## Abstract

Daily precipitation extremes are crucial in the hydrological design of major water control structures and are expected to show a changing tendency over time due to climate change. The magnitude and frequency of extreme precipitation can be assessed by studying the upper tail behavior of probability distributions of daily precipitation. Depending on the tail behavior, the distributions can be classified into two categories: heavy-tailed and light-tailed distributions. Heavier tails indicate more frequent occurrences of extreme precipitation events. In this paper, we have analyzed the temporal change in the tail behavior of daily precipitation over India from pre- to post-1970 time periods as per the global climatic shift. A modified Probability Ratio Mean Square Error norm is used to identify the best-fit distribution to the tails of daily precipitation among four theoretical distributions (e.g., Pareto-type II, Lognormal, Weibull, and Gamma distributions). The results indicate that the Lognormal distribution, which is a heavy-tailed distribution, fits the tails of daily precipitation for the majority of the grids. It is inferred from the study that there is an increase in the heaviness of tails of daily precipitation data over India from pre- to post-1970 time periods.

## HIGHLIGHTS

This paper attempts to assess the changes in the tail behavior of daily precipitation over India due to climatic shift in the 1970s.

Maps show best-suited probability distributions for pre-/post-1970s and highlight the changes.

Results highlight the use of heavy-tailed probability distributions for analyzing the extreme precipitation.

A significant increase in heaviness of tails is noticed from pre- to post-1970 time periods.

## INTRODUCTION

Among many of the well-known consequences of climate change, the intensification of hydrological variables such as precipitation and temperature is significant. Climate change leads to an increase in the intensity and frequency of rainfall events globally and locally (Groisman *et al.* 1999; Jacob & Hagemann 2007; Giorgi *et al.* 2011). The Intergovernmental Panel on Climate Change (IPCC) has put serious efforts into summarizing the impacts of extreme precipitation on floods in the Special Report on Extremes (SREX) (IPCC 2013). This shows the grave concern of the hydro-meteorologists worldwide about the hydro-climatic extremes and their consequences for mankind. The reliable estimation of daily extreme precipitation events is of utmost importance to ensure the structural safety of infrastructure projects and prevent loss of human life (Koutsoyiannis 2008). These rare, abnormal, and catastrophic events usually lie in the tail part of the probability distribution of daily precipitation data. The tail is defined as the upper part of the complementary cumulative distribution function beyond a particular threshold (e.g., Klüppelberg 1988). The conventional distribution fitting methods are unable to fit the tail of daily precipitation data adequately, which results in the exemption of extreme precipitation events (termed as improbable events or outliers), thereby underestimating their probability of occurrence (Koutsoyiannis 2004a, 2004b; Li *et al.* 2012; Papalexiou *et al.* 2013; Chen & Brissette 2014; Beskow *et al.* 2015; Papalexiou & Koutsoyiannis 2016; De Michele & Avanzi 2018; Mlyński *et al.* 2019). As per the classical extreme value theory, the block maxima (BM) extracted from a time-series resemble one of the three limiting distributions, namely, (i) Gumbel distribution (i.e., Extreme Value Type I distribution); (ii) Fréchet distribution (i.e., Extreme Value Type II distribution); and (iii) reversed Weibull (i.e., Extreme Value Type III distribution) (Fisher & Tippett 1928; Gnedenko 1943; Jenkinson 1955; Coles 2001; Langousis *et al.* 2016). In the case of annual maximum daily precipitation (i.e., BM per year), Gumbel and Fréchet distributions were found to be appropriate to model the behavior of extremes as both possess unbounded upper tail behavior (Koutsoyiannis 2004a; Papalexiou & Koutsoyiannis 2013; De Michele 2019). Generally, the reversed Weibull distribution is not considered for analyzing annual maximum daily precipitation as the distribution is bounded on the upper tail (Kotz & Nadarajah 2000; Papalexiou & Koutsoyiannis 2013; Chavan & Srinivas 2021). The inferences from the BM approach are dependent on the selection of block size (i.e., either annual or seasonal maxima, etc.). The selection of the annual maxima (AM) from daily precipitation records at a location may distort the tail behavior of their probability distribution as it might miss a few of the largest daily precipitation events from a particular year. As the BM approach discards a large portion of information from available data, the estimated distribution parameter exhibits significant variability and becomes sensitive to outliers (Coles *et al.* 2003; Koutsoyiannis 2004b; Deidda 2010; Langousis *et al.* 2016). Another way of modeling extreme precipitation is based on the peak-over-threshold (POT) approach. In the POT approach, a sample is extracted from daily precipitation series by selecting all observations above an arbitrary threshold *u* (Chow 1964). As the threshold increases, such samples tend to follow generalized Pareto distribution (GPD) (e.g., Balkema & de Haan 1974; Pickands 1975; Madsen *et al.* 1997). Many studies from the past revealed that the findings based on the POT approach are generally more efficient than the BM approach (Cunnane 1973; Madsen *et al.* 1997; Caires 2009; Villarini *et al.* 2011; Moccia *et al.* 2021). Despite its advantages, the use of POT is less prevalent than BM due to (i) the presence of serial dependence in identified peaks and (ii) ambiguity in the selection of an optimum threshold for the identification of peaks (e.g., Beguería 2005; Mailhot *et al.* 2013; Serinaldi & Kilsby 2014; Kiran & Srinivas 2021). To avoid the loss of information about the extreme precipitation as in the case of the BM approach and the selection of arbitrary threshold as in the case of the POT approach, an annual exceedance series (AES) is used for demarcating the tail of daily precipitation data in this study. An AES approach has the advantage of better representing the exact tail of the parent distribution (e.g., Chow 1964; Ben-Zvi 2009; Gupta 2011; Papalexiou *et al.* 2013). After finalizing the tail of daily precipitation data, the distribution fitting can be accomplished using the probability ratio mean square error (PRMSE) norm proposed by Papalexiou *et al.* (2013). PRMSE norm-based tail-fitting of the probability distributions yields unbiased estimates of parameters of distributions and also compares the fits of various probability distributions (e.g., Papalexiou *et al.* 2013, 2018; Moccia *et al.* 2021). In a recent study by Moccia *et al.* (2021), the equivalence between PRMSE and a conventional fitting method, i.e., Kolmogorov–Smirnov (KS) test, in identifying the best-fitting distribution was verified, and the PRMSE was found to have some additional benefits over the KS test.

