A total of 26 linear combinations (models) of location and scale parameters of GEV distribution were prepared using covariates ENSO, IOD, and AMO termed as C1, C2, and C3, respectively. Models formulated through various combinations of these covariates (M0, M1, M2, …) can be found in Table 3. Where M0 is the stationary model with constant parameters of GEV distribution; hence, independent of the LSCOs. However, total 26 nonstationary models (Table 3), starting from M1 to M26, in which scale and location parameters of GEV distribution have been represented as a function of covariates, are incorporated in this study. The shape parameter of the GEV distribution has been kept constant to avoid complexity in modelling (Coles 2001; Katz 2013; Yilmaz & Perera 2014; Das et al. 2020). The maximum-likelihood estimation (MLE) approach was used for parameter estimation from GEV distribution. Through MLE approach, the value of θ = [μ, σ, ξ] has been obtained at which maximum likelihood functions attain their maximum value. Let X = x1, x2, x3, …., x(n − 1), x(n) be the series of any selected extreme with n number of observations. Then, the log-likelihood can be defined as (Katz 2013; Bracken et al. 2018):
(2)
Here, L(θ) is the likelihood function of a particular parameter vector θ. The likelihood function is the measure of how likely the dataset represents as a function of unknown parameters of distribution (GEV, in the present case) and MLE gives the values of parameters that maximize the likelihood function (Katz 2013).
Table 3

List of total 27 models (1 stationary (M0) and 26 nonstationary) analysed in the study

Model IDDescriptionCombinations of LSCOs (For location (μ) & scale (σ) parameters of GEV distribution)
M0 X̴̴̴GEV[ μ, σ, ξNA
M1 X̴̴̴GEV[ μ0+μ1c1, σ, ξENSO
M2 X̴̴̴GEV[ μ0+μ2c2, σ, ξIOD
M3 X̴̴̴GEV[ μ0+μ3c3, σ, ξAMO
M4 X̴̴̴GEV[ μ0+μ1c1+μ2c2,σ, ξENSO, IOD
M5 X̴̴̴GEV[ μ0+μ2c2+μ3c3,σ, ξIOD, AMO
M6 X̴̴̴GEV[ μ0+μ3c3+μ1c1,σ, ξAMO, ENSO
M7 X̴̴̴GEV[ μ0+μ1c1+μ2c2+μ3c3, σ, ξENSO, IOD, AMO
M8 X̴̴̴GEV[(μ0+μ1c1), (σ0+σ1C1), ξENSO & ENSO
M9 X̴̴̴GEV[(μ0+μ1c1), (σ0+σ2C2), ξENSO & IOD
M10 X̴̴̴GEV[(μ0+μ1c1), (σ0+σ3C3), ξENSO & AMO
M11 X̴̴̴GEV[(μ0+μ2c2), (σ0+σ1C1), ξIOD & ENSO
M12 X̴̴̴GEV[(μ0+μ2c2), (σ0+σ2C2), ξIOD & IOD
M13 X̴̴̴GEV[(μ0+μ2c2), (σ0+σ3C3), ξIOD & AMO
M14 X̴̴̴GEV[(μ0+μ3c3), (σ0+σ1C1), ξAMO & ENSO
