Quantile-Based Shannon Entropy for Record Statistics

Abstract

The quantile-based entropy measures possess some unique properties than their distribution function approach. The present communication deals with the study of the quantile-based Shannon entropy for record statistics. In this regard a generalized model is considered for which cumulative distribution function or probability density function does not exist and various examples are provided for illustration purpose. Further we consider the dynamic versions of the proposed entropy measure for record statistics and also give a characterization result for that. At the end, we study $F^{\alpha }$-family of distributions for the proposed entropy measure.

Appendices

Appendix: Review of Some Simple Mathematical Results

The following definitions and mathematical results will be useful in the computations of the entropy for record value distributions.

Definition D1 Gamma function. Let $\alpha > 0$. The integral

$$\begin{aligned} \int _0^{\infty } t^{\alpha -1} e^{-t} {\mathrm{d}}t = \Gamma (\alpha ) \end{aligned}$$

(5.1)

is called a (complete) gamma function.

Definition D2 Incomplete gamma functions. The upper incomplete gamma function is defined as:

$$\begin{aligned} \Gamma (s,x) = \int _x^{\infty } t^{s-1} e^{-t} {\mathrm{d}}t, \end{aligned}$$

(5.2)

whereas the lower incomplete gamma function is defined as:

$$\begin{aligned} \gamma (s,x) = \int _0^x t^{s-1} e^{-t} {\mathrm{d}}t. \end{aligned}$$

(5.3)

Definition D3 Digamma function. A digamma function, denoted by $\Psi (z)$, called a psi function, is defined as

$$\begin{aligned} \Psi (z) = \frac{{\mathrm{d}}}{{\mathrm{d}}z}[\log (\Gamma (z))] = \frac{\Gamma '(z)}{\Gamma (z)}. \end{aligned}$$

(5.4)

• Some properties related to incomplete gamma functions

  • $\Gamma (s+1,x) = s\Gamma (s,x) + x^s e^{-x}$

  • $\gamma (s+1,x) = s\gamma (s,x) - x^s e^{-x}$

  • $\Gamma (s,x) + \gamma (s,x) = \Gamma (s)$

  • $\Gamma (s,0)=\lim _{x\rightarrow \infty } \gamma (s,x) = \Gamma (s)$

• Some properties related to digamma function

  • $\Psi (z+1) = \Psi (z) + (1/z)$

  • $\Psi (1) = -\gamma , \text {~where~} \gamma = \lim _{j\rightarrow \infty }[\{ 1/1 + 1/2 + \cdots + 1/(j-1) \} - \ln (j-1)] \approx 0.57721566 \text {~is Euler's constant}.$

  • $\Psi (n) = -\gamma + \sum _{k=1}^{n-1}\frac{1}{k},~~~~~~~ \forall \text {~integers~} n\ge 2.$

  • $\int _0^{\infty } t^{n-1} e^{-t} \ln (t){\mathrm{d}}t = \Gamma (n)\Psi (n), ~~~\forall \text {~integers~} n\ge 1$.

For proofs and further related topics on these results, see for example, Abramowitz and Stegun [2], Gradshteyn and Ryzhik [15], Chaudhry [9], and Jeffrey [18] .

Conclusion

The quantile-based entropy measures possess some unique properties than its distribution function approach. The quantile-based entropy of record statistics has several advantages. Record values can be viewed as order statistics from a sample whose size is determined by the values and the order of occurrence of observations. They are closely connected with the occurrence times of a corresponding non-homogeneous Poisson process and reliability theory. The computation of proposed measure is quite simple in cases where the distribution function is not tractable while the quantile function has a simpler form.