Quantile based entropy function

Abstract

Quantile functions are efficient and equivalent alternatives to distribution functions in modeling and analysis of statistical data (see Gilchrist, 2000; Nair and Sankaran, 2009). Motivated by this, in the present paper, we introduce a quantile based Shannon entropy function. We also introduce residual entropy function in the quantile setup and study its properties. Unlike the residual entropy function due to Ebrahimi (1996), the residual quantile entropy function determines the quantile density function uniquely through a simple relationship. The measure is used to define two nonparametric classes of distributions.

Rényi’s residual entropy: A quantile approach

“…Thus, we see that, as β → 1, we get the bounds for the residual quantile entropy function given in Sunoj and Sankaran (2012), as expected. We prove later that equality in (2.8) holds if and only if the underlying distribution is exponential.…”

“…This makes the analytical study of the properties of these distributions by means of (1.1) or (1.2) difficult. Accordingly, Sunoj and Sankaran (2012) introduced quantile versions of the Shannon entropy (1.1) and its residual form (1.2). The quantile based residual entropy is defined by…”

Quantile based entropy function in past lifetime

“…Recently, Sunoj and Sankaran (2012) obtained a quantile version of the residual entropy ξ (X; t), given by…”

“…For more properties of ψ(u), one may refer to Sunoj and Sankaran (2012). The entropy function in past lifetime (3) in terms of QF is defined by…”

“…It is interesting to note that ψ E (u) for equilibrium model is a constant (see Sunoj and Sankaran, 2012) while ψ E (u) is not.…”