In this paper, we introduce the Beta-pG family of distributions as a flexible framework for modeling long-term survival data. This family extends the traditional Beta-G distribution by incorporating a new parameter p, from which a nonzero long-term survivor fraction is induced through the limiting survival probability. This structure allows the model to capture survival patterns in which a portion of the population remains unaffected by the event of interest over extended follow-up. In particular, we focus on the case where G, the baseline distribution, is specified as the exponential distribution, leading to the Beta-pExponential model. Parameter estimation is performed using maximum likelihood method, allowing likelihood-based inference for the model parameters and the long-term survivor proportions. Monte Carlo simulation studies are conducted to evaluate the finite-sample performance of the estimators across various scenarios. The proposed model is then applied to two melanoma survival datasets, including data from a clinical trial and a population-based cancer registry. In both applications, the fitted model captures the main features of the observed survival curves and provides direct estimates of the long-term survivor proportions. These results illustrate the usefulness of the Beta-pG family as an alternative framework for analyzing survival data with long-term survivors.
Granger causality models are widely used in time series analysis to manage causal relationships between variables based on temporal precedence. Traditional Granger causality models always assume linear relationships and normality assumptions, which often fail to capture the complexities of real-world data involving nonlinear and non-normal behavior. In this work, we propose a copula-based approach that offers a novel alternative method by capturing nonlinearity and non-normality in multivariate time series modeling. We develop a copula-based joint regression model tailored for Granger causality analysis and carry out some predictions in these models. We assess and validate the performance of our method using simulation studies in different scenarios. Two applications of our proposal in real-world data analysis are also presented. Our findings highlight the superiority of copula-based approaches compared to other Granger causality models in complex time series data.
The manuscript introduces a new first-order integer-valued autoregressive process in a random environment, based on the binomial thinning operator. The proposed model is driven by two control processes: the first determines the marginal distribution, while the second governs the correlation structure and the dynamic behavior of the series. The inclusion of two control processes enhances the model’s flexibility and allows for a more accurate fit to the data. Two variants of the model are considered—one with a Poisson marginal distribution and the other with a geometric marginal distribution—and both specifications are theoretically analyzed. The fundamental properties of the model are investigated in detail, providing comprehensive theoretical insight into its structure, moment characteristics, and the dependence structure it generates. For the estimation of unknown parameters, two methods are employed: the Yule–Walker estimator and the conditional maximum likelihood estimator. Their performance is evaluated through extensive simulation studies. Finally, the practical applicability of the proposed model is illustrated using real-world data.
The Bayes factor reversal (BFR) paradox shows that for any significant result under a point-null hypothesis, there exists a prior scale τ∗—the “flip point”—where BF01=1. Below τ∗, evidence favors H1; above, it favors H0. We prove this phenomenon is universal across thirteen standard test families: z-tests, t-tests (one-sample, two-sample, paired), one-way ANOVA, chi-squared, proportion tests, correlation, regression, and nonparametric tests (Mann–Whitney and Wilcoxon signed-rank). The mathematical mechanism is the same in every case: continuity, a unique minimum, and divergence of BF01(τ) guarantee a unique flip point. We propose the flip point as a robustness diagnostic: just as power analysis quantifies frequentist fragility to sample size, the flip point quantifies Bayesian fragility to prior specification. When τ∗ lies within the range of common default priors, conclusions are sensitive to arbitrary software choices. We provide closed-form solutions where available, R code, and an interactive web-based Flip Point Calculator covering all test families, enabling researchers to assess and report the robustness of their Bayesian inferences.
This study proposes a two-dimensional index for measuring the degree of deviance from the quasi-symmetry (QS) model corresponding to the conditional QS (CQS) model for ordinal square contingency tables. The CQS model indicates the structure of asymmetry for the local odds ratios. Previous study proposed a two-dimensional index for measuring the degree of deviance from the QS model corresponding to the extended QS (EQS) which has a different probability structure from the CQS model. The asymmetry parameters of the CQS and EQS models indicate both the degree and direction of deviance from the QS model. When the CQS (or EQS) model does not fit for the presented data well, we cannot evaluate them using the asymmetry parameter. The two-dimensional index can address this issue. The proposed two-dimensional index is assembled by combining two new sub-indexes. One represents the degree of deviance from the QS model corresponding to the CQS model, and the other represents its direction. Additionally, this study derives a plug-in estimator and an approximate confidence region for the proposed two-dimensional index. For comparing several datasets, we reveal that the magnitude relationship between the degrees of deviance from the QS model changes by whether the proposed or existing two-dimensional index is used.
