Strutture uniformi e insiemi irriducibili di semidistanze

Sunto Tramite la nozione di insieme irriducibile di semidistanze, si dà la definizione di struttura uniforme associata ad un filtro su E×E. Si riconosce, in tal guisa, che uno spazio topologico è uniformizzabile se e solo se la topologia è compatibile con la struttura uniforme associata al filtro degli intorni della diagonale. Entrata in Redazione il 4 marzo 1970. Lavoro eseguito nell’ambito dei Contratti di ricerca del Comitato Nazionale per la Matematica del C.N.R. per il 1969–70.     

A test for the mean vector with fewer observations than the dimension

In this paper, we consider a test for the mean vector of independent and identically distributed multivariate normal random vectors where the dimension p is larger than or equal to the number of observations N. This test is invariant under scalar transformations of each component of the random vector. Theories and simulation results show that the proposed test is superior to other two tests available in the literature. Interest in such significance test for high-dimensional data is motivated by DNA microarrays. However, the methodology is valid for any application which involves high-dimensional data.     

Testing the equality of several covariance matrices with fewer observations than the dimension

For normally distributed data from the k populations with m×m covariance matrices Σ₁,…,Σ_k, we test the hypothesis H:Σ₁=?=Σ_k vs the alternative A≠H when the number of observations N_i, i=1,…,k from each population are less than or equal to the dimension m, N_i≤m, i=1,…,k. Two tests are proposed and compared with two other tests proposed in the literature. These tests, however, do not require that N_i≤m, and thus can be used in all situations, including when the likelihood ratio test is available. The asymptotic distributions of the test statistics are given, and the power compared by simulations with other test statistics proposed in the literature. The proposed tests perform well and better in several cases than the other two tests available in the literature.     

Minimum distance classification rules for high dimensional data

In this article, the problem of classifying a new observation vector into one of the two known groups Π_i,i=1,2, distributed as multivariate normal with common covariance matrix is considered. The total number of observation vectors from the two groups is, however, less than the dimension of the observation vectors. A sample-squared distance between the two groups, using Moore-Penrose inverse, is introduced. A classification rule based on the minimum distance is proposed to classify an observation vector into two or several groups. An expression for the error of misclassification when there are only two groups is derived for large p and n=O(p^δ),0<δ<1.     

Outliers in Multivariate Regression Models

Likelihood ratio tests for detecting a single outlier in multivariate linear models are considered, where an observation is called an outlier if there has been a shift in the mean. The test statistics are the maximum of n nonindependent statistics, where n is the number of observations. Relevant distributions to use upper and lower Bonferroni's inequalities are given.     

Seminorme semicontinue inferiormente e spazi localmente convessi

Sunto Si studiano gli spazi localmente convessi in cui ogni seminorma minore di una seminorma semicontinua inferiormente risulta semicontinua inferiormente. Entrata in Redazione il 3 aprile 1973. Lavoro eseguito nell'ambito delle attività del Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni (G.N.A.F.A.) del C.N.R.