In its 2009 manual, the WMO details several PMP estimation approaches, with hydrometeorological methods combining algorithms of storm selection, transposition, and maximization being the most popular in Canada. In regions where snow cover is important enough for floods to result from a combination of the PMP and snowmelt, the WMO typically recommends estimating both spring and summer–fall PMPs. Regarding storm selection, some authors use all rain events where the precipitation height exceeds a given threshold , while others utilize all observed precipitation data over a given period . This selection process is usually carried out by meteorologists and depends on physical factors . Since the number of selected storms is small and varies from one calculation to another and among different meteorologists, this selection process introduces significant variability into the estimation of the PMP.

To increase the number of storms used in PMP estimation, a common practice is to include storms from neighbouring areas that are likely to also affect the region of interest. Over the past decades, meteorologists have developed various techniques considering the orography and other features of the areas to realistically transpose storms . This storm transposition can be incorporated into the storm selection process of PMP estimation methods. While it increases the sample size for PMP estimation, it also introduces additional sources of variability and subjectivity.

The moisture maximization approach estimates the PMP using the relationship between the amount of precipitation and the humidity of the air. Let Yi be the precipitation of storm 1≤i≤n among the n selected storms. The PMP estimation is based on moisture maximization as follows: 1PMP=max⁡i∈{1,…,n}Yi×PWmax⁡PWi, where PWi corresponds to the precipitable water of storm i, and PWmax⁡ corresponds to the maximum precipitable water. The quantity Yi×PWmax⁡PWi is often referred to as the maximized precipitation of event i if the maximal precipitable water is available at the moment of the storm. The PMP then corresponds to the maximum of the maximized precipitation. The ratio PWmax⁡PWi is referred to as the maximization ratio and is sometimes arbitrarily set to a numerical value between 1.5 and 2.5 to avoid the overestimation of the PMP . The use of this threshold is subjective and lacks physical or mathematical justification .

A slightly different interpretation can be given for the moisture maximization equation expressed in Eq. () by rearranging the terms: the ratio YiPWi corresponds to the ratio of precipitation to precipitable water and is referred to as the precipitation efficiency of storm i. The PMP occurs when the maximum precipitation efficiency coincides with the maximum precipitable water. utilize this definition to model the dependence between extreme values of precipitation efficiency and precipitable water. They demonstrate that the comonotonicity imposed by this interpretation leads to overestimation of the PMP in North America.

In practice, the precipitable water PWi at the moment of storm i and the maximum amount of precipitation for the considered region PWmax⁡ are unknown and must be estimated in order to use the moisture maximization approach. The amount of precipitable water can be estimated using the specific humidity of the air column above the area . However, for the majority of meteorological stations, specific humidity is neither observed nor recorded. The recommended estimation of precipitable water by the uses the dew point, which is usually recorded and requires pseudo-adiabatic conditions . observed that the relation between surface dew point and precipitable water is greater when the latitude is over 25° than in lower-latitude zones. A study conducted by in the Chicago region indicates that the hypothesis of pseudo-adiabatic conditions could lead to overestimation of precipitable water, and noted that PMP estimates vary greatly depending on how the precipitable water was approximated. The uncertainty of these estimations is often neglected in PMP estimation. It is not uncommon for the uncertainty of PWi to lead to a precipitation efficiency larger than 1, which is physically impossible.

The Hershfield empirical method is an alternative to moisture maximization for PMP estimation. The method relies on a series of m precipitation annual maxima. The PMP estimate is as follows: 2PMP=x‾+Ks, where x‾ and s correspond, respectively, to the mean and the standard deviation of the series of annual maxima, and K corresponds to the frequency factor for estimating PMP at that location. proposed a method to estimate this factor. This approach is widely employed for its simplicity and ease of use but can only estimate PMP over smaller watersheds . It also has the advantage of not requiring additional hydrometeorological data such as specific humidity or dew point. Only precipitation series from which annual maxima are extracted are required.

This method is often classified as a statistical technique, but, in this paper, it is considered to be empirical due to the nature of the link between the PMP and the sample moments. It should be noted that the PMP durations considered by and available in are all less than or equal to 24 h, which is inadequate for calculating longer-duration PMP.

