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N. Then: n P(min Zi < u ¯) = 1 − i≥1 P(Xi ≥ u ¯ − bi x)ϕ(x)dx i=1 when Xi ∼ N (µi , σi2 − b2i ) independent ∀i = 1, . . , n Proof: Let X0 , X1 , . . , Xn be n + 1 independent normal probability distribution functions such that X0 ∼ N (0, 1) and Xi ∼ N (µi , σi2 − b2i ) Let Zi = bi X0 + Xi i = 1, . . 9) i=1 For the derivation of the preliminary test, we will also need the following proposition. 7 H(¯ u) = tone function of u ¯ P(Xi ≥ u ¯ − bi x)ϕ(x)dx is a mono- Proof: n H(¯ u) P(Xi ≥ u ¯ − bi x)ϕ(x)dx = i=1 n Φ = i=1 ¯ bi x + µ0i − u σi2 − b2i ϕ(x)dx 20 1 Discriminant Analysis Based on Continuous and Discrete Variables ∂H(¯ u) = ∂u ¯ n j=1 n 1 σj2 − b2j Φ i=1 j=i ⎛ ×ϕ ⎝ Therefore ∂H(¯ u) ∂u ¯ bi x + µ0i − u ¯ σi2 − b2i × ⎞ ¯⎠ bj x + µ0j − u ϕ(x)dx 2 2 σ j − bj > 0.
1) by the ordinary least squares (OLS) method makes the assumption that the residuals uj are independent random variables. Indeed, there may be correlation between the birth weights of the children of a given woman, since the corresponding pregnancies may have been influenced by the genetic characteristics of the woman and some occupational or behavioral exposures remaining constant over the woman’s reproductive life. A possible way to cope with this correlation is to use hierarchical modelling.
Accordingly, the R2 is computed as the squared correlation between yit and xit β W G + αi . In practical applications the individual specific constants may have similar size so that it is preferable to specify the model with the same constant for all groups. This assumption can be tested with an F statistic for the hypothesis α1 = α2 = · · · = αN . In order to assess the importance of the individual specific effects, their “variances” are estimated. 2 Theoretical Aspects 35 variance of αi if we assume that these constants are deterministic.
