Non-asymptotic confidence estimation of the parameters in stochastic regression models with Gaussian noises

Non-asymptotic Confidence Estimation of the Autoregressive Parameter in AR(1) Process with an Unknown Noise Variance

“…At the third stage we construct an estimator for the parameter θ on the basis of the improved sequential point estimator proposed in Konev and Vorobeychikov (2017), which is a special modification of the least squares (maximum likelihood) estimators. For each H > 0 we introduce the stopping instance…”

“…Note that, according to Theorem 6.1 in Konev and Vorobeychikov (2017), the conditional distribution of m(H) subject to F τ +l is the standard Gaussian distribution. Taking into account inequalities ( 9) and H ≤H we obtain…”

“…As ε k are standard Gaussian random variables, the sum of their squares has the χ 2 (l) distribution. According to Theorem 6.1 in Konev and Vorobeychikov (2017),…”

“…In Konev and Vorobeychikov (2017), the sequential estimation procedure of Borisov and Konev (1977) was modified; it allows obtaining a point autoregressive parameter estimator with a non-asymptotic Gaussian distribution and constructing a fixed-width confidence interval with any prescribed probability of coverage. We propose to use this estimator to construct a modified three-stage estimation procedure for AR(1) process with an unknown noise variance.…”