On cumulative residual (past) inaccuracy for truncated random variables

Some Further Results on the Fractional Cumulative Entropy

In this paper, the fractional cumulative entropy is considered to get its further properties and also its developments to dynamic cases. The measure is used to characterize a family of symmetric distributions and also another location family of distributions. The links between the fractional cumulative entropy and the classical differential entropy and some reliability quantities are also unveiled. In addition, the connection the measure has with the standard deviation is also found. We provide some examples to establish the variability property of this measure.

Bayesian Analysis of Dynamic Cumulative Residual Entropy for Lindley Distribution

Dynamic cumulative residual (DCR) entropy is a valuable randomness metric that may be used in survival analysis. The Bayesian estimator of the DCR Rényi entropy (DCRRéE) for the Lindley distribution using the gamma prior is discussed in this article. Using a number of selective loss functions, the Bayesian estimator and the Bayesian credible interval are calculated. In order to compare the theoretical results, a Monte Carlo simulation experiment is proposed. Generally, we note that for a small true value of the DCRRéE, the Bayesian estimates under the linear exponential loss function are favorable compared to the others based on this simulation study. Furthermore, for large true values of the DCRRéE, the Bayesian estimate under the precautionary loss function is more suitable than the others. The Bayesian estimates of the DCRRéE work well when increasing the sample size. Real-world data is evaluated for further clarification, allowing the theoretical results to be validated.

Uncertainty of financial time series based on discrete fractional cumulative residual entropy

Cumulative residual entropy (CRE) is a measure of uncertainty and departs from other entropy in that it is established on cumulative residual distribution function instead of density function. In this paper, we prove some important properties of discrete CRE and propose fractional multiscale cumulative residual entropy (FMCRE) as a function of fractional order α, which combines CRE with fractional calculus, probability of permutation ordinal patterns, and multiscale to overcome the limitation of CRE. After adding amplitude information through weighted permutation ordinal patterns, we get fractional weighted multiscale cumulative residual entropy (FWMCRE). FMCRE and FWMCRE extend CRE into a continuous family and can be used in more situations with a suitable parameter. Moreover, they can capture long-range phenomena more clearly and have higher sensitivity to the signal evolution. Results from simulated data verify that FMCRE and FWMCRE can identify time series accurately and have immunity to noise. We confirm that the length of time series has little effect on the accuracy of distinguishing data, and even short series can get results exactly. Finally, we apply FMCRE and FWMCRE on stock data and confirm that they can be used as metrics to measure uncertainty of the system as well as distinguishing signals. FWMCRE can also track changes in stock markets and whether adding amplitude information must be decided by the characteristics of data.