The Pearson χ²-Divergence Measure for Doubly Truncated Random Variables

The divergence information measures play a significant role in quantifying the discrimination between two probability distributions. Two conventional divergence measures, the Kullback-Leibler divergence and the Pearson χ<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup>-divergence, possess significant importance in information theory, signal processing, data science and other associated fields. In this manuscript, the Pearson χ<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup>-divergence measure between two doubly truncated non-negative random variables is proposed. The proposed measure quantifies the amount of information lost inside a truncated interval when an assumed or estimated distribution is used instead of the actual distribution. It is shown that Pearson's χ<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup>-divergence measure characterizes the distribution uniquely under certain conditions. Several upper and lower bounds have been derived. The implications of monotonic transformations on the proposed measure are thoroughly examined. Further, the doubly truncated χ<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup>-divergence for exponential populations has been estimated by a simulation study. In addition, we demonstrated the application using real world data to illustrate the utility of the new measure. Finally, we concluded our findings with possible future scopes.