The following notations will be used in this paper: Real scalar variables, whether mathematical variables or random variables, will be denoted by lower-case letters, such as $x,y$, etc.; real vector/matrix variables—mathematical and random—will be denoted by capital letters, such as $X,Y$, etc. Complex variables will be written with a tilde, such as $\tilde{x},\tilde{y},\tilde{X},\tilde{Y}$, etc. Scalar constants will be denoted by $a,b,$ etc., and vector/matrix constants by A,B, etc. No tilde will be used on constants. If $A=\left(a_{ij}\right)$ is a $p\times p$ matrix, then its determinant will be denoted by $\left|A\right|$ or $\det \left(A\right)$ if the elements of $a_{ij}$ are real or complex. The transpose of A is written as $A^{\prime }$ and the complex conjugate transpose as $A^{*}$. The absolute value of the determinant will be written as $\left|\det \left(A\right)\right|=\sqrt{\det \left(A{A}^{*}\right)}$. For example, if $\det \left(A\right)=a+ib,i=\sqrt{(-1)},a,b$ is a real scalar, then the absolute value is $\left|\det \left(A\right)\right|=\sqrt{(a^2+b^2)}$. If $X=\left(x_{ij}\right)$ is a $p\times q$ real matrix, then the wedge product of the differentials $\mathrm{d}{x}_{ij}$ is written as $\mathrm{d}X={\wedge }_{i=1}^p{\wedge }_{j=1}^q\mathrm{d}{x}_{ij}$, where, for two real scalar variables x and y with differentials $\mathrm{d}x$ and $\mathrm{d}y$, the wedge product is defined as $\mathrm{d}x\wedge \mathrm{d}y=-\mathrm{d}y\wedge \mathrm{d}x$ so that $\mathrm{d}x\wedge \mathrm{d}x=0,\mathrm{d}y\wedge \mathrm{d}y=0$. If $\tilde{X}$ in the complex domain is a $p\times q$ matrix, then we can write $\tilde{X}=X_1+i{X}_2,i=\sqrt{(-1)},X_1,X_2$, which is real; then, we define $\mathrm{d}\tilde{X}=\mathrm{d}{X}_1\wedge \mathrm{d}{X}_2$. If $f\left(X\right)$ is a real-valued scalar function of X, where X may be scalar real variable x, scalar complex variable $\tilde{x}$, vector/matrix real variable X, or vector/matrix complex variable $\tilde{X}$ such that $f\left(X\right)\ge 0$ for all X and ${\int }_{X}f\left(X\right)\mathrm{d}X=1$, then $f\left(X\right)$ will be called a statistical density.

In many disciplines, especially in physics, communication theory, engineering, and statistics, one popular method of deriving statistical distributions is the optimization of an appropriate measure of entropy under appropriate constraints. For a real scalar random variable x, [1] introduced a measure of entropy or a measure of uncertainty: where c is a constant. The corresponding measure for the discrete case is or $(p_1,\dots ,p_k)$, which is a discrete probability law. By optimizing $S\left(f\right)$, several authors have derived exponential, Gaussian, and other distributions under the constraints in terms of moments of x, such as $E\left[x\right]=$ fixed over all functional f, meaning that the first moment is given, where $E(·)$ indicates the expected value of $(·)$. This constraint will produce exponential density. If $E\left[x\right]$ and $E\left[x^2\right]$ are fixed, meaning that the first two moments are fixed, then one has a Gaussian density, etc. The basic entropy measure in (1) has been generalized by various authors. One such generalized entropy is the Havrda–Charvát entropy [2] $H_{\alpha }\left(f\right)$ for the real scalar variable x, which is given by where $f\left(x\right)$ is a density. The original $H_{\alpha }\left(f\right)$ is for the discrete case, and the corresponding continuous case is given in (2). Various properties, characterizations, and applications of the Shannon entropy and various $\alpha$-generalized entropies were discussed by [3]. A modified version of (2) was introduced by Tsallis [4], and it is known in the literature as Tsallis’ entropy, which is the following: Observe that when $\alpha \to 1$ in (2) and $q\to 1$ in (3), both of these generalized entropies in the real scalar case reduce to the Shannon entropy of (1). Tsallis developed the whole area of non-extensive statistical mechanics by deriving Tsallis’ statistics by optimizing (3) under the constraint that the first moment is fixed in an escort density, $g\left(x\right)=\frac{{\left[f\left(x\right)\right]}^q}{{\int }_x{\left[f\left(x\right)\right]}^q\mathrm{d}x}$. Hundreds of papers have been published on Tsallis’ statistics.

