# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
To how many simultaneous hypothesis tests can normal, student’s $t$ or bootstrap calibration be applied? (English) Zbl 05564448
Summary: In the analysis of microarray data, and in some other contemporary statistical problems, it is not uncommon to apply hypothesis tests in a highly simultaneous way. The number, $N$ say, of tests used can be much larger than the sample sizes, $n$, to which the tests are applied, yet we wish to calibrate the tests so that the overall level of the simultaneous test is accurate. Often the sampling distribution is quite different for each test, so there may not be an opportunity to combine data across samples. In this setting, how large can $N$ be, as a function of $n$, before level accuracy becomes poor? Here we answer this question in cases where the statistic under test is of Student’s $t$ type. We show that if either the normal or Student $t$ distribution is used for calibration, then the level of the simultaneous test is accurate provided that $logN$ increases at a strictly slower rate than ${n}^{1/3}$ as $n$ diverges. On the other hand, if bootstrap methods are used for calibration, then we may choose $logN$ almost as large as ${n}^{1/2}$ and still achieve asymptotic-level accuracy. The implications of these results are explored both theoretically and numerically.
##### MSC:
 62-99 Statistics (MSC2000)