Asymmetric cusp estimation in regression models

Confidence intervals for point-of-stabilization of content uniformity

Within the framework of continuous pharmaceutical manufacturing, we are interested in statistical modeling of the initial behavior of the production line. Assuming a gradually changing sequence of a suitable product quality characteristic (e.g., the content uniformity), we estimate the so-called point-of-stabilization (PoSt) and construct corresponding confidence regions based on appropriate asymptotic distributions and bootstrap. We investigate linear, quadratic, and nonlinear gradual change models both in homoscedastic and heteroscedastic setup. We propose a new nonlinear E_max gradual change model and show that it is applicable even if the true model is linear. Asymptotic distribution of the PoSt estimator is known only in a homoscedastic linear and quadratic model and, therefore, bootstrap approximations are used to construct one-sided PoSt confidence intervals.

Estimating a gradual parameter change in an AR(1)-process

We discuss the estimation of a change-point $$t_0$$ t 0 at which the parameter of a (non-stationary) AR(1)-process possibly changes in a gradual way. Making use of the observations $$X_1,\ldots ,X_n$$ X 1 , … , X n , we shall study the least squares estimator $$\widehat{t}_0$$ t ^ 0 for $$t_0$$ t 0 , which is obtained by minimizing the sum of squares of residuals with respect to the given parameters. As a first result it can be shown that, under certain regularity and moment assumptions, $$\widehat{t}_0/n$$ t ^ 0 / n is a consistent estimator for $$\tau _0$$ τ 0 , where $$t_0 =\lfloor n\tau _0\rfloor $$ t 0 = ⌊ n τ 0 ⌋ , with $$0<\tau _0<1$$ 0 < τ 0 < 1 , i.e., $$\widehat{t}_0/n \,{\mathop {\rightarrow }\limits ^{P}}\,\tau _0$$ t ^ 0 / n → P τ 0 $$(n\rightarrow \infty )$$ ( n → ∞ ) . Based on the rates obtained in the proof of the consistency result, a first, but rough, convergence rate statement can immediately be given. Under somewhat stronger assumptions, a precise rate can be derived via the asymptotic normality of our estimator. Some results from a small simulation study are included to give an idea of the finite sample behaviour of the proposed estimator.