RESUMO
The Markov property is widely imposed in analysis of time series data. Correspondingly, testing the Markov property, and relatedly, inferring the order of a Markov model, are of paramount importance. In this article, we propose a nonparametric test for the Markov property in high-dimensional time series via deep conditional generative learning. We also apply the test sequentially to determine the order of the Markov model. We show that the test controls the type-I error asymptotically, and has the power approaching one. Our proposal makes novel contributions in several ways. We utilise and extend state-of-the-art deep generative learning to estimate the conditional density functions, and establish a sharp upper bound on the approximation error of the estimators. We derive a doubly robust test statistic, which employs a nonparametric estimation but achieves a parametric convergence rate. We further adopt sample splitting and cross-fitting to minimise the conditions required to ensure the consistency of the test. We demonstrate the efficacy of the test through both simulations and the three data applications.
RESUMO
BACKGROUND: The majority of epidemiological studies on migraine have been conducted in a specific country or region, and there is a lack of globally comparable data. We aim to report the latest information on global migraine incidence overview trends from 1990 to 2019. METHODS: In this study, the available data were obtained from the Global Burden of Disease 2019. We present temporal trends in migraine for the world and its 204 countries and territories over the past 30 years. Meanwhile, an age-period-cohort model be used to estimate net drifts (overall annual percentage change), local drifts (annual percentage change in each age group), longitudinal age curves (expected longitudinal age-specific rate), and period (cohort) relative risks. RESULTS: In 2019, the global incidence of migraine increased to 87.6 million (95% UI: 76.6, 98.7), with an increase of 40.1% compared to 1990. India, China, United States of America, and Indonesia had the highest number of incidences, accounting for 43.6% of incidences globally. Females experienced a higher incidence than males, the highest incidence rate was observed in the 10-14 age group. However, there was a gradual transition in the age distribution of incidence from teenagers to middle-aged populations. The net drift of incidence rate ranged from 3.45% (95% CI: 2.38, 4.54) in high-middle Socio-demographic Index (SDI) regions to -4.02% (95% CI: -4.79, -3.18) in low SDI regions, 9 of 204 countries showed increasing trends (net drifts and its 95% CI were > 0) in incidence rate. The age-period-cohort analysis results showed that the relative risk of incidence rate generally showed unfavorable trends over time and in successively birth cohorts among high-, high-middle-, and middle SDI regions, but low-middle- and low-SDI regions keep stable. CONCLUSIONS: Migraine is still an important contributor to the global burden of neurological disorders worldwide. Temporal trends in migraine incidence are not commensurate with socioeconomic development and vary widely across countries. Both sexes and all age groups should get healthcare to address the growing migraine population, especially adolescents and females.