In this paper, two inferential procedures for selecting the significant predictors in the PLS1 regression model are introduced. The significant PLS components are first obtained and the two predictor selection methods, called PLS-Forward and PLS-Bootstrap, are applied to the PLS model obtained. They are also compared empirically to two other methods that exist in the literature with respect to the quality of fit of the model and to their predictive ability. Although none of the four methods is uniformly best, it is seen that PLS-Forward and PLS-Bootstrap perform well and can be very useful in practical situations in identifying the important explanatory variables.

Age replacement policies are investigated. When a unit reaches age T, a preventive replacement is made. If a failure occurs prior to age T and if the random repair cost is less than a fixed constant, a minimal repair is made. Otherwise the unit is replaced at failure.

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Reliability studies give rise to families of distribution functions F (n) defined recursively by the repeated convolution of a distribution function F with itself according to the scheme 0 tP (s) (t - x)Q (r) (x) dx where P (s) and Q (r) are the sth and rth members of families generated from distribution functions P and Q, not necessarily distinct. It is seldom possible or convenient to express the F (n) in analytical form. An algorithm based on cubic spline interpolation is given here for recursively generating continuous numerical approximations to the F (n) in a form which allows them to be convoluted together to provide useful approximation to the second of the above integrals.

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SUMMARYIn this paper we write the PLS multivariate regression model in terms of a redundancy index and obtain some properties of the successive PLS components. We study their significance in the model and build tests of hypotheses to this effect. A stopping rule is given to obtain the right number of PLS components, and a numerical measure is defined to assess the overall quality of the PLS regression. Finally, an algorithm is given explicitly as it was coded in S-Plus and is applied in some chosen examples.