $\psi(x)=f_{i:n}(x)$ and the definition of $D_{\alpha}^{\psi}(\,f_T:f_{i:n})$ measure, and making use of the transformation $u=F(x)$, we have

\begin{eqnarray*} D_{\alpha}^{\psi}(\,f_T:f_{i:n})&=& \frac{1+\alpha}{2}\int_{0}^{\infty}\frac{\big(\sum_{j=1}^{n}{\alpha_j} f_{j:n}(x)-f_{i:n}(x)\big)^2}{f_{i:n}^{1-\alpha}(x)}{\rm d}x\\ &=&\frac{1+\alpha}{2}\int_{0}^{1} \frac{\big(\sum_{j=1}^{n}s_{j}f_{U_{j:n}}(u)-f_{U_{i:n}}(u) \big)^2}{f_{U_{i:n}}^{1-\alpha}(u)}{\rm d}u\\ &=&{\chi_{\alpha}^2}\bigg(\,f_{U_{i:n}}:\sum_{i=1}^{n}s_{j}f_{U_{j:n}}\bigg), \end{eqnarray*}

as required.

Theorem 7.5. Let T ₁ and T ₂ be the lifetimes of two mixed systems with signatures s and $\textbf{s}^{\prime}$ consisting of n i.i.d. components having common c.d.f. F and p.d.f. f. Then, if $\psi(x)=f(x)$, we have

\begin{eqnarray*} D_{\alpha}^{\psi}(\,f_{T_1}:f_{T_2}) &=&\frac{1+\alpha}{2}\int_{0}^{1} {\left(\sum_{i=1}^{n}f_{U_{i:n}}(u)\big(s_{i}-s_{i}^{\prime}\big)\right)^2}f^{\alpha}(F^{-1}(u)){\rm d}u, \end{eqnarray*}

where $f_{U_{i:n}}(u)$ is the p.d.f. of a beta distribution with parameters i and $n-i+1$.

Proof. From the assumption made and use of the transformation $u=F(x)$, we have

\begin{eqnarray*} D_{\alpha}^{\psi}(\,f_{T_1}:f_{T_2})&=&\frac{1+\alpha}{2}\int_{0}^{\infty} \frac{\big(\sum_{i=1}^{n}s_{i}{f_{i:n}(x)}-\sum_{i=1}^{n}s_{i}^{\prime}{f_{i:n}(x)}\big)^2}{f^{1-\alpha}(x)}{\rm d}x\\ &=&\frac{1+\alpha}{2}\int_{0}^{1} {\left(\sum_{i=1}^{n}s_{i}f_{U_{i:n}}(u)- \sum_{i=1}^{n}s_{i}^{\prime}f_{U_{i:n}}(u)\right)^2}f^{\alpha}(F^{-1}(u)){\rm d}u\\ &=&\frac{1+\alpha}{2}\int_{0}^{1} {\left(\sum_{i=1}^{n}f_{U_{i:n}}(u)\big(s_{i}-s_{i}^{\prime}\big)\right)^2}f^{\alpha}(F^{-1}(u)){\rm d}u, \end{eqnarray*}

as required.

8. Application to image processing

In this section, we present an application of Jensen-$\chi_{\alpha}^2$ measure in the framework of image quality assessment. For pertinent details about image quality assessment, see [Reference Gonzalez10].

Figure 1 shows the original lake image that includes $512\times512$ cells, and the level of the color gray of each cell assumes a value in the interval $[0,1]$ (0 for black and 1 for white). It depicts the image labeled as X and three adjusted versions of it labeled as $Y(=X+0.3)$ (increasing brightness), $Z(=\sqrt{2\times X})$ (with increased contrast and gamma correction) and $W(=\sqrt{X})$ (gamma corrected). For pertinent details, see EBImage package in R software [Reference Oles, Pau, Smith, Sklyar and Hube19].

Figure 1. The original lake image and its three adjusted versions. Image X (top-left corner), Image Y (top-right corner), Image Z (bottom-left corner) and Image W (bottom-left corner).

