Hydrological time series are often composed by non-normally independent realizations of phenomena, and this characteristic makes the use of nonparametric trend tests very attractive (Kundzewicz and Robson, 2004). The Mann–Kendall test is a widely used rank-based tool for detecting monotonic, and not necessarily linear, trends. Given a random variable z, and assigned a sample of L independent data z=(z1, …, zL), the Kendall S statistic (Kendall, 1975) can be defined as follows: 1S=∑i=1L-1∑j=i+1Lsgnzj-zi, where “sgn” is the sign function.

The null hypothesis of this test is the absence of any statistically significant trend in the sample, whereas the presence of a trend represents an alternative hypothesis. Yilmaz and Perera (2014) reported that serial dependence can lead to a more frequent rejection of the null hypothesis. For L≥8, Mann (1945) reported that Eq. (1) is approximatively a normal variable with a zero mean and variance which, in the presence of tm ties of length m, can be expressed as V=L(L-1)(2L+5)-∑m=1ntmm(m-1)(2m+5)18. In practice, the Mann–Kendall test is performed using the Z statistic Z=S-1V(S)S>00S=0S+1V(S)S<0, which follows a standard normal distribution. Using this approach, it is simple to evaluate the p value and compare it with an assigned level of significance or, equivalently, to calculate the Zα threshold value and compare it with Z, where Zα is the (1-α) quantile of a standard normal distribution.

Yue et al. (2002b) observed that autocorrelation in time series can influence the ability of the MK test to detect trends. To avoid this problem, a correct approach with respect to trend analysis should contemplate a preliminary check for autocorrelation and, if necessary, the application of pre-whitening procedures.

A nonparametric tool for a reliable estimation of a trend in a time series with N pairs of data is the Sen's slope estimator (Sen, 1968), which is defined as the median of the set of slopes δj: 2δj=zi-zki-k,j=1,…,N, where i>k.

The likelihood ratio statistical test allows for the comparison of two candidate models. As its name suggests, it is based on the evaluation of the likelihood function of different models.

The LR test has been used multiple times (Tramblay et al., 2013; Cheng et al., 2014; Yilmaz et al., 2014) to select between stationary and nonstationary models in the context of nested models. Given a stationary model characterized by a parameter set θst and a nonstationary model, with parameter set θns, if l(θ^st) and l(θ^ns) are their respective maximized log likelihoods, the likelihood ratio test can be defined using the deviance statistic 3D=2lθ^ns-lθ^st. D is (for large L) approximately χm2 distributed, with m=dim(θns)-dim(θst) degrees of freedom. The null hypothesis of stationarity is rejected if D>Cα, where Cα is the (1-α) quantile of the χm2 distribution (Coles, 2001).

Besides the analysis of power, we also checked (in Sect. 3.3) the approximation D∼χm2 as a function of the sample size L for the evaluation of the level of significance.

Information criteria are useful tools for model selection. It is reasonable to retain that the Akaike information criterion (AIC; Akaike, 1974) is the most famous among these tools. Based on the Kullback–Leibler discrepancy measure, if θ is the parameter set of a k-dimensional model (k=dim(θ)), AIC is defined as 4AIC=-2l(θ^)+2k. The model that best fits the data has the lowest value of the AIC between candidates. It is useful to observe that the term proportional to the number of model parameters allows one to account for the increase of the estimator variance as the number of model parameters increases.

Sugiura (1978) observed that the AIC can lead to misleading results for small samples; thus, he proposed a new measure for the AIC: 5AICc=-2l(θ^)+2k(k+1)L-k-1, where a second-order bias correction is introduced. Burnham and Anderson (2004) suggested only using this version when L/kmax<40, with kmax being the maximum number of parameters between the models compared. However, for larger L, AICc converges to AIC. For a quantitative comparison between the AIC and AICc in the extreme value stationary model selection framework, the reader is referred to Laio et al. (2009).

In order to select between stationary and nonstationary candidate models, we use the ratio 6AICR=AICnsAICst, where the subscripts indicate the AIC value obtained for a stationary (st) and a nonstationary (ns) model, both fitted with maximum likelihood to the same data series.

Considering that the better fitting model has a lower AIC, if the time series arises from a nonstationary process, the AICR should be less than 1; the opposite is true if the process is stationary.

In order to provide a rigorous comparison between the use of the MK, LR, and AICR, we evaluated the AICR,α threshold value corresponding to the significance level α using numerical experiments.

