Detection of a Random Alteration in a Multivariate Observation Based on Knowledge of Probable Direction.

Abstract

The general problem of determining whether the values of some prespecified components of a single multivariate observation have been altered by the addition of a random quantity has not been addressed directly in the statistical literature. In general, researchers use techniques that do not model the random addition to a multivariate observation. In this dissertation a likelihood ratio statistic is derived to measure the "distance" of an observation from the center of the distribution in a prespecified direction. For the case of a random scalar addition, when the variances and covariances are known, the power of this likelihood ratio test is always greater than or equal to the power of the univariate test. A testing procedure based on a distribution-free tolerance interval is developed for the case where the variance-covariance matrix of the distribution must be estimated. A secondarily Bayes approach is used for the case where the variance of the random addition is unknown. The results of computer simulations for both of these cases indicate that the power of the likelihood ratio test is greater than the power of the univariate test only when the multiple correlation coefficient between the altered variate and the rest of the variates is moderate or large. When R('2) is small, the t-test has larger power. Some additional properties of the likelihood ratio test for a random scalar addition are also examined. The statistic and test procedure are also derived for the case of a random vector addition. The above methods are applied to a data set containing eight years of sample concentrations of the components of Israeli orange juice. The results are compared to the results using a chi square test (Lifshitz, Stepak and Brown, 1974) and a regression method (Rolle and Vandercook, 1963). For a simulated random scalar addition the regression approach is most powerful, but for the two examples of random vector additions the likelihood ratio test is superior.