We present a new and up-to-date analysis of the solar low-degree $p$-mode parameter shifts from the Birmingham Solar-Oscillations Network (BiSON) over the past 22 years, up to the end of 2014. We aim to demonstrate that they are not dominated by changes in the asymmetry of the resonant peak profiles of the modes and that the previously published results on the solar-cycle variations of mode parameters are reliable. We compare the results obtained using a conventional maximum likelihood estimation algorithm and a new one based on the Markov Chain Monte Carlo (MCMC) technique, both taking into account mode asymmetry. We assess the reliability of the solar-cycle trends seen in the data by applying the same analysis to artificially generated spectra. We find that the two methods are in good agreement. Both methods accurately reproduce the input frequency shifts in the artificial data and underestimate the amplitude and width changes by a small amount, around 10 per cent. We confirm earlier findings that the frequency and line width are positively correlated, and the mode amplitude anticorrelated, with the level of solar activity, with the energy supplied to the modes remaining essentially unchanged. For the mode asymmetry the correlation with activity is marginal, but the MCMC algorithm gives more robust results than the MLE. The magnitude of the parameter shifts is consistent with earlier work. There is no evidence that the frequency changes we see arise from changes in the asymmetry, which would need to be much larger than those observed in order to give the observed frequency shift.
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