More in detail, we adopted the following procedure:

N=10000 samples are generated from a stationary GEV parent distribution, with known parameters;

The cumulative distribution function of the generalized extreme value (GEV) distribution (Jenkinson, 1955) can be expressed as follows: 7Fz,θst=exp⁡-1+εz-ζσ-1/εε≠0exp⁡-exp⁡-z-ζσε=0σ>0, where ζ, σ, and ε are known as the position, scale, and shape parameters, respectively; θst=[ζ, σ, ε] is a general and comprehensive way to express the parameter set in the stationary case. The flexibility of the GEV, which accounts for the Gumbel, Fréchet, and Weibull distributions as special cases (for ε=0, ε>0 and ε<0 respectively) makes it eligible for a more general discussion about the implications of nonstationarity.

Traditional extreme value distributions can be used in a nonstationary framework, modeling their parameters as function of time or other covariates (Coles, 2001), producing θst⟶θns=[ζt, σt, εt].

In this study, only a deterministic linear dependence on the time t of the position parameter ζ has been introduced, leading Eq. (7) to be expressed as follows: 8Fz,θns=exp⁡-1+εz-ζtσ-1/εε≠0exp⁡-exp⁡-z-ζtσε=0σ>0 with 9ζt=ζ0+ζ1t and θns=[ζ0, ζ1, σ, ε].

It is important to note that Eq. (8) is a more general way of defining the GEV and has the property of degenerating into Eq. (7) for ζ1=0; in other words, Eq. (7) represents a nested model of Eq. (8) which would confirm the suitability of the likelihood ratio test for model selection.

According to Muraleedharan et al. (2010), the first three moments of the GEV distribution are as follows: 10mean=ζ+σεg1-1ε≠0,ε<1,11variance=σ2ε2g2-g12ε≠0,ε<12,and12skewness=sgn(ε)⋅g3-3g2g1+2g13g2-g123/2ε≠0,ε<13. Here, gk=Γ(1-kε), with k∈Z+ and Γ(⋅), is the gamma function. It is worth noting that, following Eqs. (10)–(12), the trend in the position parameter only affects the mean, while the variance and skewness remain constant.

In this work, we used the maximum likelihood method (ML) to estimate the GEV parameters from sample data. The ML allows one to treat ζ1 as an independent parameter, as well as ζ0, σ and ε. For this purpose, we exploited the “extRemes” R package (Gilleland and Katz, 2016).

The power of a test is related to the type II error and is the probability of correctly rejecting the null hypothesis when it is false. In particular, defining α (level of significance), the probability of a type I error, and β, the probability of a type II error, we have a power value of 1-β. The maximum value of power is 1, which correspond to β=0, i.e., no probability of a type II error. In most applications, the conventional values are α=0.05 and β=0.2, meaning that a 1-to-4 trade-off between α and β is accepted. Thus, in our experiment we always assumed a significance level of 0.05, and, for the following description of results and discussion, we considered a power level of less than 0.8 to be too low and, hence, unacceptable. In Sect. 4, we report further considerations regarding this choice. For each of the tests described in Sect. 2.1, 2.2, and 2.3, the power was numerically evaluated according to the following procedure:

N=2000 Monte Carlo synthetic series, each of length L, are generated using the nonstationary GEV in Eqs. (8) and (9) as a parent distribution with a fixed parameter set θns=[ζ0, ζ1, σ, ε] with ζ1≠0.

We used a reduced number of generations (N=2000) for the evaluation of power as a good compromise between the quality of the results and computational time. N=2000 was also used by Yue et al. (2002a).

Considering the nonstationary GEV four-parameter model, in order to satisfy the relation L/kmax<40 suggested by Burnham and Anderson (2004), a time series with a record length no less than 160 should be available. Following this simple reasoning, the AIC should be considered not to be applicable to any annual maximum series showing a changing point between the 1970s and 1980s (e.g., Kiely, 1999). In our numerical experiment, the second-order bias correction of Sugiura (1978) should always be used, as we have L/kmax=70/4=17.5 for the nonstationary GEV for a maximum sample length of L=70. Nevertheless, we checked if using the AIC or AICc may affect results. For this purpose, we evaluated the percentage differences between the power of the AICR obtained by means of the AIC and AICc from synthetic series. In Fig. 2, the empirical probability density functions of such percentage differences, grouped according to sample length, are plotted for generations with ε=0.4 and different values of σ. It is interesting to note that the error distribution only shows a regular and unbiased bell-shaped distribution for L=70. We then observe a small negative bias (about -0.02 %) for L=50, while a bias of -0.08 with a multi-peak and negatively skewed pdf is noted for L=30. The latter pdf also has a higher variance than the others. The purpose of this figure is to show that the difference between the power obtained with the AIC and the power obtained with the AICc is negligible. Different peaks in one curve (L=30) can be explained by the merging of sample errors obtained for different values of σ. Similar results were obtained for all values of ε, which always provided very low differences and allow for the conclusion to be reached that the use of the AIC or AICc does not significantly affect the power of AICR for the cases examined. This follows the combined effect of the sample size (whose minimum value considered here is 30) and the limited difference in the number of parameters in the selected models. In the following, we will refer to and show only the plots obtained for the AICR in Eq. (6) with the AIC evaluated as in Eq. (4).

