The statistical application considered here arose in epigenomics, linking the DNA methylation proportions measured at specific\r\ngenomic sites to characteristics such as phenotype or birth order. It was found that the distribution of errors in the proportions\r\nof chemical modification (methylation) on DNA, measured at CpG sites, may be successfully modelled by a Laplace distribution\r\nwhich is perturbed by a Hermite polynomial.We use a linear model with such a response function. Hence, the response function is\r\nknown, or assumed well estimated, but fails to be differentiable in the classical sense due to themodulus function. Our problem was\r\nto estimate coefficients for the linear model and the corresponding covariance matrix and to compare models with varying numbers\r\nof coefficients. The linear model coefficients may be found using the (derivative-free) simplex method, as in quantile regression.\r\nHowever, this theory does not yield a simple expression for the covariance matrix of the coefficients of the linear model. Assuming\r\nresponse functions which are C2 except where the modulus function attains zero, we derive simple formulae for the covariance\r\nmatrix and a log-likelihood ratio statistic, using generalized calculus. These original formulae enable a generalized analysis of\r\nvariance and further model comparisons.
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