The relevance of the Hershfield procedure can also be questioned. Equation () defines the PMP as an extreme quantile of the distribution, estimated using only the mean and standard deviation. This approach relies on the bulk of the distribution to extrapolate into the tail, which is inherently hazardous. Extreme-value theory EVT; see, e.g., is a branch of statistics that focuses on extreme values. It provides asymptotic parametric distributions (the generalized extreme-value and the generalized Pareto distributions) and rigorous frameworks (block maxima and peaks over threshold) for extrapolating beyond the range of observations.

As a statistical approach, it is easier to incorporate non-stationarity induced by climate change and to provide uncertainty in the estimates. However, extreme-value analysis suggests that the precipitation distribution is unbounded, which is inconsistent with the PMP definition. To reconcile the PMP definition with extreme-value theory, some authors propose using a very large return period as the PMP estimate. For example, shows that PMP estimates obtained through the Hershfield method correspond, on average, to return periods of 60 000 years when estimated using EVT. The also suggest using a quantile of an extreme-value distribution corresponding to an “extremely low annual probability of being exceeded”.

The methods for estimating PMP presented by the rely solely on precipitation data observed at hydrometeorological stations. However, the scarcity of precipitation observations and the lack of direct measurements of precipitable water have driven the development of PMP estimation methodologies based on climate simulations from regional climate models in Canada , North America , and other parts of the world .

The use of these climate simulations not only allows for the consideration of a greater number of extreme rainfall events but also enables the estimation of future PMP. Indeed, the defines the PMP as stationary values, and climate change (CC) is not taken into account in the calculations. However, it is widely acknowledged that CC has a direct impact on extreme precipitation events and should therefore be considered in their estimation. Using projected climate simulations, demonstrate a global increase in water vapour concentration in the atmosphere without a sufficient evolution in values of upward vertical motion or horizontal wind speed, factors that could counterbalance the rise in air humidity. This increase implies larger future values of PWmax⁡ and, consequently, an increase in PMP. Several papers conclude that PMP will generally increase under future climate .

PMP estimations using the non-statistical approaches described in the previous sections are highly sensitive to arbitrary choices and are generally provided without accounting for uncertainty. Statistical approaches, on the other hand, provide uncertainty quantification and facilitate the inclusion of non-stationarity if needed. However, reconciling the PMP definition as the upper bound of an unbounded distribution remains particularly challenging. The objective of this paper is to develop a statistical model for PMP estimation based on the WMO definition, which assumes an upper bound. As a statistical approach, it offers two key advantages: (1) uncertainty is quantifiable, and (2) most subjective choices are eliminated, enhancing the reproducibility of the estimation.

Starting with the principles underlying moisture maximization , we develop a sensible statistical model that assumes an upper bound for precipitation. From Eq. (), let Ỹi denote the maximized daily precipitation on day i: 3Ỹi=YiPWi×PWmax⁡. Factoring for the actual precipitation of day i gives the following expression: 4Yi=PWiPWmax⁡×Ỹi, where PWmax⁡ is the maximum of precipitable water. Hence, 0<PWiPWmax⁡≤1. Since the PMP corresponds to the maximum of the maximized precipitation Ỹi, the maximized precipitation can be viewed as a fraction of the PMP; i.e., Ỹi=riPMP, where 0<ri≤1. It is then possible to express the actual daily precipitation as follows: 5Yi=PWiPWmax⁡×riPMP.

The ratio PWiPWmax⁡×ri lies in (0,1] since each of the multiplicative factors is within (0,1]. This ratio can naturally be modelled using the Beta distribution, a flexible distribution for a random variable taking values in the unit interval. Actual precipitation Yi, which corresponds to this ratio multiplied by the PMP, can be modelled using the Beta distribution rescaled to the interval (0,PMP). The Beta distribution on the interval (a,b) for a<b is referred to as the Pearson Type-I distribution Chap. 24.