### Classification of probability distributions based on their tail behavior

*et al.*2008; Foss

*et al.*2013; Papalexiou

*et al.*2013). They have upper tails decaying as a power law (i.e., tails tend to approach zero more gently than an exponential tail). A random variable

*X*is said to have a heavy-tail when its moment-generating function becomes infinite on

*R*(the set of real number) given in Equation (1) (e.g., Panorska

*et al.*2007; Foss

*et al.*2013; Panahi 2016; Wang

*et al.*2018).

*F*to be subexponential, one of the following conditions must hold.where denotes the exceedance probability; denotes the tail of

*n*-fold convolution of

*F*(Embrechts & Goldie 1982; Embrechts

*et al.*1997; Goldie & klüppelberg 1998). Definition (a) shows the absence of any exponential moments, while condition (b) suggested by Teugel (1975) indicates the presence of enormous values (i.e., rare events) in the sample. Condition (b) is also known as the principle of ‘a single big jump’ (Foss

*et al.*2013; Hill 2019). Embrechts & Goldie (1980) showed that the equivalence of condition (a) and (b) power-law distributions like Lognormal distribution and regularly varying distributions such as Pareto-type II distribution are subsets of the subexponential distributions (Feller 1971; Bingham

*et al.*1987; El Adlouni

*et al.*2008; Foss

*et al.*2013; Voitalov

*et al.*2018). Several attempts have been made to group tails of distributions according to their limiting/asymptotic behavior (e.g., Goldie & Klüppelberg 1998; Ouarda

*et al.*1994, etc.). Werner & Upper (2004) classified the distributions in five nested classes from A to E such that A⊂ B⊂ C⊂ D⊂ E: Class A (stable distributions; distributions with Pareto tails having

*α*< 1), B (Pareto-type tail), C (regularly varying distributions), D (subexponential distributions), and the class E (the broadest class includes distributions with infinite exponential moments). El Adlouni

*et al.*(2008) combined classifications mentioned above with five graphical criteria for tail discrimination and arranged different distributions from light- to heavy-tailed. Papalexiou

*et al.*(2013, 2018) intuitively defined two broad classes of distributions based on the asymptotic tail behavior as (a) subexponential class (heavy-tailed class) and (b) the superexponential class (Nagaev & Tsitsiashvili 2006) or hyperexponential class (Vela & Rodríguez 2014) (light-tailed class). The practical implication of a heavy-tailed distribution such as a Pareto or a lognormal distribution is that the large values representing rare events are much more likely to occur than that of a light-tailed distribution like Gaussian or Exponential distribution.

Several studies like that of Mielke (1973) have shown that heavy-tailed distributions like Kappa distribution might be more suitable for modeling daily precipitation data. Panorska *et al.* (2007) verified that the daily extreme precipitation data come from Pareto distribution rather than an exponential distribution based on likelihood ratio tests for hypothesis validation. They found that the tails of daily precipitation closely resemble power law for most North American continent stations. Papalexiou *et al.* (2013) used the daily rainfall of 15,137 records and compared the performance of various probability distributions such as Pareto-type II, Weibull, Lognormal, and Gamma distributions in describing the upper tails and ranked the suitability of the distributions as per PRMSE norm. Overall findings from the study indicated that the daily rainfall extremes are better described by heavy-tailed distribution. Cavanaugh *et al.* (2015) used the statistical test of Kozubowski *et al.* (2009) and the methodology of Panorska *et al.* (2007) to show that the probability distribution of intense daily precipitation over 22,000 stations located globally exhibits heavy tails. The Pareto-type tail dominates over 65% of stations as compared with the exponential tail-type distributions. Nerantzaki & Papalexiou (2019) developed a faster algorithmic procedure for the mean excess function (MEF)-based graphical method to discriminate between exponential and subexponential tails for about 21,348 daily precipitation records all over the globe. They observed that nearly 75.8% of records showed the dominance of heavy-tail distributions. Graphical methods such as Log–log plots, MEF, Zipf plot, and Hill tail estimator are also useful in assessing the tail behavior of distributions (Hill 1975; Embrechts *et al.* 1997; Ghosh & Resnick 2010; Nieboer 2011; Cirillo 2013; Cooke *et al.* 2014). Recently, Moccia *et al.* (2021) showed that heavy-tailed distributions provide a better fit than the light-tailed distributions for about 80% of stations in two Italian regions (i.e., Lazio and Sicily). They perform the analysis for the sample selected through the AM and AES approaches.

### Effect of global climatic shift on extreme precipitation

There have been several studies that analyzed the changes in extreme rainfall over India at the national and regional scale and have drawn different conclusions (Goswami *et al.* 2006; Rajeevan *et al.* 2008; Dash *et al.* 2009; Ghosh *et al.* 2012; Kulkarni *et al.* 2012; Mondal & Mujumdar 2015; Shastri *et al.* 2015; Ghosh *et al.* 2016; Singh *et al.* 2016; Roxy *et al.* 2017; Bisht *et al.* 2018a, 2018b). The changes in extreme precipitation can be attributed to the abrupt global change of the climatic system caused by a regime shift in the 1970s in various climatic factors like the Arctic Oscillation (AO), East Asian summer monsoon (EASM), East Asian winter monsoon (EAWM), El Niño–Southern Oscillation (ENSO), North Atlantic Oscillation (NAO), Aleutian low (AL), Pacific decadal oscillation (PDO), Western Pacific subtropical high (WPSH), and Indian summer monsoon rainfall (ISMR) (Biondi *et al.* 2001; Chowdary *et al.* 2006; Zhou *et al.* 2009; O'Kane *et al.* 2014,*;*Chen *et al.* 2015; Sahana *et al.* 2015; Hsu 2016; Zuo *et al.* 2016; Weisheimer *et al.* 2017; Dai *et al.* 2018) or some local changes such as urbanization. The climate regime shift has adversely impacted the atmosphere, ecosystems, biological, and many hydro-climatic variables, such as temperature, air pressure, wind field, and rainfall, resulting in the frequent occurrence of extremes like heat, drought, heavy rainfall, and flood disasters (Graham 1994; Zhang *et al.* 1998; Wang 2001; Meehl *et al.* 2008; Jacques-coper & Garreaud 2015; Huang *et al.*2017a, 2017b). Many researchers have demonstrated that the rainfall characteristics within and beyond the monsoon period exhibit spatio-temporal changes due to climate regime shifts globally and in India (Sabeerali *et al.* 2012; Sahana *et al.* 2015). Ajayamohan & Suryachandra (2008) have also shown an increased extreme rainfall event over central India after the 1976/1977 climate shift. Vittal *et al.* (2013) showed that rainfall extremes have changed in India after 1975 and established that urbanization, in terms of change in population density, is a possible cause of change. They used a comprehensive POT approach with 95 and 99 percentile thresholds, including multiple extreme events in a year. Dash & Maity (2019) found that the precipitation-based climate change indices exhibit increasing trends over India with more spatial extent post-1975. Recently, Sarkar & Maity (2020) observed an increment of 35% in probable maximum precipitation over India in the post-1970 (1971–2010) period when compared with the pre-1970 (1901–1970) period due to climate shift.