The extracted histograms with the corresponding empirical densities for images X, Y, Z and W are plotted in Figure 2.

Figure 2. The histograms and the corresponding empirical densities for lake image (X) and its three adjusted versions (Y, Z and W).

We can see from Figures 1 and 2 that the highest degree of similarity is first related to W and then to Y, whereas Z has the highest degree of divergence from the original image X.

8.1. Nonparametric estimation of the Jensen-$\chi_{\alpha}^2$ divergence measure

Let f ₁, f ₂ and ψ be probability density functions. Suppose we draw independent and identically distributed random samples from each of these distributions, obtaining samples of sizes n ₁, n ₂ and n_ψ, respectively. Denote the resulting samples by $X_1^{(1)}, \ldots, X_{n_1}^{(1)}$ for f ₁, $X_1^{(2)}, \ldots, X_{n_2}^{(2)}$ for f ₂ and $X_1^{\psi}, \ldots, X_{n_\psi}^{\psi}$ for ψ.

To estimate the underlying probability density functions f ₁, f ₂ and ψ using kernel density estimation, we can use the following functions:

Let $\hat{f_1}(x)$ be the kernel density estimate of f ₁, based on the sample $X_1^{(1)}, \ldots, X_{n_1}^{(1)}$. Then, we have

\begin{equation*}\hat{f_1}(x) = \frac{1}{n_1h_1}\sum_{i=1}^{n}K\left(\frac{x-X_i^{(1)}}{h_1}\right).\end{equation*}

where $K(\cdot)$ is a kernel function, typically chosen to be a symmetric probability density function, and h ₁ is a bandwidth parameter that controls the smoothness of the estimate.

Similarly, let $\hat{f_2}(x)$ be the kernel density estimate for f ₂, based on the sample $X_1^{(2)}, \ldots, X_{n_2}^{(2)}$. Then, we can write

\begin{equation*}\hat{f_2}(x) = \frac{1}{n_2h_2}\sum_{i=1}^{n_2}K\left(\frac{x-X_i^{(2)}}{h_2}\right),\end{equation*}

where h ₂ is a bandwidth parameter for the kernel density estimate of $f_2.$ Finally, let $\hat{\psi}(x)$ be the kernel density estimate for ψ, based on the sample $x_1^{\psi}, \ldots, x_{n_\psi}^{\psi}$. Then, we have

\begin{equation*}\hat{\psi}(x) = \frac{1}{n_\psi h_\psi}\sum_{i=1}^{n_\psi}K\left(\frac{x-X_i^{\psi}}{h_\psi}\right),\end{equation*}

where h_ψ is a bandwidth parameter for the kernel density estimate of ψ.

For more details, see [Reference Duong, Duong and Suggests8].

Using these estimates based on Gaussian kernel, $K(u)=\frac{1}{\sqrt{2\pi}}{\rm e}^{-\frac{u^2}{2}},$ we can compute the integrated nonparametric estimate of the Jensen-$\chi_{\alpha}^2$ measure, for $0 \lt p \lt 1,$ as

\begin{eqnarray*} \widehat{{\cal {J}}_{\alpha}^{\psi}}\big({f}_{1},{f}_{2}; {\textbf{P}}\big)&=& p\chi_{\alpha}^{2}(\widehat{\psi}:\widehat{f_1})+(1-p){\chi_{\alpha}^{2}}(\widehat{\psi}:\widehat{f_2})-{{\chi_{\alpha}^{2}}}\left(\widehat{\psi}:p\widehat{f_1}+(1-p)\widehat{f_2}\right). \end{eqnarray*}

We have computed the Jensen-$\chi_{\alpha}^2$ information measure for each pair of adjusted images with respect to the original lake image, and these are presented in Table 1. The results demonstrate that the Jensen-$\chi_{\alpha}^2$ divergence is an effective measure of similarity between each pair of adjusted images and the reference original image. Specifically, the Jensen-$\chi_{\alpha}^2$ divergence highlights the high degree of similarity between images Y and Z with respect to the original image (X). Furthermore, the results in Table 1 indicate that the comparison of images Z and W with respect to the reference image X results in low similarity. Therefore, the Jensen-$\chi_{\alpha}^2$ information measure can be considered as an efficient criteria for comparing the similarity between each pair of adjusted images with respect to the reference image.

