Recent Advances in Bump Hunting

For speeding up the algorithm for the method of maximum penalized likelihood it is tempting to try to make use of the Fast Fourier Transform (FFT). This can be done by circularizing the data and making the x axis discrete. For circular data the circularization is of course unnecessary. Other methods are discussed and some comparisons are made. For multivariate data one could use a multidimensional FFT by putting the data on a torus. Apart from circularization or toroidalization one can speed up the estimation of the hyperparameter by repeatedly doubling or otherwise increasing the fineness of the grid. Roughness penalties of the form $$\beta \int {\left\{ {\left[ {\left( \text{f} \right)^\xi } \right]^{\prime \prime } } \right\}} ^2 \,\text{dx,}$$ where f(x) is the density function, are also considered, where ξ is not necessarily ½ or 1, and corresponding algorithms are suggested. For the scattering data considered in previous work we have compared the results for ξ = ½ and ξ = 1 and find that the estimates of the bumps are almost the same but two of the former bumps have each been split into a pair.