### Study goals

This study proposes to assess the temporal changes in the daily extreme precipitation over India due to climatic shift in the 1970s based on the change in the tail behavior of the probability distributions of daily precipitation. We intend to perform the assessment by considering four theoretical distributions (e.g., Pareto-type II, Lognormal, Weibull, and Gamma distributions), which belong to different classes of distributions (i.e., subexponential or hyperexponential classes) following Papalexiou *et al.* (2013). The primary goals of this study are (a) to find the best-suited distribution based on the PRMSE norm that can describe the extreme daily precipitation in changing climate over India, (b) to provide a categorical classification of grids into two broad classes of distribution, i.e., subexponential class and the hyperexponential–exponential class considering the shift in the global climatic regime in the 1970s; (c) to investigate spatial and temporal changes in the behavior of tails of the probability distribution of daily precipitation between the two time periods, namely pre-1970 (1901–1970) and post-1970 (1971–2010). The assessment is performed both at the grid and regional scale (i.e., Meteorological Subdivisions). The study would be useful to the design engineers and hydro-meteorologists for reliable planning and management of various major water-energy infrastructures in India.

## DATA AND METHOD

### Study area

India, the largest South Asian country with a wide variety of climatic regions extending from low rainfall arid regions to the heavy rainfall receiving regions, is our study area. The climatic condition of the Indian mainland is influenced by various geographical and relief features like the Himalayas in the north, Thar Desert and Arabian Sea in the west, the Bay of Bengal in the east, Western Ghats in the south-west, and the Indian Ocean in the south. The study area covers a widespread range of variations in the rainfall extremes, which motivates us to examine spatial and temporal behavior of daily extreme precipitation in terms of magnitude and frequency of occurrence between the two time periods corresponding to the shift in the global climate regime in the 1970s, i.e., pre-1970 and post-1970. Furthermore, the temporal changes are investigated at a regional scale in 34 out of 36 homogeneous Meteorological Subdivisions (see Figure S1 in the Supplementary Material) in this analysis (Guhathakurta & Rajeevan 2008).

### Rainfall data

In this study, an extensive database of daily gridded precipitation with a spatial resolution of 0.25° procured from the India Meteorological Department (IMD) is considered. The gridded rainfall data were prepared for 112 years (1901–2013) by Pai *et al.* (2014) using a varying network of 6,955 rain gauge stations. After performing a quality check, 4,789 grids each having a record length (*N*) of 110 years, i.e., from 1901 to 2010, were selected for analysis. No missing data were filled at any grids/stations. Records at each grid have been split into two parts, i.e., pre-1970 (1901–1970) and post-1970 (1971–2010), to capture the effect of the shift in the global climate regime. Despite being an unequal division of the data, the records at each grid for both pre- and post-1970 time periods have a sufficient number of non-zero daily precipitation values needed to estimate the tail behavior using the threshold-based approach for fitting probability distributions to daily precipitation data proposed by Papalexiou *et al.* (2013). The data division fulfills the condition of the availability of at least a 30-year record customary in the climate community (Arguez & Vose 2011). Individual data length for both the periods is sufficient to obtain a robust representation of the spatial pattern of the tails of the probability distribution of daily precipitation data over India.

### Threshold-based approach for fitting probability distributions

In this paper, we have adopted the threshold-based approach (i.e., AES) for fitting probability distributions to the tail part of the probability distribution of non-zero daily precipitation data proposed by Papalexiou *et al.* (2013). Since the investigation revolves around the tail behavior, it is essential first to define the part of the probability distribution known as ‘tail’. The demarcation of the tail of the empirical distribution for daily precipitation data by optimally selecting the threshold is a vital and crucial step in this approach. After demarcating the tail of the empirical distribution, fitting a theoretical probability distribution function to the daily precipitation data in the tail part can be accomplished by minimizing the difference between empirical and theoretical distributions.

### Defining tail of empirical distribution of daily precipitation

The upper or ‘right’ part of the empirical probability distribution function for non-zero rainfall is referred as the ‘tail’. The choice of a threshold needed for defining a tail is recognized as a difficult and open problem of debate to date. Hence, to avoid a priori selection of the threshold, we defined samples using the AES method in the present study. We choose a value as a threshold such that the number of extreme precipitation events above it equals the number of years of record *N* (Cunnane 1973; Ben-Zvi 2009). *N* largest daily values of the record are preferred over each year's maximum value as the latter results in the distorted tail (Papalexiou & Koutsoyiannis 2013; Papalexiou *et al.* 2013).

*p*

_{0}represents the probability of dry day. The empirical probability of exceedance is defined according to the Weibull plotting position formula (Weibull 1939; Makkonen 2006) at each station having

*N*-year record, and

*n*number of non-zero precipitation values is defined as follows:where is the rank of the precipitation equal to in the ordered sample as of the non-zero values. Thus, the empirical tail is defined by the

*N*largest non-zero precipitation values of with . Note that the threshold value for precipitation is given as .

### Theoretical distributions considered in this study

Four simple, popular, and frequently used theoretical distributions such as Weibull (W), Lognormal (LN), Pareto-type II (PII), and the Gamma (G) distributions are considered in this study following Papalexiou *et al.* (2013) and Papalexiou *et al.* (2018). Details on these four distributions are provided in Table S1 in the Supplementary Material. Distributions selected have two parameters: one is scale parameter (*β* > 0) and the other is shape parameter (*α**>* 0). The decision on the heaviness of tails of daily precipitation data is based on fitting four probability distributions to the precipitation data in the tail part of the empirical distribution. The distributions can be divided into subexponential and exponential–hyperexponential classes based on the estimates of the shape parameter, *α*. The former group comprises Pareto-type II distribution, Lognormal distribution, and Weibull distribution with *α* < 1, whereas the latter group includes Gamma distribution and Weibull distribution with *α* > 1.

### Procedure to fit probability distributions

*et al.*2013).

