ABSTRACTIn practice, the use of rounding is ubiquitous. Although researchers have looked at the implications of rounding continuous random variables, rounding may also be applied to functions of discrete random variables. For example, to infer the number of excess deaths due to falls after a national emergency, authorities may only provide a rounded average of deaths before and after the emergency started. Deaths from falling tend to be relatively low in most places, and such rounding may seriously affect inference on the change in the rate of deaths. In this paper, we study drawing inference on a parameter from the probability mass function of a non‐negative discrete random variable , when for rounding coarsening width we get as a proxy for . We show that the probability generating function of , , and capture the effect of the coarsening of the support of . Theoretical properties are explored further under some probability distributions. Moreover, we introduce two relative risks of rounding metrics to aid the numerical assessment of how sensitive the results may be to rounding. Under certain conditions, rounding has little impact. However, we also find scenarios where rounding can significantly affect statistical inference. The methods are applied to infer the probability of success of a binomial distribution and estimate the excess deaths due to Hurricane Maria. The simple methods we propose can partially counter rounding error effects.

Abstract The increasing convex order or stop‐loss order has played a significant role in various applications within actuarial science, finance, reliability, and economics. Specifically, it has been instrumental in the comparison of risks, optimal reinsurance design, capital allocation to risk assets, optimal allocation of deductibles and coverage limits, optimal retention of stop‐loss reinsurance, and risk‐reducing investments. This article introduces a novel two‐sample nonparametric test based on two independent samples for comparing two distributions with respect to the stop‐loss order. This test provides a method for comparing the expected losses incurred above some threshold or deductible for a set of risks based on independent samples. We have developed a computational technique to evaluate the proposed test statistic, ensuring its practical applicability. Our proposed test can handle variations in sample sizes and the intersection of stop‐loss functions. To assess the performance of the test statistic, a comprehensive simulation study is conducted. Additionally, the test is applied to real‐world datasets, providing practical illustrations of its utility.

Abstract Statistics on the natural log scale, such as differences, regression coefficients, and standard deviations, frequently arise when analyzing log‐transformed data or modeling binary, count, or time‐to‐event data. The statistical properties of log‐transformed estimators (for example, log odds ratio estimators) frequently appear in statistical articles. Understanding the magnitude of these quantities can be useful. Remarkably, many can be readily interpreted, without exponentiation. In this note, we introduce four new interpretations. While a log‐scale standard deviation can be interpreted as an approximate coefficient of variation describing variation about an arithmetic mean, we argue it can be more useful to interpret it exactly as a “standard relative deviation” describing variation about a geometric mean. For positive‐valued estimators, we show how the standard error of a log‐transformed estimator can be interpreted as the “standard relative error” describing the precision of the untransformed estimator. We also show how the bias and root mean squared error of a the log‐transformed estimator can be interpreted as the “mean relative error” and “root mean squared relative error” of the untransformed estimator. We illustrate the usefulness of our interpretations for (i) the usual scale parameter of a lognormal distribution, (ii) statistical properties of a log odds ratio estimator, and (iii) a single measure of the precision of an odds ratio estimator which agrees with its usual, asymmetric 95% confidence interval.

A zero‐inflated proportional odds (ZIPO) regression model is proposed as an alternative for analyzing ordinal categorical response data characterized by an excess of zeros beyond what a proportional odds regression model accounts for. To address the challenge of parameter estimation in the ZIPO regression model when covariates are missing at random, we recommend the use of the inverse probability weighting method and the nonparametric multiple imputation method. Variance estimation is conducted through Rubin‐type and two proposed bootstrap methods. Simulations are carried out to assess the finite‐sample performances of the Rubin‐type method and the two proposed bootstrap methods. The results of the simulations suggest that the performances of the two proposed bootstrap methods are comparable and generally superior to the Rubin‐type method, as indicated by the agreement between empirical standard deviation and empirical (or mean) asymptotic standard error. To illustrate the practical application of the proposed estimation methods, we utilize a survey data set capturing violations of traffic rules among motorcyclist respondents in Taiwan.

Abstract In statistical analysis, Cochran's formula plays a crucial role in disentangling the relationships between marginal and conditional regression coefficients. However, its results and implications are valid only within the linear case. Despite this, due to its simplicity and interpretability, practitioners often continue to use Cochran's formula also outside linear models. With reference to binary outcome models, we derived the approximated expression of the marginal regression coefficient when marginalization is performed over a continuous covariate and show that it mimics Cochran's formula under certain simplifying assumptions. We initially postulate a logistic link function and then show how it can be generalized. We then explore the implications of this formula in the context of sensitivity analysis and causal mediation analysis, thereby enlarging the number of circumstances where explicit parametric formulations can be used to evaluate causal direct and indirect effects, otherwise computed via numerical integration. Simulations show that our proposed estimators perform equally well as others based on numerical methods and that the additional interpretability of the explicit formulas does not compromise their precision.

Abstract In density estimation, the mean integrated squared error (MISE) is commonly used as a measure of performance. In that setting, the cross‐validation criterion provides an unbiased estimator of the MISE minus the integral of the squared density. Since the minimum MISE is known to converge to zero, this suggests that the minimum value of the cross‐validation criterion could be regarded as an estimator of minus the integrated squared density. This novel proposal presents the outstanding feature that, unlike all other existing estimators, it does not need the choice of any tuning parameter. Indeed, it is proved here that this approach results in a consistent and efficient estimator, with remarkable performance in practice. Moreover, apart from this base case, it is shown how several other problems on density functional estimation can be similarly handled using this new principle, thus demonstrating full potential for further applications.

This paper is concerned with deriving a new test on a covariance matrix which is based on its nonlinear shrinkage estimator. The distribution of the test statistic is deduced under the null hypothesis in the large‐dimensional setting, that is, when with variables and samples both tending to infinity. The theoretical results are illustrated by means of an extensive simulation study where the new nonlinear shrinkage‐based test is compared with existing approaches, in particular with the commonly used corrected likelihood ratio test, the corrected John test, and the test based on the linear shrinkage approach. It is demonstrated that the new nonlinear shrinkage test possesses better power properties under heteroscedastic alternative.

We analyze nonparametric estimators for the distribution function of the incubation time in the singly and doubly interval censoring model. The classical approach is to use parametric families like Weibull, log‐normal or gamma distributions in the estimation procedure. We propose nonparametric estimates for functions of the observations, which stay closer to the data than the classical parametric methods. We also give explicit limit distributions for discrete versions of the models and apply this to compute confidence intervals. The methods complement the analysis of the continuous model in Groeneboom (2021, 2023). R scripts for computation of the estimates are provided in Groeneboom (2020).

In this paper we study how the expectations of power means behave asymptotically as some relevant parameter approaches infinity and how to approximate them accurately for general nonnegative continuous probability distributions. We derive approximation formulae for such expectations as distribution mean increases, and apply them to some commonly used distributions in statistics and financial mathematics. By numerical computations we demonstrate the accuracy of the proposed formulae which behave well even for smaller sample sizes. Furthermore, analysis of behavior depending on sample size contributes to interesting connections with the power mean of probability distribution.