Solution: Using Equations (6.3) and (6.4), we can graph the Plackett copula function and its density function in Figure 6.1 using u, v ∈ [0, 1]u,v∈01. From the copula density function plots with different parameters in Figure 6.1, it is seen that (i) the density is higher if both u and v take on smaller or bigger values at the same time for θ = 20θ=20, i.e., high follows high and low follows low as the representation of positive dependence; (ii) the density is constant, i.e., 1, if θ = 1θ=1 for the independent random variables; and (iii) the negative dependence is observed from the density function plot for θ = 0.5θ=0.5, in this case, smaller u and bigger v reach higher density and vice versa.

Figure 6.1 Plackett copula function and its density function plot for θ = 20, θ = 1 and θ = 0.5θ=20,θ=1andθ=0.5.

To generate the variables, use the following information:

1. 
   

   1. Simulate Plackett random variables from the uniformly distributed independent random variables w₁ = 0.1645w1=0.1645, w₂ = 0.9629w2=0.9629, and θ = 50θ=50.

   
   
2. 
   

   2. Given θ = 50θ=50, θ = 2.5θ=2.5, and θ = 0.1θ=0.1, graph the the random variables generated from the Plackett copula with a sample size of 100.

Solution: We can use the procedure discussed in Section 6.1.2 to generate the random variables from Plackett copula:

1. 
   

   1. w₁ = 0.1645w1=0.1645, w₂ = 0.9629w2=0.9629, and θ = 50θ=50.

   
   

   Set u = w₁ = 0.1645u=w1=0.1645. We may then compute the random variate vv using w₂ = C(v| U = u; θ)w2=CvU=uθ.

   
   

   Solving Equation (6.8), we have the following:

   
   
   
   

   S = 0.0357; b = 135.7723; c = 75.8700; d = 69.6972

   S=0.0357;b=135.7723;c=75.8700;d=69.6972.
   Then we have the following:
   
   v=c−1−2w2d2b=75.8700−1−20.962969.69722135.7723=0.5170
   
   
   
   

   Thus, the generated random variables are (u, v) = (0.1645, 0.5170)uv=0.16450.5170.

   
   
2. 
   

   2. Set θ = 50θ=50, θ = 2.5θ=2.5 and θ = 0.1θ=0.1 with a sample size of 100.

   
   

   Using the same procedure as in step 1, we graph the simulated random variables with a sample size of 100 in Figure 6.2. Again, Figure 6.2 clearly shows that (i) the random variables generated are positively dependent with θ = 50θ=50; (ii) the random variables generated are negatively dependent with θ = 0.1θ=0.1; and (iii) the random variables generated are more scattered within [0, 1]² that are near independent when θ = 2.5θ=2.5.

Figure 6.2 Scatter plot of simulated random variables from the Plackett copula.

Solution: With the assumption of X following the Gumbel distribution (Equation (2.10)) and Y following the gamma distribution (Equation (2.8)), applying MLE, we can initially estimate the parameters of random variables X and Y as follows:

• 
  

  Random variable X: μ_X = 14.9358; σ_X = 3.8484μX=14.9358;σX=3.8484.

  
  
• 
  

  Random variable Y: α_Y = 4.0031; β_Y = 0.3668αY=4.0031;βY=0.3668.

In addition, using Equation (3.72), we can compute the sample Kendall correlation coefficient as τ_n = 0.3690τn=0.3690.

1. 
   

   1. Full ML Method:

   
   

   As discussed in Section 3.6.1, we will need to estimate the parameters of marginal distributions and copula function simultaneously with the full log-likelihood function given as follows:

   
   
   LL=∑ilncplackett(FXNormalxiμXσXFYGammayiαYβYθ+∑ilnfXNormalxiμXσX+∑ilnfYGammayiαYβY
   
   
   
   

   Using the parameters initially estimated for marginal distributions and assuming the initial estimate of the Plackett copula parameter θ = 10θ=10, we can use optimization toolbox in MATLAB to estimate the full set of parameters. The fitted marginal distribution is listed in Table 6.3 with the estimated parameters listed in Table 6.4.

   
   
2. 
   

   2. IFM Method:

   
   

   As discussed in Section 3.6.2, the parameters of marginal distributions and copulas are estimated separately with the use of IFM method. We will first compute the cumulative probability using the parameters initially estimated for the marginal distributions listed in Table 6.3. Then we will estimate the parameter of the Plackett copula using the ML method (the optimization toolbox in MATLAB) and the computed cumulative probabilities as random variates as follows:

   
   
   LL=∑ilncplackettF̂Xxiμ̂Xσ̂XF̂Yyiα̂Yβ̂Yθ
   
   
   
   

   The estimated copula parameter is listed in Table 6.4.

   
   
3. 
   

   3. Semiparametric Method:

   
   

   As discussed in Section 3.6.3, the semiparametric method is also called the pseudo-ML method. The marginal distributions are estimated nonparametrically using the Weibull plotting-position formula (Equation (3.92)) as listed in Table 6.3. Now with the use of the probability estimated nonparametrically, the pseudo-log-likelihood function can be written as follows:

   
   
   LL=∑ilncplackettF̂nxiF̂nyiθ
   
   
   
   

   The estimated parameter is again estimated using the optimization toolbox in MATLAB and listed in Table 6.4. From Table 6.4, it is seen that there is minimal difference in regard to the parameters of the marginal distributions estimated separately from the copula using the IFM method and those estimated simultaneously using the full ML method. Figure 6.3 further indicates this similarity through the univariate probability density comparison. Figure 6.4 compares the observed variates with the simulated variates from the fitted copula function. Figure 6.4 shows that the performances are very similar for the copulas with parameters estimated using three different techniques.