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Neyman smooth goodness-of-fit tests for the marginal distribution of dependent data. (English) Zbl 1441.62224

Summary: We establish a data-driven version of Neyman’s smooth goodness-of-fit test for the marginal distribution of observations generated by an \(\alpha \)-mixing discrete time stochastic process \((X_t)_{t \in \mathbb {Z}}\). This is a simple extension of the test for independent data introduced by T. Ledwina [J. Am. Stat. Assoc. 89, No. 427, 1000–1005 (1994; Zbl 0805.62022)]. Our method only requires additional estimation of the cumulative autocovariance. Consistency of the test will be shown at essentially any alternative. A brief simulation study shows that the test performs reasonable especially for the case of positive dependence. Finally, we illustrate our approach by analyzing the validity of a forecasting method (“historical simulation”) for the implied volatilities of traded options.

MSC:

62M07 Non-Markovian processes: hypothesis testing
62G10 Nonparametric hypothesis testing
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P05 Applications of statistics to actuarial sciences and financial mathematics

Citations:

Zbl 0805.62022
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