Quadratic forms are ubiquitous and intensively studied in statistics, often in time series analysis, including those formed out of wavelet coefficients. Most wavelet transform methods in statistics assume regularly-spaced and complete data, which does not always occur in real problems where observations are sometimes missing, resulting in a non-regular design. To handle this, we use second-generation wavelets (lifting) which are explicitly designed to handle non-regular situations: we introduce a new estimator of the second-generation wavelet spectrum and show that it is consistent in the case of an underlying locally stationary wavelet process where the observations are subject to a random drop-out model. Our new estimator is then used to construct a new lifting-based stationarity test with significance assessed by the bootstrap. The simulation study shows excellent results, not only on time series with missing observations, but in the complete data settings too.
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