Spectral analysis and stationarity tests for time series with missing values

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.