M15 X̴̴̴GEV[(μ0+μ3c3), (σ0+σ2C2), ξAMO & IOD
M16 X̴̴̴GEV[(μ0+μ3c3), (σ0+σ3C3), ξAMO & AMO
M17 X̴̴̴GEV[(μ0+μ1c1+μ2c2), (σ0+σ1C1+σ2C2), ξENSO, IOD & ENSO, IOD
M18 X̴̴̴GEV[(μ0+μ1c1+μ2c2), (σ0+σ2C2+σ3C3), ξENSO, IOD & IOD, AMO
M19 X̴̴̴GEV[(μ0+μ1c1+μ2c2), (σ0+σ3C3+σ1C1), ξENSO, IOD & AMO, ENSO
M20 X̴̴̴GEV[(μ0+μ2c2+μ3c3), (σ0+σ2C2+σ1C1), ξIOD, AMO & IOD, ENSO
M21 X̴̴̴GEV[(μ0+μ2c2+μ3c3), (σ0+σ2C2+σ3C3), ξIOD, AMO & IOD, AMO
M22 X̴̴̴GEV[(μ0+μ2c2+μ3c3), (σ0+σ3C3+σ1C1), ξIOD, AMO & AMO, ENSO
M23 X̴̴̴GEV[(μ0+μ3c3+μ1c1), (σ0+σ1C1+σ2C2), ξAMO, ENSO & ENSO, IOD
M24 X̴̴̴GEV[(μ0+μ3c3+μ3c1), (σ0+σ2C2+σ3C3), ξAMO, ENSO & IOD, AMO
M25 X̴̴̴GEV[(μ0+μ3c3+μ1c1), (σ0+σ3C3+σ1C1), ξAMO, ENSO & AMO, ENSO
M26 X̴̴̴GEV[(μ0+μ1c1+μ2c2+μ3c3), (σ0+σ1C1+σ2C2+σ3C3), ξENSO, IOD, AMO & ENSO, IOD, AMO
Model IDDescriptionCombinations of LSCOs (For location (μ) & scale (σ) parameters of GEV distribution)
M0 X̴̴̴GEV[ μ, σ, ξNA
M1 X̴̴̴GEV[ μ0+μ1c1, σ, ξENSO
M2 X̴̴̴GEV[ μ0+μ2c2, σ, ξIOD
M3 X̴̴̴GEV[ μ0+μ3c3, σ, ξAMO
M4 X̴̴̴GEV[ μ0+μ1c1+μ2c2,σ, ξENSO, IOD
M5 X̴̴̴GEV[ μ0+μ2c2+μ3c3,σ, ξIOD, AMO
M6 X̴̴̴GEV[ μ0+μ3c3+μ1c1,σ, ξAMO, ENSO
M7 X̴̴̴GEV[ μ0+μ1c1+μ2c2+μ3c3, σ, ξENSO, IOD, AMO
M8 X̴̴̴GEV[(μ0+μ1c1), (σ0+σ1C1), ξENSO & ENSO
M9 X̴̴̴GEV[(μ0+μ1c1), (σ0+σ2C2), ξENSO & IOD
M10 X̴̴̴GEV[(μ0+μ1c1), (σ0+σ3C3), ξENSO & AMO
M11 X̴̴̴GEV[(μ0+μ2c2), (σ0+σ1C1), ξIOD & ENSO
M12 X̴̴̴GEV[(μ0+μ2c2), (σ0+σ2C2), ξIOD & IOD
M13 X̴̴̴GEV[(μ0+μ2c2), (σ0+σ3C3), ξIOD & AMO
M14 X̴̴̴GEV[(μ0+μ3c3), (σ0+σ1C1), ξAMO & ENSO
M15 X̴̴̴GEV[(μ0+μ3c3), (σ0+σ2C2), ξAMO & IOD
M16 X̴̴̴GEV[(μ0+μ3c3), (σ0+σ3C3), ξAMO & AMO
M17 X̴̴̴GEV[(μ0+μ1c1+μ2c2), (σ0+σ1C1+σ2C2), ξENSO, IOD & ENSO, IOD
M18 X̴̴̴GEV[(μ0+μ1c1+μ2c2), (σ0+σ2C2+σ3C3), ξENSO, IOD & IOD, AMO
M19 X̴̴̴GEV[(μ0+μ1c1+μ2c2), (σ0+σ3C3+σ1C1), ξENSO, IOD & AMO, ENSO
M20 X̴̴̴GEV[(μ0+μ2c2+μ3c3), (σ0+σ2C2+σ1C1), ξIOD, AMO & IOD, ENSO
M21 X̴̴̴GEV[(μ0+μ2c2+μ3c3), (σ0+σ2C2+σ3C3), ξIOD, AMO & IOD, AMO
M22 X̴̴̴GEV[(μ0+μ2c2+μ3c3), (σ0+σ3C3+σ1C1), ξIOD, AMO & AMO, ENSO
M23 X̴̴̴GEV[(μ0+μ3c3+μ1c1), (σ0+σ1C1+σ2C2), ξAMO, ENSO & ENSO, IOD
M24 X̴̴̴GEV[(μ0+μ3c3+μ3c1), (σ0+σ2C2+σ3C3), ξAMO, ENSO & IOD, AMO
M25 X̴̴̴GEV[(μ0+μ3c3+μ1c1), (σ0+σ3C3+σ1C1), ξAMO, ENSO & AMO, ENSO
M26 X̴̴̴GEV[(μ0+μ1c1+μ2c2+μ3c3), (σ0+σ1C1+σ2C2+σ3C3), ξENSO, IOD, AMO & ENSO, IOD, AMO
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