Given independent and identically distributed observations X(1), …, X(n) from a density f, provides five methods to perform a multiscale analysis about f as well as the necessary critical values. The first method, introduced in Duembgen and Walther (2008), provides simultaneous confidence statements for the existence and location of local increases (or decreases) of f, based on all intervals I(all) spanned by any two observations X(j), X(k). The second method approximates the latter approach by using only a subset of I(all) and is therefore computationally much more efficient, but asymptotically equivalent. Omitting the additive correction term Gamma in either method offers another two approaches which are more powerful on small scales and less powerful on large scales, however, not asymptotically minimax optimal anymore. Finally, the block procedure is a compromise between adding Gamma or not, having intermediate power properties. The latter is again asymptotically equivalent to the first and was introduced in Rufibach and Walther (2010).
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