Evaluation of Tweedie exponential dispersion model densities by Fourier inversion

The Tweedie family of distributions is a family of exponential dispersion models with power variance functions V (μ) = μ^p for p not between (0, 1). These distributions do not generally have density functions that can be written in closed form. However, they have simple moment generating functions, so the densities can be evaluated numerically by Fourier inversion of the characteristic functions. This paper develops numerical methods to make this inversion fast and accurate. Acceleration techniques are used to handle oscillating integrands. A range of analytic results are used to ensure convergent computations and to reduce the complexity of the parameter space. The Fourier inversion method is compared to a series evaluation method and the two methods are found to be complementary in that they perform well in different regions of the parameter space

A note on modelling cross correlations: hyperbolic secant regression

The problem of determining if a bivariate normal correlation changes with respect to time or some other covariate is considered. It is assumed that the means and standard deviations of the normal random variables can be consistently estimated from the entire data run, and do not need to be re-estimated for each covariate value. A new estimator of a bivariate normal correlation is given that has useful performance down to samples of size one. This allows regression type modelling of the correlation without unnecessary loss of resolution. The arc-tanh transformation of this estimator has a symmetric Fisher’s z-distribution about the arc-tanh correlation. A method of smoothing the correlation estimates is given using moving average smoothers of the sufficient statistics from which the correlation estimator is calculated

Camera: a competitive gene set test accounting for inter-gene correlation

Competitive gene set tests are commonly used in molecular pathway analysis to test for enrichment of a particular gene annotation category amongst the differential expression results from a microarray experiment. Existing gene set tests that rely on gene permutation are shown here to be extremely sensitive to inter-gene correlation. Several data sets are analyzed to show that inter-gene correlation is non-ignorable even for experiments on homogeneous cell populations using genetically identical model organisms. A new gene set test procedure (CAMERA) is proposed based on the idea of estimating the inter-gene correlation from the data, and using it to adjust the gene set test statistic. An efficient procedure is developed for estimating the inter-gene correlation and characterizing its precision. CAMERA is shown to control the type I error rate correctly regardless of inter-gene correlations, yet retains excellent power for detecting genuine differential expression. Analysis of breast cancer data shows that CAMERA recovers known relationships between tumor subtypes in very convincing terms. CAMERA can be used to analyze specified sets or as a pathway analysis tool using a database of molecular signatures.Statistic

A Note on Modelling Cross Correlaions: Hyperbolic Secant Regression

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