The PRMSE norm is a function of the parameters *β* and *α* of the theoretical distributions. The norm is selected because (i) it is unbiased and suitable for subexponential distributions, (ii) it is easy to use and allows direct comparison of different distribution tails, and (iii) it gives equal weightage to each point in the tail, which contributes to the sum as relative errors between theoretical and empirical values (Papalexiou *et al.* 2013, 2018). In this study, the approach proposed by Papalexiou *et al.* (2013) is slightly modified by using a genetic algorithm (GA) (Goldberg 1989; Michalewicz *et al.* 1992) for parameter estimation of the distributions. A GA is a heuristic, stochastic, combinatorial, optimization technique based on the biological process of natural evolution (reproduction, crossover, and mutation). The heuristic is applied probabilistically to the discrete decision variables coded into binary strings. GA has been utilized effectively to minimize the PRMSE given in Equation (5) in two ways: (i) by fitting theoretical distribution to the entire precipitation data observed at a grid and (ii) by fitting theoretical distribution to *N* largest values of precipitation at a grid. Figure 1 depicts the approach to fit different probability distributions, namely Lognormal, Pareto-type II, Weibull, and Gamma distributions to the precipitation events in the tail part for both pre- and post-1970 periods (i.e., 1901–1970 and 1971–2010). Grids for which the parameters change but the distribution remains unchanged are shown in Figure 1. It can be inferred from the figures that the first approach where distribution is fitted to entire non-zero precipitation data does not adequately describe the tail (refer to the black dashed line). On the other hand, the solid red line representing fitting of the distributions only to the events in the tail part appears to describe the tail adequately.

## RESULTS

This paper investigates the temporal and spatial changes in the behavior of daily extreme precipitation over India in terms of its frequency of occurrence due to the shift in the global climatic regime in the 1970s. The temporal changes are assessed between the two time periods, pre-1970 and post-1970, both at grid scale and regional scale. Further, a categorical classification of grids based on the change in average rainfall above threshold (increase or decrease in magnitude) and the change in the nature of the tails (i.e., from light to heavy or heavy to light) over the two time periods is also presented in this study.

### Assessment of spatial and temporal changes in tail behavior of probability distribution of daily precipitation at the grid scale

In this section, the spatial and temporal changes in the behavior of tails of probability distributions of daily precipitation over India were analyzed for pre- and post-1970 periods. Following the procedure described earlier, Lognormal, Pareto-type II, Weibull, and Gamma distributions were considered to fit the non-zero daily precipitation data at 4,789 grids over India from 1901 to 1970 and 1971 to 2010, respectively. The distributions were fitted either by considering entire precipitation data at a grid or considering either 70 or 40 largest precipitation data values depicting the tail part of the distribution for the time periods of 1901–1970 and 1971–2010. Visual investigation of the fits at all 4,789 grids indicated that the fit based on the largest values in the precipitation data adequately described the tail part of the empirical distribution. This shows the advantage of the threshold-based approach proposed by Papalexiou *et al.* (2013) for fitting probability distributions to daily precipitation data, especially while analyzing the daily extreme precipitation events.

To find the best-fitted distribution of the four fitted distributions at each grid, the PRMSE norm was considered in this study. In the case of each grid, the distribution function yielding the least estimate for the PRMSE norm was declared to be the best-suited distribution for that grid. Figure 2 shows the geographical or spatial variation of best-suited distribution over India for both pre- and post-1970 periods. For the pre-1970 period, out of 4,789 grids over India, Lognormal distribution was found to be better suited for nearly 41.87% grids, followed by Pareto (32.43%), Weibull (18.56%), and Gamma (7.14%) distributions. For the post-1970s, the sequence remains the same with Lognormal as the best-fitted distribution for most grids over India. It has been observed that 45.86% of grids exhibit lognormal distribution as best-suited distribution, followed by Pareto for 32.20%, Weibull for 17.06%, and Gamma for 4.88% grids. Given these overall percentages, one may conclude that the Lognormal and Pareto-type II distributions (both heavy-tailed distributions) are the most suitable distributions for modeling the tails of probability distributions of daily precipitation data. Overall, it can be seen that there is a 4% increment and a 2.26% decrement in the number of grids following Lognormal distribution and Gamma distribution, respectively. The Weibull distribution with a shape parameter less than 1 (which is a heavy-tailed distribution) was found to be suitable for 96.40 and 98.53% of the grids, which showed Weibull as the best-suited distribution for pre-and post-1970 periods, respectively. It can be concluded from the results that there exists a dominance of heavy-tailed distributions over light-tailed Gamma and Weibull distributions along with the increase in the tail heaviness of precipitation data over India due to climatic shifts. This indicates that extreme precipitation events in India have become more frequent.

Details on the shape (*α*) and scale (*β*) parameters of the best-suited distributions are provided in Table 1(a) and 1(b) for both pre- and post-1970 periods, respectively. The shape parameter is a scalar measure of tail behavior, and its histogram constructed based on estimates at all grids can be helpful in providing essential information about the tail heaviness. Figure 3(a) and 3(b) shows the empirical histograms of the shape parameters of four distributions considered in this study for pre- and post-1970 periods. Modal values of the histograms represent the most probable values of the shape parameters for each of the distributions. For Pareto-type II distribution, the modes were observed as 0.19 and 0.176 for the pre- and post-1970 periods. Low modal values for Pareto distribution imply the nonexistence of statistical moments for higher orders, i.e., greater than 5.26 and 5.88 (Papalexiou *et al.* 2013). The mode value of the shape parameter for Lognormal distribution was about 1.1 for both pre- and post-1970 periods. In the case of Weibull distribution, the modes of the histograms were observed to be around 0.84 for pre-1970 and 0.82 for post-1970, both implying the presence of heavier tails of the distribution as the shape parameter is less than 1. Histograms for the shape parameter of Gamma distribution show low modal values of 0.67, 0.73 for pre- and post-1970s, respectively, which indicate the presence of hyperexponential tails representing a lesser frequency of occurrence for extreme precipitation events. Histograms of shape parameters did reveal a lot about the basic nature of the tail of four distributions but to further investigate the tail relevances in describing daily precipitation, the average ranking was also considered. All four distributions were ranked in the ascending order of the PRMSE norm, i.e., the distribution yielding the least PRMSE was declared as Rank 1 distribution, while the distribution with the highest PRMSE was ranked as 4. Figure 4 illustrates the average rank of the four probability distributions for pre- and post-1970 periods. A lower average rank of a probability distribution indicates better suitability of the distribution in describing the tails of precipitation data as compared with those with higher ranks. Lognormal distribution was the best-fitted distribution with an average rank of 1.9 and 1.7 for both pre-1970 and post-1970 periods. The best-fitted distributions were ordered as Lognormal, Pareto-type II, Weibull, and Gamma based on their ranks for both pre- and post-1970 periods. Conventionally, Gamma distribution is the most commonly used probability distribution for representing daily precipitation. However, results from this study inferred that the Gamma distribution was the worst performer for both periods. It can be noted from Figure 4 that the average ranking of Lognormal and Pareto-type II distributions is decreased from 1.9 to 1.78 and 2.31 to 2.2 over the pre- and post-1970 periods, respectively. This shows that the gridded daily precipitation for the post-1970 period over India exhibits heavier tails than the pre-1970 period. Additionally, an increase in the tail heaviness of the distributions in the post-1970 period was also evident from the increase in the average rank of Gamma distribution.

