We introduce the so-called naive tests and give a brief review of the new developments. Naive testing methods are easy to understand and perform robustly, especially when the dimension is large. We focus mainly on reviewing some naive testing methods for the mean vectors and covariance matrices of high-dimensional populations, and we believe that this naive testing approach can be used widely in many other testing problems.
Some other related work on tests of high-dimensional locations can be found in @cite_12 @cite_47 @cite_33 @cite_43 @cite_52 @cite_37 @cite_16 , which we do not discuss at length in this paper.
@cite_37 @cite_33 @cite_52 @cite_43 @cite_47 @cite_16 @cite_12
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In this article, we propose new multivariate two-sample tests based on nearest neighbor type coincidences. While several existing tests for the multivariate two-sample problem perform poorly for high dimensional data, and many of them are not applicable when the dimension exceeds the sample size, these proposed tests can be conveniently used in the high dimension low sample size (HDLSS) situations. Unlike Schilling (1986) [26] and Henze’s (1988) test based on nearest neighbors, under fairly general conditions, these new tests are found to be consistent in HDLSS asymptotic regime, where the sample size remains fixed and the dimension grows to infinity. Several high dimensional simulated and real data sets are analyzed to compare their empirical performance with some popular two-sample tests available in the literature. We further investigate the behavior of these proposed tests in classical asymptotic regime, where the dimension of the data remains fixed and the sample size tends to infinity. In such cases, they turn out to be asymptotically distribution-free and consistent under general alternatives. The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50). We explore the effect of dimensionality on the "nearest neighbor" problem. We show that under a broad set of conditions (much broader than independent and identically distributed dimensions), as dimensionality increases, the distance to the nearest data point approaches the distance to the farthest data point. To provide a practical perspective, we present empirical results on both real and synthetic data sets that demonstrate that this effect can occur for as few as 10-15 dimensions. These results should not be interpreted to mean that high-dimensional indexing is never meaningful; we illustrate this point by identifying some high-dimensional workloads for which this effect does not occur. However, our results do emphasize that the methodology used almost universally in the database literature to evaluate high-dimensional indexing techniques is flawed, and should be modified. In particular, most such techniques proposed in the literature are not evaluated versus simple linear scan, and are evaluated over workloads for which nearest neighbor is not meaningful. Often, even the reported experiments, when analyzed carefully, show that linear scan would outperform the techniques being proposed on the workloads studied in high (10-15) dimensionality! We discuss a one-sample location test that can be used in the case of highdimensional data. For high-dimensional data, the power of Hotelling’s test decreases when the dimension is close to the sample size. To address this loss of power, some non-exact approaches were proposed, e.g., Dempster (1958, 1960), Bai and Saranadasa (1996) and Srivastava and Du (2006). In this paper, we focus on Hotelling’s test and Dempster’s test. The comparative merits and demerits of these two tests vary according to the local parameters. In particular, we consider the situation where it is difficult to determine which test should be used, that is, where the two tests are asymptotically equivalent in terms of local power. We propose a new statistic based on the weighted averaging of Hotelling’s T 2 statistic and Dempster’s statistic that can be applied in such a situation. Our weight is determined on the basis of the maximum local asymptotic power on a restricted parameter space that induces local asymptotic equivalence between Hotelling’s test and Dempster’s test. In addition, some good asymptotic properties with respect to the local power are shown. Numerical results show that our test is more stable than Hotelling’s T 2 statistic and Dempster’s statistic in most parameter settings.