In early 2000, the second author introduced a generalized entropy of the following form: where $f\left(X\right)$ is a statistical density, $f\left(X\right)\ge 0,{\int }_{X}f\left(X\right)\mathrm{d}X=1$, where X may be real scalar x, complex scalar $\tilde{x}$, real vector/matrix X, or complex vector/matrix $\tilde{X}$, a is a fixed real scalar anchoring point, $\alpha$ is a real scalar parameter, and $\eta >0$ is a real scalar constant so that the deviation of $\alpha$ from a is measured in $\eta$ units. In the real scalar case, we can see that when $\alpha \to a$, then (4) goes to the Shannon entropy in (1). Therefore, for vector/matrix variables in the real and complex domain, one has a generalization of the Shannon entropy in (4). If (3) is optimized under the constraint that the first moment $E\left[x\right]$ in $f\left(x\right)$ is fixed; then, it does not lead directly to Tsallis’ statistics. One must optimize (3) in the escort density mentioned above under the restriction that the first moment in the escort density is fixed. Then, one obtains Tsallis’ statistics. If (4) is used, then one can derive various real and complex, scalar, vector, or matrix-variate distributions directly from $f\left(X\right)$ by imposing moment-like restrictions in $f\left(X\right)$. A particular case of (4) for $a=1,\eta =1$, introduced by the second author was applied by [5] in time-series analysis, fractional calculus, and other areas. The researchers in [6] used a particular case of (4) in record values, ordered random variables, and derived some properties, including characterization theorems. In [7] discussed the analytical properties of the classical Mittag–Leffler function as being derived as the solution of the simplest fractional differential equation governing relaxation processes. In [8] studied the complexity of the ultraslow diffusion process using both the classical Shannon entropy and its general case with the inverse Mittag–Leffler function in conjunction with the structural derivative.

In the present article, the term “entropy” is used as a mathematical measure of uncertainty or information characterized by some basic axioms, as illustrated by [3]. Thus, it is a functional resulting from a set of axioms, that is, a function that can be interpreted in terms of a statistical density in the continuous case and in terms of multinomial probabilities in the discrete case. A general discussion of “entropy” is not attempted here because, as per Von Neumann, “whoever uses the term ‘entropy’ in a discussion always wins since no one knows what entropy really is, so in a debate, one always has the advantage”. An overview of various entropic functional forms used so far in the literature is available from [9], along with their historical backgrounds and an account of the numbers of citations of these various functional forms. Hence, no detailed discussion of various entropic functional forms is attempted in the present paper. The concept of entropy is applied in general physics, information theory, chaos theory, time series, computer science, data mining, statistics, engineering, mathematical linguistics, stochastic processes, etc. An account of the entropic universe was given by [10], along with answers to the following questions: How different concepts of entropy arose, what the mathematical definitions of each entropy are, how entropies are related to each other, which entropy is appropriate in which areas of application, and their impacts on the scientific community. Hence, the present article does not attempt to repeat the answers to these questions again. The present paper is about one entropy measure on a real scalar variable, its generalizations to vector/matrix variables in the real and complex domains, and an illustration of how this entropy can be optimized under various constraints to derive various statistical densities in the scalar, vector, and matrix variables in the real and complex domains. Because the entropy measure to be considered in the present article does not contain derivatives, the method of calculus of variation is used for optimization so that the resulting Euler equations will be simple. Mathematical variables and random variables are treated in the same way so that the double notations used for random variables are avoided. In order to avoid having too many symbols and the resulting confusion, scalar variables are denoted by lower-case letters and vector/matrix variables are denoted by capital letters so that the presentation is concise, consistent, and reader-friendly.