(a) Pre-1970 time period . | ||||||
---|---|---|---|---|---|---|

. | Pareto . | Lognormal . | ||||

. | MSE . | β
. | α
. | MSE . | β
. | α
. |

Minimum | 0.0025 | 0.8866 | 0.0000 | 0.0026 | 1.5062 | 0.1878 |

Mean | 0.1014 | 8.1352 | 0.2513 | 0.0581 | 10.2748 | 1.2208 |

Maximum | 0.6572 | 25.6076 | 0.6910 | 0.8410 | 24.5317 | 2.1150 |

Median | 0.0438 | 7.6815 | 0.2340 | 0.0335 | 10.0345 | 1.1873 |

SD | 0.1224 | 4.2759 | 0.1206 | 0.0718 | 4.7997 | 0.2572 |

Skew | 1.8099 | 0.5574 | 0.4701 | 3.4391 | 0.2278 | 0.4291 |

. | Weibull . | Gamma . | ||||

. | MSE . | β
. | α
. | MSE . | β
. | α
. |

Minimum | 0.0041 | 0.3476 | 0.3087 | 0.0037 | 3.4304 | 0.0379 |

Mean | 0.1108 | 12.2075 | 0.9143 | 0.1339 | 17.3272 | 1.0737 |

Maximum | 0.9855 | 31.5581 | 17.7532 | 0.4853 | 30.1988 | 7.0028 |

Median | 0.0737 | 12.4927 | 0.8501 | 0.1199 | 17.6201 | 0.9060 |

SD | 0.1221 | 5.0817 | 0.7085 | 0.0826 | 4.2256 | 0.7182 |

Skew | 3.4180 | −0.1809 | 14.4425 | 0.8034 | −0.2213 | 1.9645 |

(b) Post-1970 time periods . | ||||||

. | Pareto . | Lognormal . | ||||

. | MSE . | β
. | α
. | MSE . | β
. | α
. |

Minimum | 0.0033 | 0.7920 | 0.0033 | 0.0040 | 1.9162 | 0.6009 |

Mean | 0.0892 | 8.1501 | 0.2842 | 0.0551 | 9.6792 | 1.3141 |

Maximum | 0.5203 | 31.8910 | 0.7445 | 0.4819 | 24.0319 | 2.2519 |

Median | 0.0487 | 7.5730 | 0.2660 | 0.0377 | 9.1881 | 1.2814 |

SD | 0.0967 | 4.2906 | 0.1314 | 0.0535 | 4.6877 | 0.2856 |

Skew | 1.8672 | 0.6598 | 0.4807 | 2.6614 | 0.3851 | 0.4276 |

. | Weibull . | Gamma . | ||||

MSE . | β
. | α
. | MSE . | β
. | α
. | |

Minimum | 0.0030 | 0.3850 | 0.3066 | 0.0040 | 5.1253 | 0.1139 |

Mean | 0.1173 | 12.2138 | 0.8655 | 0.1788 | 18.1197 | 1.1743 |

Maximum | 0.9744 | 30.8359 | 19.2964 | 0.6899 | 32.7686 | 10.6443 |

Median | 0.0862 | 12.6215 | 0.8158 | 0.1684 | 18.3819 | 0.9731 |

SD | 0.1111 | 5.1673 | 0.6941 | 0.1021 | 4.3883 | 0.8208 |

Skew | 3.3131 | −0.1866 | 16.2781 | 0.6251 | −0.1592 | 2.6873 |

(a) Pre-1970 time period . | ||||||
---|---|---|---|---|---|---|

. | Pareto . | Lognormal . | ||||

. | MSE . | β
. | α
. | MSE . | β
. | α
. |

Minimum | 0.0025 | 0.8866 | 0.0000 | 0.0026 | 1.5062 | 0.1878 |

Mean | 0.1014 | 8.1352 | 0.2513 | 0.0581 | 10.2748 | 1.2208 |

Maximum | 0.6572 | 25.6076 | 0.6910 | 0.8410 | 24.5317 | 2.1150 |

Median | 0.0438 | 7.6815 | 0.2340 | 0.0335 | 10.0345 | 1.1873 |

SD | 0.1224 | 4.2759 | 0.1206 | 0.0718 | 4.7997 | 0.2572 |

Skew | 1.8099 | 0.5574 | 0.4701 | 3.4391 | 0.2278 | 0.4291 |

. | Weibull . | Gamma . | ||||

. | MSE . | β
. | α
. | MSE . | β
. | α
. |

Minimum | 0.0041 | 0.3476 | 0.3087 | 0.0037 | 3.4304 | 0.0379 |

Mean | 0.1108 | 12.2075 | 0.9143 | 0.1339 | 17.3272 | 1.0737 |

Maximum | 0.9855 | 31.5581 | 17.7532 | 0.4853 | 30.1988 | 7.0028 |

Median | 0.0737 | 12.4927 | 0.8501 | 0.1199 | 17.6201 | 0.9060 |

SD | 0.1221 | 5.0817 | 0.7085 | 0.0826 | 4.2256 | 0.7182 |

Skew | 3.4180 | −0.1809 | 14.4425 | 0.8034 | −0.2213 | 1.9645 |

(b) Post-1970 time periods . | ||||||

. | Pareto . | Lognormal . | ||||

. | MSE . | β
. | α
. | MSE . | β
. | α
. |

Minimum | 0.0033 | 0.7920 | 0.0033 | 0.0040 | 1.9162 | 0.6009 |

Mean | 0.0892 | 8.1501 | 0.2842 | 0.0551 | 9.6792 | 1.3141 |

Maximum | 0.5203 | 31.8910 | 0.7445 | 0.4819 | 24.0319 | 2.2519 |

Median | 0.0487 | 7.5730 | 0.2660 | 0.0377 | 9.1881 | 1.2814 |

SD | 0.0967 | 4.2906 | 0.1314 | 0.0535 | 4.6877 | 0.2856 |

Skew | 1.8672 | 0.6598 | 0.4807 | 2.6614 | 0.3851 | 0.4276 |

. | Weibull . | Gamma . | ||||

MSE . | β
. | α
. | MSE . | β
. | α
. | |

Minimum | 0.0030 | 0.3850 | 0.3066 | 0.0040 | 5.1253 | 0.1139 |

Mean | 0.1173 | 12.2138 | 0.8655 | 0.1788 | 18.1197 | 1.1743 |

Maximum | 0.9744 | 30.8359 | 19.2964 | 0.6899 | 32.7686 | 10.6443 |

Median | 0.0862 | 12.6215 | 0.8158 | 0.1684 | 18.3819 | 0.9731 |

SD | 0.1111 | 5.1673 | 0.6941 | 0.1021 | 4.3883 | 0.8208 |

Skew | 3.3131 | −0.1866 | 16.2781 | 0.6251 | −0.1592 | 2.6873 |

Another method adopted for assessing the temporal change in the tails of probability distributions during the pre- and post-1970 periods is achieved by comparing the tails of the distributions in couples or pairs. Various pairs of distributions considered in this study are ‘Lognormal vs. Pareto’, ‘Pareto vs. Weibull’, ‘Pareto vs. Gamma’, ‘Lognormal vs. Weibull’, ‘Lognormal vs. Gamma’, and ‘Weibull vs. Gamma’. The best-fitted distribution among the pair (any two distributions) was selected based on the PRMSE norm for each grid. The distribution with a lesser PRMSE value was considered as the best fit. Figure 5 illustrates the comparison between two probability distributions in pairs for pre- and post-1970 periods. The figure presents the percentage of grids found suitable for each probability distribution compared in pairs. It can be deduced from the figure that Lognormal distribution (which is a heavy-tailed distribution) fits the extreme daily precipitation data for 58.66 and 60.51% grids in the pre- and post-1970 periods when compared with Pareto-type II distribution. Further, the Lognormal distribution was found to be better suited than Weibull and Gamma distributions for both periods. It can be noted that the percentage of grids where daily precipitation is well represented by the Lognormal distribution against the Weibull and Gamma distributions has increased from 70.49 to 74.23% and 81.23 to 87.7%, respectively, over the pre- and post-1970 