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Possibilities of the determination of sample size in regression analysis: A review. (English) Zbl 0661.62059

Sequential methods in statistics, Banach Cent. Publ. 16, 291-308 (1985).
[For the entire collection see Zbl 0652.00012.]
We consider the following problem: X is a random vector with \(X\sim N_{n+1}(\mu,\Sigma)\), \[ X^ T=(X_{n+1},X^{(1)T}),\quad X^{(1)T}=(X_ 1,...,X_ n),\quad \mu^ T=(\mu_{n+1},\mu^{(1)T}), \]
\[ \Sigma =\left( \begin{matrix} \sigma^ 2_{n+1}\\ \sigma_{n+1(1)}\end{matrix} \begin{matrix} \sigma^ T_{n+1(1)}\\ \Sigma_{11}\end{matrix}\right), \] where \(\sigma_{n+1(1)}\) is the vector of covariances between \(X_{n+1}\) and the subvector \(X^{(1)}\) and \(\Sigma_{11}\) is the covariance matrix of \(X^{(1)}\). We are interested in the conditional moments of \(X_{n+1}\) under the condition of \(X^{(1)}=x^{(1)}:\)
(1) \(E(X_{n+1} | X^{(1)}=x^{(1)})=\mu_{n+1}+\sigma^ T_{n+1(1)} \Sigma^{-1}_{11}(x^{(1)}- \mu^{(1)})=:\mu_{n+1/(1)}\) and
(2) \(var(X_{n+1} | X^{(1)})=\sigma^ 2_{n+1}-\sigma^ T_{n+1(1)} \Sigma^{-1}_{11} \sigma_{n+1(1)}=:\sigma^ 2_{n+1/(1)}.\)
These moments should be estimated on the basis of a (mathematical) sample \(X^ T_{(N)}=(X^{(N)}_{n+1},X_{(N)}^{(1)T})\) of size N.

MSC:

62J05 Linear regression; mixed models
62L10 Sequential statistical analysis
62L12 Sequential estimation

Citations:

Zbl 0652.00012