Constraining the number of positive responses in adaptive,non-adaptive,and two-stage group testing

Group testing is a well known search problem that consists in detecting the defective members of a set of objects O by performing tests on properly chosen subsets (pools) of the given set O. In classical group testing the goal is to find all defectives by using as few tests as possible. We consider a variant of classical group testing in which one is concerned not only with minimizing the total number of tests but aims also at reducing the number of tests involving defective elements. The rationale behind this search model is that in many practical applications the devices used for the tests are subject to deterioration due to exposure to or interaction with the defective elements. In this paper we consider adaptive, non-adaptive and two-stage group testing. For all three considered scenarios, we derive upper and lower bounds on the number of “yes” responses that must be admitted by any strategy performing at most a certain number t of tests. In particular, for the adaptive case we provide an algorithm that uses a number of “yes” responses that exceeds the given lower bound by a small constant. Interestingly, this bound can be asymptotically attained also by our two-stage algorithm, which is a phenomenon analogous to the one occurring in classical group testing. For the non-adaptive scenario we give almost matching upper and lower bounds on the number of “yes” responses. In particular, we give two constructions both achieving the same asymptotic bound. An interesting feature of one of these constructions is that it is an explicit construction. The bounds for the non-adaptive and the two-stage cases follow from the bounds on the optimal sizes of new variants of d-cover free families and (p, d)-cover free families introduced in this paper, which we believe may be of interest also in other contexts.     

A performance counterexample of Billor–Chatterjee–Hadi procedure and an improvement proposal for robust regression

The BCH procedure introduced by Billor, Chatterjee, and Hadi for fitting linear models was found to be inefficient for y-outliers in the presence of a high perturbation level. We propose to modify the first step of the BCH procedure, so that the robust distances are computed on the matrix Z = (y, X) of the basic subset. The performance of the present note procedure (PNP), as compared to the BCH procedure and the ordinary least-square (OLS) method, was studied by processing several datasets used in the literature for robust regression and by performing a Monte Carlo experiment. PNP performs better particularly with datasets having high perturbation.     

A pseudo-values regression model for non-fatal event free survival in the presence of semi-competing risks

Statistical Methods & Applications - During follow-up patients may experience non-fatal events related to disease progression and death. This is a “semi-competing risks” setting, as...     

Hidden information problems in the design of family allowances

We consider the optimal determination of family allowances in a model where some of the parents have higher ability to raise children than others. First-best policy gives both types of parents the same level of utility. If neither parental ability nor parental actions are fully observable, however, the policy maker has to take into account the incentive-compatibility constraint that more able parents should not find it profitable to mis-represent their true ability by investing less in their children, and having a lower number of children. The second-best policy induces more able parents to have the first-best number of children, and to invest in each child at the first-best level. Less able parents are induced to have fewer children than in first best, and will underinvest in each child. Whether the government should subsidize more the more able parents, or the less able ones, depends on the properties of the cost function. In second best, however, less able parents will end up with lower utility than more able parents whatever the cost function.All correspondence to Annalisa Luporini. The paper has benefitted from comments by three anonymous referees. Responsible editor: Junsen Zhang.