Therefore, we propose modelling the actual precipitation of day i with the Pearson Type-I distribution as follows: 6Yi∼PearsonType1(0,ψ,α,β), where the Beta parameters α>0 and β>0 govern the ratio PWiPWmax⁡×ri and where ψ>0 corresponds to the PMP. The lower bound is set at 0 because only non-zero precipitation events are considered. When 0<α<1 and β≥1, the Pearson Type-I density is monotonically decreasing and convex, resembling the precipitation histogram shown in Fig. . With this proposed statistical model, the PMP constitutes a distribution parameter to be estimated with the data. Uncertainty can then be provided using the usual statistical methods, as described in the next section.

Three methods are considered for estimating the parameters of the proposed model: the method of moments, maximum likelihood estimation, and the Bayesian method.

When 0<α<1 and β≥1, i.e., when the density is convex, a non-identifiability issue occurs between β and ψ. Indeed, these parameters can compensate for each other. For example, let Z1∼PearsonType110,110,9910 and Z2∼PearsonType1100,110,99910 be two random variables with very different upper bounds – 10 for Z1 and 100 for Z2 – but whose moments are similar, as shown in Table . Both variables have the same mean and approximately the same variance. Although there are slight differences in skewness and kurtosis, these differences are not large enough to overcome the sampling uncertainty of these higher-order-moment estimates.

This non-identifiability issue is even more critical for parameter estimation using the model likelihood (both maximum likelihood and Bayesian methods). For example, consider the variable Z1 again and generate a large random sample of size 5000. The log-likelihood of the model evaluated at the true parameter vector 10,110,9910 is 35 678.3. The log-likelihood evaluated at another parameter vector 100,110,99910 is 35 678.0, which is practically the same, even though the parameters are quite different. The impact of this non-identifiability issue is assessed for parameter estimation with the method of moments, maximum likelihood, and Bayesian method with a simulation study provided in the following section.

For each of the 100 random samples, with sizes ranging from 100 to 8000, parameters were estimated using the method of moments, maximum likelihood, and Bayesian method. Figure  displays the mean of the 100 parameter estimates for the upper bound ψ as a function of the sample size, as well as the 95 % empirical confidence interval. For the Bayesian method, both a Gibbs sampling scheme and the No-U-Turn Sampler (NUTS) algorithm were implemented, yielding similar results.

The three estimation procedures perform very well in estimating the upper bound, which is the parameter of interest in this paper. The mean estimate hovers around the true value of 50, and the confidence intervals include the true value. Estimation remains accurate even for relatively small sample sizes of 2000, which corresponds to approximately 20 years of precipitation data. However, the methods based on likelihood yield more precise results than the method of moments.

Figure  shows the mean and the 95 % empirical confidence intervals for the samples generated from the Pearson Type-I distribution with a convex density. For the method of moments, the estimation of the upper bound is close to the true value of 50. The confidence intervals, wider compared to those associated with the concave density, include the true value. However, estimation variability is very large. It is very sensitive to the sample. For example, for moderate sample sizes around 4000, the upper-bound estimate average is around 50, but for some samples, the estimate exceeds 100, which is 2 times larger than the true value. For other samples, the estimate is smaller than 25, which is half the true value.

Upper-bound estimates using the maximum likelihood and Bayesian methods are not useful, as shown in Fig. . The non-identifiability issue arises because the shape parameter β compensates for the larger upper bound. While an informative prior for the upper bound could be introduced to control this issue, it would need to be highly informative. However, this approach was not pursued because using such a restrictive prior defeats the purpose of removing subjectivity in PMP estimation.

The sensitivity to the sample and the non-identifiability issue are resolved with very large sample sizes, as shown in Fig. . The estimates are well stabilized around a sample size of 40 000. For precipitation in Canada, this corresponds to approximately 400 years of data.

For the Pearson Type-I distribution with a concave density, parameter estimates are precise with all three estimation methods considered. For a convex density, the non-identifiability issue in the likelihood is too severe to obtain usable upper-bound estimates for common sample sizes using the maximum likelihood and Bayesian methods. However, these two methods could be used with very large sample sizes.

Although the method of moments is sensitive to the data, it provides realistic estimates of the upper bound, even for common sample sizes. This method should be favoured for parameter estimation of non-concave Pearson Type-I distributions.

