  This paper has three main contributions. The first is the construction of
wavelet transforms from B-spline scaling functions defined on a grid of
non-equispaced knots. The new construction extends the equispaced,
biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new
construction is based on the factorisation of wavelet transforms into lifting
steps. The second and third contributions are new insights on how to use these
and other wavelets in statistical applications. The second contribution is
related to the bias of a wavelet representation. It is investigated how the
fine scaling coefficients should be derived from the observations. In the
context of equispaced data, it is common practice to simply take the
observations as fine scale coefficients. It is argued in this paper that this
is not acceptable for non-interpolating wavelets on non-equidistant data.
Finally, the third contribution is the study of the variance in a
non-orthogonal wavelet transform in a new framework, replacing the numerical
condition as a measure for non-orthogonality. By controlling the variances of
the reconstruction from the wavelet coefficients, the new framework allows us
to design wavelet transforms on irregular point sets with a focus on their use
for smoothing or other applications in statistics.