The effect of the parent distribution parameters and the sample size on the numerical evaluation of the power and significance level of the MK, LR, and AICR tests for different values of ε, σ, and ζ1 is shown in Fig. 3. The curves represent both the significance level, which is shown for ζ1=0 (true parent is the stationary GEV), and the power, which is shown for all other values ζ1≠0 (true parent is the nonstationary GEV). Each panel in Fig. 3 shows the dependence of the power and significance level of MK, LR and AICR on the trend coefficient for one set of parameter values and different sample sizes. In all panels, the test power strongly depends on the trend coefficient and sample size. This dependence is also affected by the parent parameter values. In all cases, the power reaches 1 for a strong trend and approaches 0.05 (the chosen level of significance) for a weak trend (ζ1 close to 0). In all combinations of the shape and scale parameters (and especially for short samples) for a wide range of trend values, the power exhibits values well below the conventional value of 0.8. The curves' slope between 0.05 and 1 is sharp for long samples and gentle for short samples. It also depends on the parameter set, with slopes generally being gentler for higher values of the scale (σ) and shape (ε) parameters of the parent distribution. A significant difference in the power between the MK, LR, and AICR tests is observable when the sample size is smaller and even more so when the parent distribution is heavy-tailed (ε=+0.4).

In particular, for ε=0, -0.4 and L=50, 70, it is possible to report a slightly larger power of LR with respect to the AICR and MK, but values are very close to each other. However, the reciprocal position of MK and AICR power curves is interesting; in fact, the AICR power is always larger than that of the MK, except when ε=-0.4, for all values of the scale parameter.

A higher difference is found for a heavy-tailed parent distribution (ε=+0.4). While LR still has the largest power value, the difference with respect to AICR remains small and the MK power value almost always collapses to values smaller than 0.5.

The practical consequences of such patterns are very important and are discussed in Sect. 4.

We evaluated the threshold values (corresponding to a significance level of 0.05) for accepting/rejecting the null hypothesis of stationarity according to the methodologies recalled in Sect. 2.1 and 2.2 for the MK and LR tests and introduced in Sect. 2.3 for AICR. Based on these thresholds, we exploited the generation of series from a stationary model (ζ1=0) in order to numerically evaluate the rate of rejection of the null hypothesis, i.e., the actual significance level of the tests considered in the numerical experiment, following the procedure described in Sect. 2.5.

Table 1 shows the numerical values of the actual level of significance, obtained numerically, to be compared with the theoretical value of 0.05 for all of the sets of parameters and sample sizes considered. Among the three measures for trend detection, the LR shows the worst performance. The results in Table 1 show that the rejection rate of the (true) null hypothesis is systematically higher than it should be, and it is also dependent on parent parameter values. This effect is exalted when the parent distribution has an upper boundary (ε=-0.4) and for shorter series (L=30). In practice, this implies that when using the LR test, as described in Sect. 2.2, there is a higher probability of rejecting the null hypothesis of stationarity (if it is true) than expected or designed.

Conversely, the performance of MK with respect to the designed level of significance is less biased and is independent of the parameter set. Similar good performance is trivially obtained for the AICR, whose rejection threshold is numerically evaluated.

The plot in Fig. 4 is displayed in order to focus on the actual value of the level of significance and, in particular, on the LR approximation D∼χm2 as a function of the sample length L. The difference between the theoretical and numerical values of the significance level is represented by the distance between the bottom value of the curve (obtained for ζ1=0, i.e., the stationary GEV model) and the chosen level of significance 0.05, represented by the horizontal dotted line. In particular, in Fig. 4, results for the parameter set (σ=15, ε=-0.4) show that the actual rate of rejection is always higher than the theoretical one and changes significantly with the sample size; this means that the χm2 approximation leads to a significant underestimation of the rejection threshold of the D statistic. Moreover, it seems that the LR power curves (in red) are shifted toward higher values as a consequence of the significance level overestimation, meaning that the LR test power is also overestimated due to the approximation D∼χm2. These results suggest the use of a numerical procedure for the LR test (such as that introduced for AICR in Sect. 2.3) for evaluating the D distribution and the rejection threshold.

