The Kerridge [(1961). Inaccuracy and inference. Journal of the Royal Statistical Society: Series B 23(1): 184-194] inaccuracy measure is the mathematical expectation of the information content of the true distribution with respect to an assumed distribution, reflecting the inaccuracy introduced when the assumed distribution is used. Analyzing the dispersion of information around such measures helps us understand their consistency. The study of dispersion of information around the inaccuracy measure is termed varinaccuracy. Recently, Balakrishnan et al. [(2024). Dispersion indices based on Kerridge inaccuracy measure and Kullback–Leibler divergence. Communications in Statistics – Theory and Methods 53(15): 5574-5592] introduced varinaccuracy, to compare models where lower variance indicates greater precision. As interval inaccuracy is crucial for analyzing the evolution of system reliability over time, examining its variability strengthens the validity of the extracted information. This article introduces the varinaccuracy measure for doubly truncated random variables and demonstrates its significance. The measure has been studied under transformations, and bounds are also provided to broaden the applicability of the measure where direct evaluation is challenging. Additionally, an estimator for the measure is proposed, and its consistency is analyzed using simulated data through a kernel-smoothed nonparametric estimation technique. The estimator is validated on real data sets of COVID-19 mortality rates for Mexico and Italy. Furthermore, the article illustrates the practical value of the measure in selecting the best alternative to a given distribution within an interval, following the minimum information discrimination principle, thereby highlighting the effectiveness of the study.

In this paper, we study ordering properties of vectors of order statistics and sample ranges arising from bivariate Pareto random variables. Assume that $(X_1,X_2)\sim\mathcal{BP}(\alpha,\lambda_1,\lambda_2)$ and $(Y_1,Y_2)\sim\mathcal{BP}(\alpha,\mu_1,\mu_2).$ We then show that $(\lambda_1,\lambda_2)\stackrel{m}{\succ}(\mu_1,\mu_2)$ implies $(X_{1:2},X_{2:2})\ge_{st}(Y_{1:2},Y_{2:2}).$ Under bivariate Pareto distributions, we prove that the reciprocal majorization order between the two vectors of parameters is equivalent to the hazard rate and usual stochastic orders between sample ranges. We also show that the weak majorization order between two vectors of parameters is equivalent to the likelihood ratio and reversed hazard rate orders between sample ranges.

A company with n geographically widely dispersed sites seeks an insurance policy that pays off if m out of the n sites experience rarely occurring catastrophes (e.g., earthquakes) during a year. This study compares three strategies for an insurance company wishing to offer such an m-out-of-n policy, assuming the existence of markets for insurance on the individual sites with coverage periods of various lengths of a year or less. Strategy A is static: at the beginning of the year it buys a reinsurance policy on each individual site covering the entire year and makes no later adjustments. By contrast, Strategies S and C are dynamic and adaptive, exploiting the availability of individual-site policies for shorter periods than a year to make changes in the coverage on individual sites as quakes occur during the year. Strategy S uses the payoff from reinsurance when a quake occurs at a particular site to increase coverage for the remainder of the year on the sites that have not yet had quakes. Strategy C buys individual-site policies covering successive time periods of fixed length, observing the system at the beginning of each period and using cash on hand plus cash obtained from a reinsurance payoff (if any) during the previous period to decide how much cash to retain and how much reinsurance to purchase for the current period. The study relies on expected utility to determine indifference premiums and compare the premiums and loss probabilities for the three strategies.

This paper considers two supercritical branching processes with immigration in different random environments, denoted by $\{Z_{1,n}\}$ and $\{Z_{2,m}\}$, with criticality parameters µ1 and µ2, respectively. Under certain conditions, it is known that $\frac{1}{n} \log Z_{1,n} \to \mu_1$ and $\frac{1}{m} \log Z_{2,m} \to \mu_2$ converge in probability as $m, n \to \infty$. We present basic properties about a central limit theorem, a non-uniform Berry–Esseen’s bound, and Cramér’s moderate deviations for $\frac{1}{n} \log Z_{1,n} - \frac{1}{m} \log Z_{2,m}$ as $m, n \to \infty$. To this end, applications to construction of confidence intervals and simulations are also given.

Measure of uncertainty in past lifetime distribution plays an important role in the context of information theory, forensic science and other related fields. In the present work, we propose non-parametric kernel type estimator for generalized past entropy function, which was introduced by Gupta and Nanda [9], under $\alpha$-mixing sample. The resulting estimator is shown to be weak and strong consistent and asymptotically normally distributed under certain regularity conditions. The performance of the estimator is validated through simulation study and a real data set.

In this paper, we study the joint distribution of the forward and backward recurrence times in a delayed renewal process, as well as their marginal distributions. We obtain several exact results and bounds for these quantities. Some of these bounds are “general,” in the sense that the bounds are valid for any arbitrary distributions of the inter-arrival times, and some are based on aging properties of the distributions of the interarrival times of the renewals. Finally, several numerical examples are presented to illustrate the results.