The proposed method for estimating the PMP is demonstrated using data from two meteorological stations in Quebec (QC), Canada, located 26 km apart: the Montréal Pierre-Elliott-Trudeau International Airport station (1953–2024) and the St. Hubert Airport station (1949–2024). The data are available from the Environment and Climate Change Canada (ECCC) website. Daily precipitation (in mm) and dew point (in degrees Celsius) were extracted from 1 May to 31 October each year to focus on liquid rainfall and to minimize the effect of seasonality. While it may still be present, it appears to be negligible compared to the natural variability of precipitation and precipitable water, as illustrated in Fig.  for the Montréal data. Descriptive statistics of recorded precipitation at these two stations are provided in Table . Figure  shows the histogram of non-zero daily rainfall for each station. Typically, for precipitation at the considered locations, autocorrelation exists in daily non-zero series but is very weak (0.0092 for Montréal and 0.0095 for St. Hubert) and short range.

As a point of comparison, summer–fall PMP estimates for both stations are calculated using the moisture maximization method, Hershfield's approach, and the 60 000-year return level estimated with EVT. For the moisture maximization method, daily precipitation amounts from the top 10 % of storms for each year are selected, as proposed by . A sensitivity analysis was performed on the storm selection percentage (10 %, 1 %, or 0.1 %), but the PMP estimates remained unchanged because the maximized events were the same for both locations. Since precipitable water was not directly observed, it was estimated using the dew point over 12 h, as described by , which may have affected the quality of the PMP estimates. The corresponding PMP estimates for both stations are provided in Table . The PMP estimates are 282 and 436 mm for Montréal and St. Hubert, respectively, corresponding to maximization ratios of 4.4 and 4.9. Some authors suggest limiting this ratio to a value between 1.5 and 2.5 to constrain PMP estimation. Setting the maximization ratio to 2.0 would yield PMP estimates of 128 and 178 mm for Montreal and St. Hubert. While this would improve the moisture maximization results, it merely conceals the methodology's flaws, particularly its high variability, rather than addressing them.

Note that, for this approach, it is also possible to estimate the PMP using the 100-year return level of precipitable water instead of the sample maxima e.g.,, but with our data, the estimated PMP values were similar: 284 mm and 427 mm for Montréal and St. Hubert, respectively.

For Hershfield's approach, the frequency factor of K=15 is employed as proposed by . The adjustment based on the number of data points, as suggested by , is unnecessary. Hershfield's PMP estimates are 261 mm for Montréal and 322 mm for St. Hubert and are also provided in Table .

The 60 000-year return level was estimated using the peaks-over-threshold model see, e.g.,, which adjusts the excesses over a high threshold to a generalized Pareto distribution (GPD). The threshold of 30 mm for both Montréal and St. Hubert is selected using the mean residual life plot as described by Chap. 4. The estimated return levels are 185 mm (114,363) and 200 mm (120,385) for the Montréal and St. Hubert data, respectively, where the values in parentheses represent the 95 % confidence intervals for the 60 000-year return level estimates.

The proposed statistical approach to estimate the PMP translates the usual definition of the PMP into a statistical distribution for the recorded precipitation. The PMP constitutes one of the three parameters, and the remaining two concern the shape of the distribution. By estimating the parameters using standard statistical approaches, such as the moment, maximum likelihood, and Bayesian methods, it is possible to adequately describe the uncertainty, particularly for the PMP parameter. Additionally, the proposed approach uses all of the precipitation recorded at the station rather than only a subset from the stations and neighbouring stations. This reduces the subjectivity present in standard approaches.

In the simulation study, it is shown that the non-identifiability issue vanished with very large sample sizes. Figure  shows that the maximum likelihood estimation is stable with a sample size of 45 000. If we consider that 100 storms occur during a year, such a sample size would correspond to 450 years of observation. Of course, no meteorological record is that long, but it could be possible to have such a sample size of synthetic storms generated with a storm generator. However, such data augmentation should be carefully implemented to avoid overconfidence. For instance, if 40 000 daily precipitation data points generated from a weather generator are used to estimate the model, do these 40 000 data points contain 400 times more information than an actual recorded series of size 100? At this point, this is beyond the scope of the present paper, but it could be an interesting avenue for future investigation.