Let x be a real scalar variable and let $f\left(x\right)$ be a density function, that is, $f\left(x\right)\ge 0$ for all x and ${\int }_{x}f\left(x\right)\mathrm{d}x=1$. Consider the optimization of (4) under the following moment-like constraints: over all possible densities $f\left(x\right)$. Then, if we use calculus of variation for the optimization of (4), the Euler equation is the following: where ${\lambda }_1$ and ${\lambda }_2$ are Lagrangian multipliers and $\frac{{\lambda }_2}{{\lambda }_1}$ is taken as $b(a-\alpha )$ for convenience for $\alpha <a,b>0,\gamma >0,\delta >0,\eta >0$; a is a fixed real scalar constant, $1-b(a-\alpha )x^{\delta }>0$, and $c_1$ is the normalizing constant. For $\alpha >a$, $f_1\left(x\right)$ changes into for $\alpha >a$, $b>0,\eta >0,\delta >0,\gamma >0,x\ge 0$. When $\alpha \to a$, both $f_1\left(x\right)$ and $f_2\left(x\right)$ go to for $b>0,\eta >0,\delta >0,\gamma >0,x\ge 0$. Observe that all three functions $f_i\left(x\right),i=1,2,3$ can be reached through the pathway parameter $\alpha$. From $f_1\left(x\right)$, one can go to $f_2\left(x\right)$ and $f_3\left(x\right)$. Similarly, from $f_2\left(x\right)$, one can obtain $f_1\left(x\right)$ and $f_3\left(x\right)$. Hence, $f_1\left(x\right)$ or $f_2\left(x\right)$ is Mathai’s pathway model for the real scalar positive variable x as a mathematical model or as a statistical model. The model $f_1\left(x\right)$ is a generalized type-1 beta model, $f_2\left(x\right)$ is a generalized type-2 beta model, and $f_3\left(x\right)$ is a generalized gamma model. For $\delta =2,\gamma =0$, $f_3\left(x\right)$ is a real scalar Gaussian model. For $\gamma =2,\delta =2$, $f_3\left(x\right)$ is a Maxwell–Boltzmann density for $x\ge 0$, and for $\gamma =\frac{1}{2},\delta =2,x\ge 0$, $f_3\left(x\right)$ is the Rayleigh density for the real scalar positive variable case. If a location parameter is desired, then x is replaced by $x-m$ in all of the above models, where m is the relocation parameter. For $\gamma =0,\delta =1,\eta =1,a=1,\alpha =q$, $f_i\left(x\right),i=1,2,3$ is Tsallis’ statistic of non-extensive statistical mechanics; see [4] Tsallis (1988). Hundreds of articles have been published on Tsallis’ statistics. For $\delta =1,\eta =1,a=1,\alpha =1$, $f_2\left(x\right)$ and $f_3\left(x\right)$—but not $f_1\left(x\right)$—provide superstatistics of statistical mechanics. Several articles have been published on superstatistics.

Fermi–Dirac and Bose–Einstein densities are also available from the same procedure. In this case, the second factor $x^{\delta }$ in the constraint is replaced by ${\mathrm{e}}^{cx},c>0,x\ge 0$, and the Lagrangian multipliers are taken as $-{\lambda }_1$ and $-{\lambda }_2$ so that the second factor in Equation (6) becomes ${({\lambda }_1+{\lambda }_2{\mathrm{e}}^{cx})}^{-\eta /(\alpha -a)}$ for $\alpha >a$ with $({\lambda }_1+{\lambda }_2{\mathrm{e}}^{cx})>0$ to create a density function. Now, take $\gamma =0,\eta =1,\alpha -a=1$. Then, for ${\lambda }_1=1,{\lambda }_2={\mathrm{e}}^d$ and for some constant d, this gives the Fermi–Dirac density, and for ${\lambda }_1=-1$ and ${\lambda }_2={\mathrm{e}}^d$, this gives Bose–Einstein density.

In model-building situations, if $f_3\left(x\right)$ is the generalized gamma model, Maxwell–Botlzmann model $(\gamma =2,\delta =2)$, Rayleigh model $(\gamma =\frac{1}{2},\delta =2)$, or Gaussian model $(\gamma =0,\delta =2)$ and is the stable or ideal situation in a physical system, then $f_1\left(x\right)$ and $f_2\left(x\right)$ provide the unstable or chaotic neighborhoods, and through the pathway parameter $\alpha$, one can model the stable situation, the unstable neighborhoods, and the transitional stages in a data analysis situation. This is the pathway idea of Mathai.

Let X be a $p\times 1$ real vector with distinct real scalar variables $x_j$ as elements; $X^{\prime }=(x_1,\dots ,x_p)$, where a prime denotes the transpose. Let $\mu$ be a $p\times 1$ location vector. Let the covariance matrix in X be $\Sigma =E\left[(X-\mu ){(X-\mu )}^{\prime }\right],\mu =E\left[X\right]$; then, $\Sigma ={\Sigma }^{\prime }$, and let $\Sigma >O$ (real positive definite). Then, the square of the Euclidean distance of X from the point of location $\mu$ is ${(X-\mu )}^{\prime }(X-\mu )$, and the generalized distance of X from $\mu$ is ${(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )$. Because $\Sigma$ is real positive definite, $u={(X-\mu )}^{\prime }{\Sigma }^{-1}(X-\mu )$ is known as the ellipsoid of concentration. The probability content of this ellipsoid of concentration is an important quantity in statistical analysis. Let us consider constraints in terms of moments of the ellipsoid of concentration u. Consider the following constraints: over all possible densities $f\left(X\right)$, where X is a $p\times 1$ vector random variable. Then, optimizing Mathai’s entropy in (4) for all possible densities $f\left(X\right)$ and proceeding as in Section 2, we have the following three densities: For $\alpha <a$, for $\alpha <a,b>0,\gamma >0,\delta >0,\Sigma >O,\eta >0$. For $\alpha >a$, the model in (9) changes into the model for $\alpha >a,b>0,\gamma >0,\delta >0,\eta >0,\Sigma >O$, and for $\alpha \to a$, the models in both (9) and (10) go to the model for $b>0,\eta >0,\Sigma >O$, where $C_i,i=1,2,3$ are the normalizing constants. These normalizing constants can be evaluated, and further properties of the models can be studied with the help of the following results from [11]:

For the proof of this result, as well as for other similar results, see [11]. We will state one more result from [11] here without proof.

Let $X=\left(x_{ij}\right)$ be a real $p\times q,p\le q$, and rank p matrix with distinct real scalar variables $x_{ij}$ as elements. Let $A>O$ be a $p\times p$ constant positive definite matrix and let $B>O$ be a $q\times q$ constant positive definite matrix. Let $u=\mathrm{tr}\left(A^{\frac{1}{2}}XB{X}^{\prime }{A}^{\frac{1}{2}}\right)$. This u is an important quantity in statistical literature. Hence, we will impose restrictions in terms of moments of u. Consider the optimization of Mathai’s entropy in (4) over all densities $f\left(X\right)$, where X is a $p\times q$ matrix, as defined above, subject to the constraints: over all possible densities $f\left(X\right)$. Then, proceeding as in Section 3, we end up with the following densities, where we use the same notations of $f_i\left(X\right),C_i,i=1,2,3$ in order to avoid having too many symbols: For $\alpha <a$, For $\alpha >a$, and for $\alpha \to a$, For evaluating the normalizing constants, we use the following transformations: $Y=A^{\frac{1}{2}}X{B}^{\frac{1}{2}}\Rightarrow \mathrm{d}Y={\left|A\right|}^{-\frac{q}{2}}{\left|B\right|}^{-\frac{p}{2}}\mathrm{d}X$, $s=\mathrm{tr}\left(Y{Y}^{\prime }\right)=$ the sum of squares of all the $pq$ elements in Y, and, hence, $\mathrm{tr}\left(Y{Y}^{\prime }\right)=Z{Z}^{\prime }$, where Z is a $1\times pq$ vector. Then, from Lemma 2, $s=Z{Z}^{\prime }\Rightarrow \mathrm{d}Y=\mathrm{d}Z=\frac{{\pi }^{\frac{pq}{2}}}{\mathsf{\Gamma }\left(\frac{pq}{2}\right)}{s}^{\frac{pq}{2}-1}\mathrm{d}s.$ Then, for $\alpha <a$, we evaluate the s-integral by using a real scalar type-1 beta integral; for $\alpha >a$, we evaluate the s-integral by using a real scalar type-2 beta integral; for $\alpha \to a$, the s-integral is evaluated by using a real scalar gamma integral. Then, the normalizing constants are the following: where, in (23), the conditions are $\alpha <a,A>O,B>O,b>0,\eta >0,\delta >0,\gamma >0$; in (24), the conditions are $\alpha >a,A>O,B>O,b>0,\eta >0,\delta >0,\gamma >0,\frac{\eta }{\alpha -a}-\frac{1}{\delta }(\gamma +\frac{pq}{2})>0$; in (25), the conditions are $A>O,B>O,b>0,\eta >0,\gamma >0,\delta >0$.