periods. This indicates that the probability distributions of daily precipitation in the post-1970 period exhibit heavier tails than the pre-1970 period. We have also compared Pareto-type II distribution with Lognormal, Gamma, and Weibull distributions, and it emerged as the second best-fitted distribution after Lognormal distribution. Similar to Lognormal distribution, Pareto-type II distribution was inferred as a better fitting distribution against Gamma and Weibull distributions for both pre- and post-1970 periods. The analysis revealed that the percentage of grids where the tails of probability distributions of daily precipitation are better fitted by the Pareto-type II distribution against the Weibull and Gamma distributions has increased from 58.17 to 62.98% and 69.05 to 77.34% over the periods. Among the Weibull and Gamma distributions, the Weibull distribution was better suited for describing the tails of daily precipitation data over India in both pre- and post-1970 periods. Interestingly, a heavier tailed distribution was better fitted in each case during both periods. These findings highlight that the heavier tailed distributions should be preferred over their counterparts while representing the tails of daily precipitation data over India.

We have also investigated the existence of any geographical/spatial pattern of best-suited distributions over India. The maps shown in Figure 2 illustrate the spatial distribution of best-fitted distributions for pre- and post-1970 periods. These maps do not unveil any regular patterns; instead, they seem to follow a random spatial variation. Hence, to reveal some meaningful conclusions, we categorized the best-suited distributions into either subexponential or exponential–hyperexponential classes based on the estimates of the shape parameter (*α*), following El Adlouni *et al.* (2008) and Papalexiou *et al.* (2013). The subexponential class includes Pareto-type II distribution, Lognormal distribution, and Weibull distribution with shape parameter <1, while the exponential–hyperexponential class comprises the Gamma distribution and Weibull distribution with shape parameter >1. Figure 6 represents maps showing the spatial distribution of subexponential and exponential–hyperexponential distributions over India for pre- and post-1970 periods. For the pre-1970 period, subexponential distributions were better suited for 4,415 out of 4,789 grids (i.e., about 92.19%) over India. In comparison, the latter class was found to be applicable for merely 374 grids (i.e., only 7.81% of all the grids). Similarly, for the post-1970 period, subexponential distributions were found to be adequate to model daily precipitation data at 4,543 grids (i.e., 94.86% of all the grids while exponential–hyperexponential tails were found to be appropriate for the remaining 246 grids) (i.e., 5.14% of the grids). It can be observed from maps that the heaviness in the tails of probability distributions of daily precipitation over India has increased post-1970s climatic shift. With the dominance of heavy tails all over India, few pockets comprising lighter tails were observed in the northeast region and along the western coastal plain of the Indian Peninsula for both periods. It can be seen very well from the maps that the subexponential tails are much more dominant in the Indian region than the exponential–hyperexponential tails.

Overall, the comparison is made in terms of the difference in the percentage of the number of grids belonging to one category and the values of shape parameter (*α*). The maps showing the distribution and the two broad classes of tail behavior (i.e., subexponential and exponential–hyperexponential) are compared for both pre- and post-1970 periods to examine the impact of the global shift in climate regime in the 1970s on extreme precipitation. The presence of heavy tails in the daily precipitation data points to the fact that the extreme precipitation events over India are no longer rare.

### Assessment of temporal changes in tail behavior of probability distribution of daily precipitation at the regional scale

Analysis at the grid scale revealed some essential inferences about the tail behavior of daily precipitation over India. However, to make the analysis more interpretable and usable at the regional scale, the temporal changes in tail behavior of the probability distribution of daily precipitation were assessed considering 34 Meteorological Subdivisions over India. Table 2 provides details about the percentage of grids having heavy or subexponential tails in each subdivision, considering the pre-1970 and post-1970 periods. Figure 7 shows the percentage of subexponential tails in each Meteorological Subdivision over India for both periods using color codes. In the case of the pre-1970 period, Saurashtra Kutch & Diu (subdivision 22) showed a complete dominance of heavy tails, followed by Gujarat (subdivision 21), Gangetic West Bengal (subdivision 6), and Orissa (subdivision 7). Nearly 22 subdivisions had more than 90% of grids showing heavy-tailed behavior. For the post-1970 period, Sub Him West Bengal and Sikkim (subdivision 5), Coastal Andhra Pradesh (subdivision 28), and Gangetic West Bengal (subdivision 6) have 100% heavy tails. In the post-1970s, 30 subdivisions were found to have more than 90% of grids showing heavy-tail behavior.

