127 NTU Management Review Vol. 31 No. 1 Apr. 2021 2 { >0} log ( 2| ) 365 =1 1 = , (2.18) where n = (t modulus 365) + 1, since the period is 1 year (365 days). There are N years of data. The function fn (Φ2│Mt ) is the probability density function (PDF) of the mixedexponential distribution, in which α(t), β1(t), and β2(t) are truncated Fourier series as (2.12), specified by coefficients Φ2. The indicator function 1{Mt > 0} is defined as (2.19), depending on whether it rains or not. 0, Mt = 0, 1{Mt > 0} = ⎧ ⎨ ⎩ (2.19) 1, Mt > 0. 2.3.2 Order Determination of Fourier Series When fitting models, it is possible that we increase the likelihood by adding parameters, but doing so may result in overfitting. The criteria for model selection are to find the best-fitted model with the minimum number of parameters by introducing a penalty term. For both models, the orders of the Fourier series in parameters θj are determined by the AIC and BIC: AIC = 2k - 2 l n(Lmax) , (2.20) BIC = ln(n)k - 2 l n(Lmax) , (2.21) where k is the number of estimated coefficients, n is the number of data points (sample size), and Lmax is the maximum value of the likelihood function. The difference between the two criteria is the penalty term. For either the AIC or BIC, the model with the lowest criterion is preferred. According to Stowasser (2012) and Wilks (2011), the BIC is generally more preferable. 3. The Model of Pricing Precipitation Derivatives 3.1 Simulation Methods of Precipitation 3.1.1 Simulation of the Occurrence Model The occurrence model could be simulated recursively with the help of a uniform
RkJQdWJsaXNoZXIy ODg3MDU=