Statistical inference of low-frequency time series is a challenge present in various fields, such as financial risk management and weather forecasting. Practical difficulties arise due to the scarcity of non-overlapping observations. The “direct method”, which directly uses the available low-frequency data to construct estimators, often results in inaccurate estimations.

In this thesis, we propose a novel “simulation-based method” for statistical inference of low-frequency time series that result from the aggregation of a higher-frequency time series over a period of time. We start by estimating the distribution of this higher-frequency process. We then simulate a large number of paths from this estimated distribution. By independently aggregating each simulated path, we generate corresponding low-frequency data. This provides us with a large simulated dataset of the low-frequency process, which enables us to apply estimation procedures and bypass the limitations posed by the shortage of original low-frequency data.

We also provide a theoretical framework and propose three families of estimators constructed from the estimated higher-frequency distribution, analyzing their properties under additional assumptions. Through a comprehensive simulation study, we compare the simulation-based method with the traditional direct method across different scenarios and objectives. While our study focuses on the marginal distributions of low-frequency processes, the simulation-based method’s applicability extends to joint distributions across multiple time points. This research offers a robust method for parameter estimation when faced with limited low-frequency data.