ID . | Metrological region . | Percentage of grids having heavy tails . | |
---|---|---|---|

Pre-1970 (1901–1970) . | Post-1970 (1971–2010) . | ||

2 | Arunachal Pradesh | 68.54 | 95.51 |

3 | Assam and Meghalaya | 77.71 | 90.36 |

4 | Naga Mani Mizo and Tripura | 88.89 | 90.00 |

5 | Sub Him W Bengal Sikkim | 96.67 | 100.00 |

6 | Gangetic West Bengal | 98.92 | 100.00 |

7 | Orissa | 98.15 | 99.07 |

8 | Jharkhand | 93.86 | 97.37 |

9 | Bihar | 94.90 | 96.82 |

10 | East Uttar Pradesh | 95.48 | 99.10 |

11 | West Uttar Pradesh | 98.05 | 94.81 |

12 | Uttaranchal | 97.59 | 93.98 |

13 | Haryana Chandigarh and Delhi | 93.20 | 94.20 |

14 | Punjab | 96.67 | 97.78 |

15 | Himachal Pradesh | 87.64 | 93.18 |

16 | Jammu and Kashmir | 93.83 | 93.09 |

17 | West Rajasthan | 93.18 | 92.80 |

18 | East Rajasthan | 97.09 | 98.06 |

19 | West Madhya Pradesh | 95.95 | 98.38 |

20 | East Madhya Pradesh | 97.45 | 97.45 |

21 | Gujarat | 99.19 | 97.58 |

22 | Saurashtra Kutch and Diu | 100.00 | 99.36 |

23 | Konkan and Goa | 71.43 | 90.00 |

24 | Madhya Maharastra | 87.50 | 88.82 |

25 | Marathwada | 93.26 | 92.13 |

26 | Vidarbha | 94.89 | 97.08 |

27 | Chhatisgarh | 95.00 | 97.78 |

28 | Coastal Andhra Pradesh | 96.80 | 100.00 |

29 | Telangana | 92.67 | 98.00 |

30 | Rayalaseema | 84.44 | 95.56 |

31 | Tamil Nadu and Pondicherry | 88.57 | 93.71 |

32 | Coastal Karnataka | 58.97 | 74.36 |

33 | North Interior Karnataka | 82.69 | 81.73 |

34 | South Interior Karnataka | 78.05 | 78.05 |

35 | Kerala | 86.15 | 95.38 |

ID . | Metrological region . | Percentage of grids having heavy tails . | |
---|---|---|---|

Pre-1970 (1901–1970) . | Post-1970 (1971–2010) . | ||

2 | Arunachal Pradesh | 68.54 | 95.51 |

3 | Assam and Meghalaya | 77.71 | 90.36 |

4 | Naga Mani Mizo and Tripura | 88.89 | 90.00 |

5 | Sub Him W Bengal Sikkim | 96.67 | 100.00 |

6 | Gangetic West Bengal | 98.92 | 100.00 |

7 | Orissa | 98.15 | 99.07 |

8 | Jharkhand | 93.86 | 97.37 |

9 | Bihar | 94.90 | 96.82 |

10 | East Uttar Pradesh | 95.48 | 99.10 |

11 | West Uttar Pradesh | 98.05 | 94.81 |

12 | Uttaranchal | 97.59 | 93.98 |

13 | Haryana Chandigarh and Delhi | 93.20 | 94.20 |

14 | Punjab | 96.67 | 97.78 |

15 | Himachal Pradesh | 87.64 | 93.18 |

16 | Jammu and Kashmir | 93.83 | 93.09 |

17 | West Rajasthan | 93.18 | 92.80 |

18 | East Rajasthan | 97.09 | 98.06 |

19 | West Madhya Pradesh | 95.95 | 98.38 |

20 | East Madhya Pradesh | 97.45 | 97.45 |

21 | Gujarat | 99.19 | 97.58 |

22 | Saurashtra Kutch and Diu | 100.00 | 99.36 |

23 | Konkan and Goa | 71.43 | 90.00 |

24 | Madhya Maharastra | 87.50 | 88.82 |

25 | Marathwada | 93.26 | 92.13 |

26 | Vidarbha | 94.89 | 97.08 |

27 | Chhatisgarh | 95.00 | 97.78 |

28 | Coastal Andhra Pradesh | 96.80 | 100.00 |

29 | Telangana | 92.67 | 98.00 |

30 | Rayalaseema | 84.44 | 95.56 |

31 | Tamil Nadu and Pondicherry | 88.57 | 93.71 |

32 | Coastal Karnataka | 58.97 | 74.36 |

33 | North Interior Karnataka | 82.69 | 81.73 |

34 | South Interior Karnataka | 78.05 | 78.05 |

35 | Kerala | 86.15 | 95.38 |

The maps showing the percentage of heavy tails in each subdivision for pre- and post-1970s were compared to find the change in the number of grids comprising heavy tails. The changes in the percentage of heavy tails over time for each subdivision are presented in Figure 7(c). In nearly 23 out of 34 subdivisions, the percentage of grids with heavy tails is observed to be increased over time. Out of those 23, nine subdivisions, namely Arunachal Pradesh, Assam and Meghalaya, Himachal Pradesh, Konkan and Goa, Telangana, Tamil Nadu, and Pondicherry, Rayalaseema, Coastal Karnataka, and Kerala, showed an increase of about 5% or above. The substantially higher percentage of grids exhibiting heavy tails post-1970 compared with that of pre-1970 might be a possible consequence of climate change and global climatic shift in the 1970s.

### Assessment of temporal changes in magnitude and frequency of extreme precipitation over India

The classification of grids exhibiting the severity in terms of increase in the magnitude and frequency of extreme precipitation events due to the climatic shift is achieved by considering the combined effect of change in average precipitation above threshold and change in the tail behavior. The rainfall values above the threshold are the ones that belong to the tail. We considered the average of these values at each grid, which served as an indicator of the magnitude of the extreme precipitation. The average rainfall values above the threshold vary from a minimum value of 25.19–287.51 mm for the pre-1970 period. On the other hand, average rainfall values above the threshold range from 39.65 to 646.71 mm. Eight categories of severity were proposed by considering an increase or decrease of average rainfall above threshold and change in the nature of tail over the period from pre-1970 to post-1970. Table 3 describes the categories along with the number and percentage of grids falling in them. The categories are ranked from 1 to 8, with 1 being the most severe case and 8 being the least severe case. Figure 8 shows the spatial pattern of grids belonging to each category. Figure 8 and Table 3 show that nearly 64.5% of grids belong to Category 1, representing the most severe case. Category 2 comprises 5.42% grids where an increase in the magnitude of extreme precipitation and transition in the tail behavior from light to heavy were observed. Category 3 includes nearly 23.34% grids where the magnitude of extreme precipitation has decreased with heavy-tailed behavior during pre- and post-1970. Overall, most of the grids in India show a tendency of transition towards heavier tails along with an increase in the magnitude of extreme precipitation.