Another alternative to increase the sample size would be to include information from nearby stations. This could be achieved within the Bayesian framework described in Sect.  by replacing the parameter prior distribution with a spatial prior. However, the dependence between stations would need to be modelled in the likelihood as a single storm can generate precipitation across multiple stations. Accounting for this dependence would decrease the effective sample size of the pooled stations, and we believe that this effective sample size might not reach the level where the estimates are stable. However, we could be wrong.

PMP estimation, whether with the method proposed in this paper or with more standard approaches, is very sensitive to the data due to non-identifiability. For the two nearby meteorological stations considered, i.e., the Montréal Pierre-Elliott-Trudeau International Airport and the St. Hubert Airport stations, located 26 km apart, the PMP estimates are very different. However, these two stations do not experience significantly different climates and storms. Furthermore, in several estimates provided by engineering consulting firms, storms from even more distant stations are combined to estimate the PMP, a practice known as storm transposition. Moreover, the extreme-value analysis of the precipitation at these two stations, presented later in Sect. , yields consistent return level estimates. Therefore, the difference in the PMP for these two stations is more a numerical problem related to the PMP definition than a genuine difference in the PMP.

Another drawback is fitting the proposed short-tailed statistical model to heavy-tailed precipitation data, as shown in the following section. This inconsistency could explain why the model does not fit the St. Hubert data well, as shown in Fig. .

For the purpose of comparison, an extreme-value analysis has been performed on the precipitation data of Montréal and St. Hubert. As mentioned in Sect. 4.2, the peaks-over-threshold extreme-value model has been fitted by maximum likelihood to the Montréal data, with a 30 mm threshold. The estimated parameters of the generalized Pareto distribution modelling the excesses above the threshold can be found in Table . The model fits the data very well, as shown by the return level plot in Fig. . Note that the shape parameter estimate is positive, indicating an unbounded heavy-tailed distribution, which is typical for precipitation but inconsistent with the PMP existence assumption. Nevertheless, a short-tailed distribution cannot be excluded as the shape parameter confidence interval includes negative values. As an indication, the 60 000-year return level estimated with the POT model is 185 mm (114,363), and the PMP value of 270 mm corresponds to a return period longer than 10×106 years.

The POT model has also been fitted to the St. Hubert data, and parameter estimates are shown in Table , and Fig.  shows the model fit to the data. Again, the model fits the data very well, and the shape estimate is positive, indicating a heavy-tailed distribution. Using the fitted POT model, the 60 000-year return level estimate is 200 mm (120,385), which is consistent with the corresponding estimate of 185 mm for Montréal, located 26 km away. The PMP estimate of 416 mm corresponds to a return period longer than 1×109 years.

To reduce the impact of model non-identifiability, we also developed a new parameterization for the Pearson Type-I distribution, replacing the shape parameters with a location parameter μ>0 and a concentration parameter ν>0: μ=αψα+β, ν=α+β. We therefore have α=μνψ and β=νψ(ψ-μ). However, even with this parameterization, non-identifiability remained an issue for maximum likelihood and Bayesian inference.

Another approach to imposing a short-tailed model compliant with the PMP concept would be to consider the reverse Weibull distribution. The reverse Weibull is obtained by imposing a negative shape parameter on an extreme-value distribution. While this choice results in a short-tailed distribution, it is difficult to justify this constraint beyond the fact that it produces an upper bound. Moreover, we are concerned that using an extreme-value distribution in an inappropriate context, such as by imposing a negative shape parameter when the data suggest a positive one, could give practitioners a false sense of security. They might believe they are operating within the extreme-value framework when they are not.

The major drawback of the proposed approach lies in the issue of non-identifiability when the data distribution is convex, as is the case for precipitation. Maximum likelihood and Bayesian methods become highly unstable, and, although the method of moments is less affected, it is still impacted. Regularized maximum likelihood or informative priors could be employed to address non-identifiability. For example, a penalizing term corresponding to the log-density of an exponential distribution with parameter λ>0 can be introduced into the log-likelihood as a regularization term. This is equivalent to considering an exponential distribution with parameter λ as an informative prior for the PMP. When λ→0, the penalty vanishes and the regularized estimates converge to the maximum likelihood estimates. As λ increases, the penalty becomes more influential in the regularized likelihood. Choosing the penalty value λ involves a trade-off between bias and variance: λ should be small enough to minimize the added bias but large enough to help control non-identifiability.