Table 1. The Jensen-$\chi_{\alpha}^2$ divergence measure between each pair of adjusted images with respected to the original image for the choices $\alpha=0.5, 1.5$ and p = 0.5

9. Concluding remarks

In this paper, by considering the $\chi_{\alpha}^2$ divergence measure, we have proposed relative-$\chi_{\alpha}^2$, Jensen-$\chi_{\alpha}^2$ and (p, w)-Jensen-$\chi_{\alpha}^2$ divergence measures. We have first shown that the $\chi_{\alpha}^2$ divergence measure has a close relationship with q-Fisher information of mixing parameter of an arithmetic mixture distribution. We have then shown that the proposed relative-$\chi_{\alpha}^2$ divergence measure includes some other well-known versions of chi-square divergence such as the usual chi-square (χ ²), generalized-χ ² ($\chi_{\alpha}^{2}$), triangular and Balakrishnan–Sanghavi divergence measures all as special cases. We have shown that the Jensen-$\chi_{\alpha}^2$ divergence is a mixture of relative-$\chi_{\alpha}^2$ divergence measures. A lower bound for Jensen-$\chi_{\alpha}^2$ divergence has been obtained in terms of Jensen–Shannon entropy measure. We have also introduced (p, w)-Jensen-$\chi_{\alpha}^2$ divergence measure and have then established some of its properties. Further, we have studied the relative-$\chi_{\alpha}^2$ divergence measure of escort and arithmetic mixture densities. Next, we have introduced $(p,\eta)$-mixture density, which includes arithmetic-mixture and harmonic-mixture densities as special cases. Interestingly, we have shown that the proposed mixture density possesses optimal information under three different optimization problems associated with the $\chi_{\alpha}^2$ divergence measure. We have also provided a discussion about the relative-$\chi_{\alpha}^2$ divergence measure of order statistics and mixed reliability systems. Finally, we have described an application of the Jensen-$\chi_{\alpha}^2$ measure in image processing.

In summary, in this paper, some extensions of the chi-square divergence measure such as the relative-$\chi_{\alpha}^2$, $D_{\alpha}^{\psi}$, Jensen-$\chi_{\alpha}^2$ and (p, w)-Jensen-$\chi_{\alpha}^2$ divergence measures have been proposed. Particularly, it has been shown that the relative-$\chi_{\alpha}^2$ divergence measure includes the well-known divergence measures, such as L ₂, χ ², triangular, symmetric χ ², $\chi_{\alpha}^2$ and Balakrishnan–Sanghvi divergence measures, all as special cases, and provides a flexible and powerful divergence measure for comparing probability distributions in a wide rage of problems. The choice of α and the weight function $\psi(x)$ can be tailored to suit the specific characteristics and the features of the data for the models that are being compared, allowing for greater sensitivity and flexibility in the comparison process.

Furthermore, the proposed Jensen-$\chi_{\alpha}^2$ and (p, w)-Jensen-$\chi_{\alpha}^2$ divergence measures are extensions of $D_{{\alpha}}^{\psi}(\,f:g)$ that are based on a convex combination. These extensions allow for the incorporation of additional divergence measures into the framework, further increasing the flexibility and applicability of the method.

There are, of course, several areas of the proposed information measures that require more study with regard to its theoretical as well as experimental analysis. Additionally, with the incorporation of the idea of relative-$\chi_{\alpha}^2$ divergence and Jensen-$\chi_{\alpha}^2$ divergence measures, there is an opportunity to broaden and explore the discrete and cumulative versions of the established divergence measures, utilizing the properties of convexity or concavity. It will also be of great interest to study cumulative versions of these measures, and we plan to do this in our future work. Finally, there is also a potential to extend the idea to relative Fisher information measure. We are currently working on these problems and hope to report the findings in a future paper.