# Semi-parametric estimation and prediction intervals in state space models

## TL;DR

This paper develops semi-parametric methods for estimating unknown components of state space models and constructing reliable prediction intervals, addressing limitations of traditional parametric approaches.

## Contribution

It introduces consistent estimation techniques for unknown state transition matrices and noise distributions, along with an algorithm for prediction intervals in semi-parametric state space models.

## Key findings

- Estimation methods are consistent under unknown parameters.
- Constructed prediction intervals are asymptotically valid.
- Numerical experiments demonstrate effectiveness of the proposed methods.

## Abstract

Literatures in state space models focus on parametric inference and prediction, which fail if the state space model is not fully specified and the maximum likelihood estimation does not work. In this paper, we assume the state transition matrix and the distribution of state noises are unknown. Under this assumption, we provide methods to consistently estimate these terms. In addition, we introduce an algorithm to construct consistent prediction intervals for state vectors and future observations. We complement the asymptotic results with several numerical experiments.