For the Montréal data, the PMP estimate from the maximum likelihood exceeds 2×1013 mm, an absurd value explained by the non-identifiability issue, as demonstrated in the simulation study. Figure  shows the regularized maximum likelihood PMP estimate as a function of the penalty λ. If λ^=0.09 is selected using the subjective elbow method, the resulting PMP estimate is 272 mm. This value is more reasonable, but it is extremely sensitive to the choice of λ^. We felt that this introduced too much subjectivity into the proposed approach and diminished its advantages over standard PMP estimation methods.

For the observed data considered, there is no evidence of a trend in either the precipitable water or precipitation, as shown in Fig. . Non-stationarity might be present in long series of simulated data from a climate model. In such cases, the proposed statistical model could easily be extended to account for non-stationarity and also for seasonality if needed. Equation () could be generalized to incorporate seasonality and non-stationarity by allowing either or both the PMP and the shape parameters to evolve over time. For example, precipitation Ytsi of year t, season s, and event i can be modelled as a function of the year and the event i as follows: Ytsi∼PearsonType1(0,ψts,αts,βts), where the year t and season s could serve as covariates.

Although translating the definition of the PMP into a statistical model is interesting and despite the possibility of including non-stationarity and estimating uncertainty, we do not recommend using the Pearson Type-I distribution to estimate the PMP. The non-identifiability makes the model too sensitive to the data, and the PMP estimate becomes too volatile. This problem is also present in the standard moisture maximization method. Therefore, we align with the conclusions of the , which recommends an extreme-value analysis instead. Additionally, the extreme-value theory allows for a genuine estimation of the uncertainty of extreme values, even when extrapolating to return periods that exceed the range of the data. Furthermore, it is easily generalizable to non-stationary cases, allowing the integration of the effects of climate change.

It should be noted that the recommendations of the address the North American context, but they stem from the broader observation that the definition and assumptions underlying PMP are outdated. This criticism is globally relevant as the definition and assumptions of PMP are consistent worldwide. Therefore, we believe that the recommendation can be considered to be general and not limited to the North American context.

More generally, the fact that PMP estimates using either moisture maximization, Hershfield's method, or the Pearson Type-I method are so sensitive to the data is a critical concern from an engineering standpoint. Specifically, for the Pearson Type-I method with a convex density, depending on the data, the PMP estimate can range from half to more than twice the true value. In the former case, using the estimate would result in under-dimensioning the infrastructure, putting the public at risk. In the latter case, using it would result in over-dimensioning the infrastructure, thereby increasing costs and environmental impacts. Although uncertainty estimates are not available with the moisture maximization and Hershfield's methods, the fact that the corresponding PMP estimates for Montréal and St. Hubert were so different is an important indication of the methods' sensitivity.

In the case of this article, we have seen that the POT model fits the data from both stations very well and that the estimates of the 60 000-year precipitation were consistent. Moreover, the extreme-value analysis indicates an unbounded and heavy-tailed distribution of precipitation, which is consistent with numerous results in the literature e.g.,. Therefore, it is better to design infrastructure by setting an appropriate level of risk and evaluating the uncertainty of the estimate.

In our opinion, EVT-based PMP estimates are preferable, but they do not resolve all challenges. Extrapolating beyond the data range, especially for large return periods associated with PMP estimates, remains difficult and introduces substantial uncertainty. Such return level estimates should be accompanied by uncertainty evaluations (e.g., confidence intervals) to clearly communicate to end-users that PMP estimates carry significant uncertainty inherent to extrapolation. The methodology based on simulated data, presented in Sect.  to address data scarcity, could also be adapted within the extreme-value framework to reduce uncertainty to some extent. However, a degree of uncertainty will always remain as it is inherent to extrapolation.