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In large scale multiple testing, the use of an empirical null distributionrather than the theoretical null distribution can be critical for correctinference. This paper proposes a ``mode matching'' method for fitting anempirical null when the theoretical null belongs to any exponential family.Based on the central matching method for $z$-scores, mode matching estimatesthe null density by fitting an appropriate exponential family to the histogramof the test statistics by Poisson regression in a region surrounding the mode.The empirical null estimate is then used to estimate local and tail falsediscovery rate (FDR) for inference. Delta-method covariance formulas andapproximate asymptotic bias formulas are provided, as well as simulationstudies of the effect of the tuning parameters of the procedure on thebias-variance trade-off. The standard FDR estimates are found to be biased downat the far tails. Correlation between test statistics is taken into account inthe covariance estimates, providing a generalization of Efron's ``wingfunction'' for exponential families. Applications with $\chi^2$ statistics areshown in a family-based genome-wide association study from the Framingham HeartStudy and an anatomical brain imaging study of dyslexia in children.

Empirical null and false discovery rate inference for exponential families