It has been shown by Hickernell and Niederreiter that there exist generating vectors for integration lattices which yield small integration errors for n = p, p2, . . . for all integers p ≥ 2. This paper provides algorithms for the construction of generating vectors which are finitely extensible for n = p, p2, . . . for all integers p ≥ 2. The proofs which show that our algorithms yield good ext...

Let Γ ⊂ Rs be a lattice, obtained from a module in a totally real algebraic number field. Let G be an axis parallel parallelepiped, and let |G| be a volume of G. In this paper we prove that lim sup |G|→∞ (det Γ#(Γ ∩G)− |G|)/ ln |G| > 0. Thus the known estimate det Γ#(Γ ∩ G) = |G| + O(ln |G|) is exact. We obtain also a similar result for the low discrepancy sequence corresponding to Γ.

In discrepancy minimization problems, we are given a family of sets S = {S1, . . . , Sm}, with each Si ∈ S a subset of some universe U = {u1, . . . , un} of n elements. The goal is to find a coloring χ : U → {−1,+1} of the elements of U such that each set S ∈ S is colored as evenly as possible. Two classic measures of discrepancy are l∞-discrepancy defined as disc∞(S , χ) := maxS∈S | ∑ ui∈S χ(u...

According to the authors, time-modified confounding occurs when the causal relation between a time-fixed or time-varying confounder and the treatment or outcome changes over time. A key difference between previously described time-varying confounding and the proposed time-modified confounding is that, in the former, the values of the confounding variable change over time while, in the latter, t...

where u ∼ N(0, G) and 2 ∼ N(0, R) with u and 2 independent. This model can be interpreted as Y |β, u ∼ N(Xβ+Zu,R), u ∼ N(0, G), yielding (on integrating out u) the marginal model Y |β ∼ N(Xβ,ZGZ + R). Let V = ZGZ + R. Then the maximum penalized likelihood estimates of β and u given G and R are given by θ̂ = [ β̂ û ] = (CTR−1C + B)−1CTR−1y (A.2) where C = [X Z] and B is the block diagonal matrix w...

Longitudinal studies, where data are repeatedly collected on one subject over a period, are common in medical research. When effect of a time-varying exposure on an outcome of interest is measured at different time points, standard statistical methods fail to give robust estimate in the presence of time-dependent confounders. There is alternative method avoid, that is, inverse probability weigh...

A concept known as ‘marginal gains’, an approach to performance improvement in elite sport came to prominence during the London 2012 Olympics when Team GB cycling won a record eight gold medals. Since National Lottery funding was awarded in 1997 to develop British cycling, the ethos of making aggregated small gains in many areas to improve overall performance has been central to its success. Th...

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