Category (Severity decreases top to bottom) . | Change in average rainfall above the threshold from pre-1970 to post-1970 . | Change in tail type from pre-1970 to post-1970 . | Number of grids (out of 4789) . | Percentage of grids (%) . |
---|---|---|---|---|

Category 1 | Increases | Heavy to heavy | 3093 | 64.58 |

Category 2 | Increases | Light to heavy | 260 | 5.42 |

Category 3 | Decreases | Heavy to heavy | 1118 | 23.34 |

Category 4 | Decreases | Light to heavy | 72 | 1.50 |

Category 5 | Increases | Heavy to light | 98 | 2.04 |

Category 6 | Decreases | Heavy to light | 106 | 2.21 |

Category 7 | Increases | Light to light | 28 | 0.58 |

Category 8 | Decreases | Light to light | 14 | 0.29 |

Category (Severity decreases top to bottom) . | Change in average rainfall above the threshold from pre-1970 to post-1970 . | Change in tail type from pre-1970 to post-1970 . | Number of grids (out of 4789) . | Percentage of grids (%) . |
---|---|---|---|---|

Category 1 | Increases | Heavy to heavy | 3093 | 64.58 |

Category 2 | Increases | Light to heavy | 260 | 5.42 |

Category 3 | Decreases | Heavy to heavy | 1118 | 23.34 |

Category 4 | Decreases | Light to heavy | 72 | 1.50 |

Category 5 | Increases | Heavy to light | 98 | 2.04 |

Category 6 | Decreases | Heavy to light | 106 | 2.21 |

Category 7 | Increases | Light to light | 28 | 0.58 |

Category 8 | Decreases | Light to light | 14 | 0.29 |

## SUMMARY AND CONCLUSIONS

In this paper, we have analyzed the temporal and spatial changes in the tail behavior of daily precipitation over India from pre- to post-1970 time periods as per the global climatic shift. The tail behavior of precipitation data is assessed by identifying the best-fitted distribution out of four theoretical distributions to the sample obtained by the AES approach (e.g., Pareto-type II, Lognormal, Weibull, and Gamma distributions) based on the PRMSE norm. The approach is found to be easy to use and effective in diagnosing the tail behavior of daily precipitation data. Maps showing the geographical variation in the percentage of best-fitted subexponential tails over 34 Meteorological Subdivisions in India are given in this study. Also, the categorical classification of grids in terms of severity by considering the combined effect of an increase or decrease in average rainfall above threshold and change in the nature of tail over the period from pre-1970 to post-1970. Results from this study emphasize the importance of heavy-tailed distributions for reliable estimation of the frequency of extreme precipitation events in India. Important highlights from this study are as follows.

- (i)
Lognormal and Pareto-type II distributions (both heavy-tailed distributions) are found to be better suited for daily precipitation over India for both pre- and post-1970 periods. It can be concluded from the results that there exists a dominance of heavy-tailed distributions over light-tailed Gamma and Weibull distributions along with the increase in the tail heaviness of precipitation data over India due to climatic shifts. This directs us to the fact that the extreme precipitation events in India have become more frequent during both the pre- and post-1970 periods.

- (ii)
Gamma distribution, in general, underestimates the frequency and magnitude of extreme events. Hence, the distribution should not be considered for modeling the extreme precipitation events over India.

- (iii)
Histograms of shape parameters of the four probability distributions revealed that the tails of daily precipitation data have become heavier from pre- to post-1970 periods.

- (iv)
Heavy-tailed distributions can describe the observed precipitation extremes more effectively than light-tailed distributions. About 92.19% of the records in the pre-1970s and 94.86% in the post-1970s are better characterized by subexponential tails. Exponential–hyperexponential tails are found to be better suited for only 7.81 and 5.14% of records for the pre- and post-1970 periods. It can be seen that increasing trends of heavy tails persist in the later period, indicating a rising trend of more frequent and ‘severe events’ of precipitation.

- (v)
Twenty-three Meteorological Subdivisions in India show an increase in the percentage of heavy tails in the post-1970s compared with the pre-1970s. Further, nine subdivisions out of those 23, namely Arunachal Pradesh, Assam and Meghalaya, Himachal Pradesh, Konkan and Goa, Telangana, Tamil Nadu, and Pondicherry, Rayalaseema, Coastal Karnataka, and Kerala, showed a substantial increase in the percentage of grids exhibiting heavy tails.

- (vi)
Eight categories of severity are proposed by considering an increase or decrease in average rainfall above threshold and change in the nature of tail over the period from pre-1970 to post-1970. Nearly 70% of grids in India belong to Category 1 and Category 2 which are deemed to indicate severe/critical categories in terms of increase in the magnitude of extreme precipitation and the presence of heavier tails over pre- to post-1970 periods.

An important inference from this analysis is that the frequency and the magnitude of extreme precipitation events have generally been undervalued in the past. The use of light-tailed distributions for modeling daily precipitation can lead to a serious underestimation of the frequency and the magnitude of design extreme precipitation, which is highly undesirable for the design of water control structures. It can be noted that the results obtained from the present study are dependent on the length of precipitation records (e.g., Arguez & Vose 2011; Cavanaugh *et al.* 2015) available at each grid and the presence of serial dependence among the peaks/extreme precipitation events selected using the AES approach (e.g., Koutsoyiannis 2008). Extended research is underway to alleviate the limitation of serial dependence among the selected extreme precipitation events in the AES by exploring the strategies that can form a sample with independent events (e.g., Adams *et al.* 1986).

## ACKNOWLEDGEMENTS

The authors express their gratitude to the Editor and anonymous reviewers for the constructive reviews that improved the quality of the work. The authors are also thankful to the IMD, which provided the gridded rainfall data for the study.

## DATA AVAILABILITY STATEMENT

Data cannot be made publicly available; readers should contact the corresponding author for details.

## REFERENCES

*Heuristics of Heavy-Tailed Distributions and the Obesity Index*

*.*