Projection error term in Gower's interpolation

New points can be superimposed on a Euclidean configuration obtained as a result of a metric multidimensional scaling at coordinates given by Gower's interpolation formula. The procedure amounts to discarding a, possibly nonnull, coordinate along an additional dimension. We derive an analytical formula for this projection error term and, for real data problems, we describe a statistical method for testing its significance, as a cautionary device prior to further distance-based predictions.     

Family transitions and juvenile delinquency

There is a large body of research that shows children from non-intact homes show higher rates of juvenile delinquency than children from intact homes, partially due to weaker parental control and supervision in non-intact homes. What has not been adequately addressed in the research is the influence of changes in family structure among individual adolescents over time on delinquent offending. Using the first and third waves of the National Youth Study, we assess the effect of family structure changes on changes in delinquent offending between waves through the intermediate process of changes in family time and parental attachment. Although prior research has documented adolescents in broken homes are more delinquent than youth in intact homes, the process of family dissolution is not associated with concurrent increases in offending. In contrast, family formation through marriage or cohabitation is associated with simultaneous increases in offending. Changes in family time and parental attachment account for a portion of the family formation effect on delinquency, and prior parental attachment and juvenile offending significantly condition the effect of family formation on offending.     

Karhunen–Loève Basis in Goodness-of-Fit Tests Decomposition: An Evaluation

In a previous article (Grané and Fortiana, 2006 Grané , A. , Fortiana , J. ( 2006 ). An adaptive goodness-of-fit test . Commun. Statist. A. Theor. Meth 35 ( 6 ): 1141 – 1155 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), we studied a flexible class of goodness-of-fit tests associated with an orthogonal sequence, the Karhunen–Loève decomposition of a stochastic process derived from the null hypothesis. Generally speaking, these tests outperform Kolmogorov–Smirnov and Cramér–von Mises, but we registered several exceptions. In this work we investigate the cause of these anomalies and, more precisely, whether and when such poor behavior may be attributed to the orthogonal sequence itself, by replacing it with the Legendre polynomials, a commonly used basis for smooth tests. We find an easily computable formula for the Bahadur asymptotic relative efficiency, a helpful quantity in choosing an adequate basis.     

Asymptotic properties of a goodness-of-fit test based on maximum correlations

We study the efficiency properties of the goodness-of-fit test based on the Q _n statistic introduced in Fortiana and Grané [Goodness-of-fit tests based on maximum correlations and their orthogonal decompositions, J. R. Stat. Soc. B 65 (2003), pp. 115–126] using the concepts of Bahadur asymptotic relative efficiency and Bahadur asymptotic optimality. We compare the test based on this statistic with those based on the Kolmogorov–Smirnov, the Cramér-von Mises criterion and the Anderson–Darling statistics. We also describe the distribution families for which the test based on Q _n is locally asymptotically optimal in the Bahadur sense and, as an application, we use this test to detect the presence of hidden periodicities in a stationary time series.     

An Adaptive Goodness-of-Fit Test

In Fortiana and Grané (2003 Fortiana , J. , Grané , A. ( 2003 ). Goodness-of-fit tests based on maximum correlations and their orthogonal decompositions . J. Roy. Statist. Soc. B 65 ( 1 ): 115 – 126 . [CSA] [CROSSREF] [Crossref] , [Google Scholar]), we introduced the statistic Q _n , based on Hoeffding's maximum correlation, as a general-purpose goodness-of-fit test of uniformity. It admits an expansion along a countable set of orthogonal axes, originating a sequence of statistics. Linear combinations of a given number p of terms in this sequence have easy-to-compute probability distributions, either the exact ones for a finite sample or their normal asymptotic approximations for a large sample. In this article we develop an algorithm for tailoring a statistic within this class of linear combinations to test uniformity with optimal power against a specific alternative or family of alternatives.