Other considerations can be made regarding the use of AICR. As explained in Sect. 2.3, we empirically evaluated the AICR,α threshold value using numerical generations with a significance level 0.05 for each of the parameter sets and sample sizes considered. Similar results were obtained using the AICc, which are not shown for brevity. We found a significant dependence of AICR,α on the sample size. Figure 5 shows the AICR,α curves obtained for each of the parameter sets vs. sample size. It is also worth noting that all curves asymptotically trend to 1 as L increases. This property is due to the structure of the AIC and the peculiarity of the nested models used in this paper: while using a sample generated with weak nonstationarity (i.e., when ζ1→0 in Eq. 9), the maximum likelihood of the model shown in Eq. 7, l(θ^st), tends toward l(θ^ns) of the model shown in Eq. 8, leaving only the bias correction (2k in equation 4) to discriminate between competing AIC values in model selection applications. As a consequence, the AICR,α should always be lower than 1; however, when increasing the sample size, both the likelihood terms -2l(θ^st) and -2l(θ^ns) in Eq. (4) will also increase, pushing AICR toward the limit 1. Conversely, Fig. 5 shows that the threshold value AICR,α is significantly smaller than 1 up to L values well beyond the length usually available in this kind of analysis. Hence, the numerical evaluation of the threshold has to be considered as a required task in order to provide an assigned significance level to model selection. In contrast, the simple adoption of the selection criteria AICR<1 (i.e., AICR,α=1) would correspond to an unknown significance level that is dependent on the parent distribution and sample size. In order to highlight this point, we evaluated the significance level α corresponding to AICR,α=1, following the procedure described in Sect. 2.5, by generating N=10000 synthetic series (from a stationary model) for any parameter set and sample length. The results, provided in Table 2, show that α ranges between 0.16 and 0.26 in the explored GEV parameter domain and mainly depends on the sample length and the shape parameter of the parent distribution.

In our opinion, the results shown above, with respect to the performance of parametric and nonparametric tests, are quite surprising and important. It is proved that the preference widely accorded to nonparametric tests, due to the fact that their statistics are allegedly independent from the parent distribution, is not well founded. Conversely, the use of parametric procedures raises the problem of correctly estimating the parent distribution and, for the purpose of this paper, its parameters. Moreover, as the trend coefficient ζ1 is a parameter of the parent distribution under nonstationary conditions, the proposed parametric approach provides a maximum likelihood-based estimation of the same trend coefficient, which is hereafter referred to as ML-ζ1. For a comparison with nonparametric approaches, we also evaluated the sample variability of the Sen's slope measure (δ) of the imposed linear trend. Furthermore, in order to provide insights into these issues, we analyzed the sample variability of the maximum likelihood estimates ML-ε and ML-σ (from the same sets of generations exploited above) for different parameter sets and sample lengths.

We evaluated sample variability s[⋅], as the standard deviation of the ML estimates of parameter values obtained from synthetic series. In the upper panels of Fig. 6, we show s[ML-ζ1], and in the lower panels, we show the Sen's slope median s[δ]. In both cases, the sample variability of the linear trend is strongly dependent on sample size and independent from the true ζ1 value in the range examined [-1, 1]. It reaches high values for short samples and, in such cases, its dependence on the scale and shape parent parameters is also relevant. The ML estimation of the trend coefficient is always more efficient than Sen's slope, and this is observed for heavy-tailed distributions in particular.

In Fig. 7, we show the empirical distributions of the Sen's slope δ and ML-ζ1 estimates obtained from samples of size L=30 with a parent distribution characterized by σ=15 and ε=[-0.4, 0, 0.4], providing visual information about the range of trend values that may result from a local evaluation. Similar results, characterized by smaller sample variability, as shown in Fig. 6, are obtained for L=50 and L=70 and are not shown for brevity.

Figure 8 shows the sample variability of ML-ε and ML-σ, which is still independent of the true ζ1 for values of ε=0 and 0.4, whereas for the upper-bounded GEV distributions (ε=-0.4) it shows a significant increase for higher values of σ and high trend coefficients (|ζ1|>0.5). The randomness of results for L=30 and σ=[15, 20] is probably due to the reduced efficiency of the algorithm that maximizes the log⁡-likelihood function for heavy-tailed distributions.

In order to better analyze such patterns, for the scale and shape parent parameters we also report the distribution of their empirical ML estimates for different parameter sets vs. the true ζ1 value used in generation. The sample distribution of ML-ε for σ=15 is shown in Fig. 9 for L=30 and L=70. The sample distribution of ML-σ for σ=15 is shown in Fig. 10 for L=30 and L=70. The panels show that the presence of a strong trend coefficient may produce significant loss in the estimator efficiency, which is probably due to deviation from the normal distribution of the sample estimates for long samples. This suggests the need for more robust estimation procedures that provide higher efficiency for estimates of ε and σ in the case of a strong observed trend. It should be highlighted that efficiency in the parameter estimation increases with sample size for ε=[0, 0.4], whereas it decreases for both ε and σ in the case of ε=-0.4, where the trend of the location parameter implies a shift in time of the distribution upper bound.