By Sbihi A.

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**Extra info for A best first search exact algorithm for the Multiple-choice Multidimensional Knapsack Problem**

**Sample text**

34) 21 A p × 1 random vector (X , . . , X )T is said to have a multivariate Gaussian, density, with p × 1 p 1 mean vector μ and p × p covariance matrix Σ if f (x) = ((2π ) p |Σ |)−1/2 exp −(1/2)(x − μ )T Σ −1 (x − μ ) . 40 2 Preliminaries In principle, application of the maximum likelihood method here requires us to maximize the joint density22 fY,X as a function of the unknown parameters (β , σ ). However, this density may be written as the product of the conditional density fY|X and the marginal density fX , and only the conditional density is a function of these parameters.

Yn , xn ), where xi = (xi1 , . . , xip )T is now a p × 1 vector. 30) j=1 where E(εi |Xi = xi ) = 0 and V(εi |Xi = xi ) = σ 2 . 28). Letting X = [x1 , . . , xn ]T be the n × p matrix obtained by stacking the n vectors xi in rows, and writing β = (β1 , . . , β p )T , ε = (ε1 , . . , εn )T , and y = (y1 , . . 30) can be re-expressed as y = Xβ + ε . 30) specifies the yi to be ‘noisy’ measurements of the hyperplane defined by r(x) = xT β , where the argument x is now a p × 1 vector. Therefore, we again seek a value for β that minimizes the size of the residuals.

This phenomenon is the socalled multiple testing problem (also referred to as the multiple comparisons problem). Suppose there are m independent samples x(1) , . . , x(m) , and m corresponding test statistics T j = T (x( j) ), one for each sample j = 1, . . m. While we know how to control each individual test so as to have an individual significance level α , in some settings it may be desirable to instead control some analogous notion of error for the family of m tests as a whole. Note that if each test T j is controlled at the α significance level, then the probability of making at least one false rejection over the set of m tests is 1 − (1 − α )m .