As well summarized in the title, the main goal of this paper is to demonstrate the importance of accounting for spatio-temporal dependence on the frequency of precipitation extremes when investigating the possible presence of non-stationarity. The paper is motivated, in part, by a recent study by Farris et al. (2021) who performed analyses of long-term daily precipitation records covering several regions of the world to investigate the importance of serial correlation and field significance in trend analysis. In the manuscript under review here, most of the analyses of Farris et al. (2021) are repeated using alternative models and approaches that the author believes are more correct for the purpose. The main critiques raised by the author against the work of Farris et al. (2021) are (1) the adoption of the INAR(1)-Poisson and Non-Homogeneous Poisson (NHP) models to simulate time series that reproduce realistic stationary autocorrelated series and uncorrelated series with trends, respectively; and (2) the use of statistical tests to detect trends (and in general the ‘standard’ approaches used in the literature on trend analyses). To pursue his main goal, the author presents figures and analyses aimed at showing (and erroneous proving) that:

• The INAR(1) and NHP used in Farris et al. (2021) are not proper models of count time series of over-threshold (OT) daily precipitation (P) series since they do not capture the marginal distribution of the dataset.
• The hypothesis of no-trend is verified in all cases using the Iterative Amplitude Adjusted Fourier Transform (IAAFT) model after the power spectrum is bias corrected to account for the sample size and the field significance is considered.
• The Beta-Binomial (BB) distribution parameterized through the empirical spatial, temporal, and spatiotemporal linear correlation structure of the binary process of daily OT P occurrences captures the distribution of annual OT counts. Since the binary process used to parameterize the BB is assumed stationary by the author, it is concluded that there is no need to apply any trend test.

I carefully read the paper by Dr. Serinaldi to properly understand the critiques of some of the results of Farris et al. (2021), of which I am a co-author, and whether the conclusions reached in this paper are supported by the data. As I demonstrate in the attached document, while some of the conclusions are reasonable, there are a number of assumptions made by the author that are (1) not empirically supported, and (2) similar in nature to the subjectivity that the author criticizes in the ‘standard’ approaches used in the literature on trend analyses. My main concerns are on:

1. The value of the BB distribution for the time series of annual OT counts.
2. The lack of details on the application of the IAAFT model and the power of tests based on this model.
3. The assumption of stationarity for the BB model applied to count time series.

These issues are explained in detail in the attached document. Based on these issues, I recommend the paper rejection.

Giuseppe Mascaro

I have made some comments and suggestions for improvement of the paper in the attached pdf file (Supplement)

In this study, the authors pose the question of whether there is a negligible or stronger impact of the spatio-temporal dependence on the frequency of precipitation extremes. In my opinion, the analysis and the discussion are intriguing and I believe it fits the purpose of the journal. Please see some minor comments below that I hope can serve as an open discussion and exchange of scientific opinions between the Reviewer and the Authors:

1) One of the main conclusions of this study is that "... most of this literature resorts to methods based on the same unrealistic assumption of independence and corresponding trivial models...", "On the other hand, when dependence is accounted for, its true consequences on the entire inference procedure are commonly underestimated, partially missed, or neglected.", and "...we focus on the iterated underestimation and misinterpretation of the role of spatio-temporal dependence in statistical analysis of hydro-climatological processes.". Although this conclusion is very common to me, since my main research is based on the long-term persistence, I strongly agree that dependence (and especially in its simplest form of the long-term persistence, which basically includes only 1 key parameter) is often neglected in the modern literature, not being tested at all in some studies (although there are many papers that have shown its existence, from subdaily records to climatic scales; see for example, one of the largest global-scale station-analyses by Dimitriadis et al., 2021, where short- and long-term dependence is traced in all key hydrological-cycle processes; or the work by Koutsoyiannis 2020, where there are extended discussions on how observed random trends could be misleading and can be easily explained by the LTP behaviour, as well as applications and strong traces of LTP to global-scale gridded ground observations, satellite data and reanalyses), and being replaced by others more complicated methods that some have been proven to be unrealistic (like the classical detection of trends, which by definition can support only the past values, and it is almost certain that will fail many times even in the near future). Therefore, I believe that, even solely based on the main analysis and conclusion of this study, this is a robust work that deserves publication.

Dimitriadis, P., D. Koutsoyiannis, T. Iliopoulou, and P. Papanicolaou, A global-scale investigation of stochastic similarities in marginal distribution and dependence structure of key hydrological-cycle processes, Hydrology, 8 (2), 59, doi:10.3390/hydrology8020059, 2021.

Koutsoyiannis, D., Revisiting the global hydrological cycle: is it intensifying?, Hydrology and Earth System Sciences, 24, 3899–3932, doi:10.5194/hess-24-3899-2020, 2020.

