The adaptive Wynn algorithm in generalized linear models with univariate response

Convergence of least squares estimators in the adaptive Wynn algorithm for some classes of nonlinear regression models

“…An appropriate stochastic model will be stated in Sect. 3 which was used in Freise et al (2019) and goes back to Lai and Wei (1982), Lai (1994), andChen et al (1999). In particular, the generated sequence x i , i ∈ N, and the observed responses y i , are viewed as values of random variables X i , i ∈ N, and Y i , i ∈ N, respectively, following a stochastic model which we call an 'adaptive regression model'.…”

“…The other (alternative) condition states that the model is essentially a generalized linear model. We call this condition (GLM * ) where the 'star' is to distinguish it from a similar 'condition (GLM)' employed in Freise et al (2019) which addressed only to the family f θ ignoring the mean response function μ. and by z 1 < z 2 this yields θ 2 = θ 2 , and hence θ 1 = θ 1 . So θ = θ and (SI) has been verified for model (b).…”

“…In general, however, Lemma 1 and its proof does not give a practical way of finding a value of r and thus of n st = r p, or even an upper bound on them. Another condition ensuring the existence of an initial design with n st = p is condition (GLM) considered in Freise et al (2019) which is the second half of condition (GLM * ) from Sect. 1.…”

“…The classical algorithm of Wynn (1970) for D-optimal design in linear regression models has motivated a particular scheme for sequential adaptive design in nonlinear regression models, see Freise (2016), Pronzato (2010) and Freise et al (2019). We refer to this scheme as 'the adaptive Wynn algorithm'.…”

“…We refer to this scheme as 'the adaptive Wynn algorithm'. In a previous paper (Freise et al 2019) of the authors the asymptotics of the sequences of designs and maximum likelihood estimators under the adaptive Wynn algorithm was studied, for the important class of generalized linear models with univariate response. In the present paper the asymptotics of least squares estimators (LSEs) under the adaptive Wynn algorithm is studied, firstly, for the class of nonlinear models satisfying a condition of 'saturated identifiability' and, secondly, for a class of generalized linear models.…”