  For testing a group of hypotheses, tremendous $p$-value combination methods
have been developed and widely applied since 1930's. Some methods (e.g., the
minimal $p$-value) are optimal for sparse signals, and some others (e.g.,
Fisher's combination) are optimal for dense signals. To address a wide spectrum
of signal patterns, this paper proposes a unifying family of statistics, called
TFisher, with general $p$-value truncation and weighting schemes. Analytical
calculations for the $p$-value and the statistical power of TFisher under
general hypotheses are given. Optimal truncation and weighting parameters are
studied based on Bahadur Efficiency (BE) and the proposed Asymptotic Power
Efficiency (APE), which is superior to BE for studying the signal detection
problem. A soft-thresholding scheme is shown to be optimal for signal detection
in a large space of signal patterns. When prior information of signal pattern
is unavailable, an omnibus test, oTFisher, can adapt to the given data.
Simulations evidenced the accuracy of calculations and validated the
theoretical properties. The TFisher tests were applied to analyzing a whole
exome sequencing data of amyotrophic lateral sclerosis. Relevant tests and
calculations have been implemented into an R package $TFisher$ and published on
the CRAN.