2) Regarding the motivation of this paper by Farris et al. (2021), I also agree with the other Reviewers that more focus should be given to the main conclusion "Therefore, accounting for the effect of dependence in the analysis of the frequency of extreme precipitation has huge impact that cannot be ignored.". I also understand that there are some differences between the work by Farris et al. (2021) and the current study, such as that more data-values are used and additional methods are tested, and therefore, differences in the results should be expected, making it difficult to exactly replicate their results (that again, as I stated, there is no need to focus this research on that, but trying, as a unique study, to support independently the main conclusion shown above, while discussing the similarities and differences from other scientists' work in the literature; in this way, I believe the paper could benefit and be more unambiguous to the Readers).

Farris, S., R. Deidda, F. Viola, F., and G. Mascaro, On the role of serial correlation and field significance in detecting changes in extreme precipitation frequency,Water Resources Research, 57, e2021WR030 172, 2021.

3) One of the main highlights of the current study is the clear separation between the model options independence-nonstationarity and dependence-stationarity in extreme precipitation since a mix between nonstationatity and dependence could lead to erroneous results due to the validation of the ergodicity property. For example, I much enjoyed statements like "Similar remarks hold for nonstationarity. Dealing with it does not mean just adding time dependent parameters to a stationary model, using for instance Generalized Linear Models (GMLs) and their available extensions: it means that the ergodicity property, which is key in the interpretation of statistical inference, is no longer valid. In these cases, any estimate of whatever summary statistics, such as the sample mean, is uninformative as it does not have a corresponding unique population counter part, as the latter does not exist anymore." or "Stationarity and nonstationarity are not properties of the observed hydro-climatic processes (finite observed time series), but assumptions of the models we deem suitable to describe physical processes." or "... the underlying question is whether possible monotonic fluctuations are deterministic (resulting from a well identifiable mechanism) or stochastic (as an effect of dependence, for instance). In the former case, we work under the assumption of independence and nonstationarity, whereas in the latter under the assumption of dependence and stationarity.", since this is a very common misinterpretation in the literature (several such studies are cited in the paper), and so, the highlighting of such issues is very useful. I would also recommend looking Iliopoulou and Koutsoyiannis (2019; 2020) works, which I think are very much related and focused on similar subjects and conclusions.

Iliopoulou, T., and D. Koutsoyiannis, Revealing hidden persistence in maximum rainfall records, Hydrological Sciences Journal, 64 (14), 1673–1689, doi:10.1080/02626667.2019.1657578, 2019.

Iliopoulou, T., and D. Koutsoyiannis, Projecting the future of rainfall extremes: better classic than trendy, Journal of Hydrology, 588, doi:10.1016/j.jhydrol.2020.125005, 2020.

4) Regarding the statements "This approach allows us to highlight the actual impact of spatio-temporal dependence and finite sample size on statistical inference, resulting in over-dispersed marginal distributions and biased estimates of dependence properties, such as autocorrelation and power spectrum density. These issues also affect the outcome and interpretation of statistical tests for trend detection." or "Generally, dependence implies information redundancy and reduced effective sample size, along with variance inflation and bias of standard estimators of summary statistics such as marginal and joint moments.", etc., please see if found interesting and helpful, the study by Koutsoyiannis (2004), which includes an analysis on the marginal-estimation-bias of moments and how it can hide the true nature of the distribution, and the study by Dimitriadis and Koutsoyiannis (2015), where is theoretically/mathematically derived (and linked to the climacogram, thus uniting all three metrics) the dependence-estimation-bias and impact to size-sample of autocorrelation and power-spectrum-density, while shown how one can handle it to perform robust estimations for interpretation and simulation, and thus tackiling variance inflation (in any scale) that "... affects any sampling summary statistics, including sample variance and auto-/cross-correlation."

Dimitriadis, P., and D. Koutsoyiannis, Climacogram versus autocovariance and power spectrum in stochastic modelling for Markovian and Hurst–Kolmogorov processes, Stochastic Environmental Research & Risk Assessment, 29 (6), 1649–1669, doi:10.1007/s00477-015-1023-7, 2015.

Koutsoyiannis, D., Statistics of extremes and estimation of extreme rainfall, 1, Theoretical investigation, Hydrological Sciences Journal, 49 (4), 575–590, doi:10.1623/hysj.49.4.575.54430, 2004.

5) Regarding the trace of long-term persistence and its impact on the high-order moments of skewness and kurtosis and the tail-heaviness that "imply possible non existence of the moments of high order as well as bias and/or high variability in the estimates of the moments themselves, including variance, covariance, and autocorrelation, as well as long range dependence", if the authors find it useful, please see the recently introduced K-moments by Koutsoyiannis (2023) that are found to tackle many of the, aforementioned by the authors, issues, as also shown in Dimitriadis et al. (2021) applications to a massive big-data analysis of subdaily water-cycle processes (see for example, Fig. 10 in this analysis).

Koutsoyiannis, D., Stochastics of Hydroclimatic Extremes - A Cool Look at Risk, Edition 3, ISBN: 978-618-85370-0-2, 391 pages, doi:10.57713/kallipos-1, Kallipos Open Academic Editions, Athens, 2023.