Observe that (21) and (22) are available from (20). Similarly, (20) and (22) are available from (21). In other words, all densities in (20)–(22) are available through the pathway parameter $\alpha$. Note that (22) for $\delta =1,\gamma =1$ can be taken as a multivariate version of Maxwell–Boltzmann density coming from a rectangular matrix-variate real random variable. Similarly, for $\delta =1,\gamma =\frac{1}{2}$, one can take (22) as a version of the multivariate real Rayleigh density coming from a rectangular matrix-variate real random variable. For $\gamma =0,\delta =1$, (20) is a real rectangular matrix-variate Gaussian density. One can consider (20) as a generalized real multivariate type-1 beta density, (21) as a generalized real multivariate type-2 beta density, and (22) as the corresponding gamma density. For $\gamma =0$, the model in (20) is a suitable model for reliability analysis for a real multivariate situation. As observed in Section 3, one can see that $\mathrm{tr}\left(A^{\frac{1}{2}}XB{X}^{\prime }{A}^{\frac{1}{2}}\right)$ has an H-function distribution for $\alpha <a,\alpha >a,\alpha \to a$. In addition, for $\alpha <a$, ${\left[b(\alpha -a)\mathrm{tr}\left(A^{\frac{1}{2}}XB{X}^{\prime }{A}^{\frac{1}{2}}\right)\right]}^{\frac{1}{\delta }}$ is a real scalar type-1 beta distributed with the parameters $(\gamma +\frac{pq}{2},1+\frac{\eta }{a-\alpha })$; for $\alpha >a$, ${\left[b(\alpha -a)\mathrm{tr}\left(A^{\frac{1}{2}}XB{X}^{\prime }{A}^{\frac{1}{2}}\right)\right]}^{\frac{1}{\delta }}$ is a real scalar type-2 beta distributed with the parameters $(\gamma +\frac{pq}{2},\frac{\eta }{\alpha -a}-(\gamma +\frac{pq}{2}))$; for $\alpha \to a$, ${\left[b\eta \mathrm{tr}\left(A^{\frac{1}{2}}XB{X}^{\prime }{A}^{\frac{1}{2}}\right)\right]}^{\frac{1}{\delta }}$ is a real scalar gamma distributed with the parameters $(\gamma +\frac{pq}{2},1)$.

Let $X=\left(x_{ij}\right)$ be a $p\times q,p\le q$, and rank p matrix with distinct elements $x_{ij}$. Let $A>O$ be $p\times p$ and $B>O$ be $q\times q$ constant positive definite matrices. Consider the optimization of Mathai’s entropy (4) under the constraint over all real $p\times q,p\le q$, and rank p matrix-variate densities $f\left(X\right)$. Then, following the same procedure as in the above cases, we end up with the density for $\alpha <a,b>0,\eta >0,A>O,B>O,I-b(a-\alpha )A^{\frac{1}{2}}XB{X}^{\prime }{A}^{\frac{1}{2}}>O$, and a is a fixed scalar constant. In order to avoid having too many symbols, we will use the same notations of $f_i\left(X\right),C_i,i=1,2,3$ in this section. For $\alpha >a$, the model in (26) changes into for $\alpha >a,b>0,\eta >0,A>O,B>O$. When $\alpha \to a$, both $f_1\left(X\right)$ and $f_2\left(X\right)$ go to for $b>0,\eta >0,A>O,B>O$. The transition of (26) and (27) to (28) can be seen from the following properties. Let ${\lambda }_1,\dots ,{\lambda }_p$ be the eigenvalues of $A^{\frac{1}{2}}XB{X}^{\prime }{A}^{\frac{1}{2}}$. Then, However, from the definition of the mathematical constant e, we have ${lim}_{\alpha \to a_{-}}{(1-b(a-\alpha ){\lambda }_j)}^{\frac{\eta }{a-\alpha }}={\mathrm{e}}^{-b\eta {\lambda }_j}$. In a similar manner, ${lim}_{\alpha \to a_{+}}{(1+b(\alpha -a){\lambda }_j)}^{-\frac{\eta }{\alpha -a}}={\mathrm{e}}^{-b\eta {\lambda }_j}$. Then, the product gives the sum of the eigenvalues or the trace in the exponent and, hence, the result. The normalizing constants $C_i,i=1,2,3$ can be evaluated by using the following transformations: $Y=A^{\frac{1}{2}}X{B}^{\frac{1}{2}}\Rightarrow \mathrm{d}Y={\left|A\right|}^{\frac{q}{2}}{\left|B\right|}^{\frac{p}{2}}\mathrm{d}X$ by using Lemma 1 or $A=Y{Y}^{\prime }\Rightarrow \mathrm{d}Y=\frac{{\pi }^{\frac{pq}{2}}}{{\mathsf{\Gamma }}_p\left(\frac{q}{2}\right)}{\left|S\right|}^{\frac{q}{2}-\frac{p+1}{2}}\mathrm{d}S$ by using Lemma 2. Then, evaluating the S-integral by using a real matrix-variate type-1 beta integral for $\alpha <a$, by using a real matrix-variate type-2 beta integral for $\alpha >a$, or by using a real matrix-variate gamma integral for $\alpha \to a$, we obtain the results, where, for example, ${\mathsf{\Gamma }}_p\left(\alpha \right)$ is the real matrix-variate gamma defined earlier in (14).