6.1.1 Definition of Bivariate Plackett Copula

As discussed in Chapter 3, the Plackett copula is constructed using the algebraic method. The cross-product ratio θθ, or odds ratio, is a measure of “association” or “dependence” in 2 × 22×2 contingency tables. Here, we label the categories for each variable as “low” and “high” and give four categories in Table 6.1, where a, b, c, and d represent the observed counts in the four categories, respectively. From Table 6.1, the cross-product ratio (θ : θ>0)θ:θ>0) is defined as θ=adbc. Following Palaro and Hotta (2006), the dependence may be explained through θθ as follows:

1. 
   

   1. 0 < θ < 10<θ<1 corresponds to negative dependence, i.e., observations are more concentrated in the “low-high” and “high-low” cells.

   
   
2. 
   

   2. θ = 1θ=1 corresponds to independence, each “observed” entry; for example, a is equal to its “expected value” under independence [i.e.,a+ba+ca+b+c+d].

   
   
3. 
   

   3. θ>1θ>1 corresponds to positive dependence, i.e., observations are more concentrated in the “low-low” and “high-high” cells.

With the use of the 2 × 22×2 contingency table, Plackett (1965) developed what is now called the Plackett copula for bivariate continuous random variables. Assuming the continuous random variables X and Y with marginals F_X and F_YFXandFY and the joint distribution function H(x, y) = P(X ≤ x, Y ≤ y)Hxy=PX≤xY≤y, then the “low” and “high” categories for the column and row variables are replaced by events X ≤ x, X>xX≤x,X>x and Y ≤ y, Y>yY≤y,Y>y, respectively. According to the definition of cross-product ratio θ=adbc, it is clear that a, b, c, and d denote the probabilities of P(X ≤ x, Y ≤ y), P(X>x, Y ≤ y), P(X ≤ x, Y>y),PX≤xY≤y,PX>xY≤y,PX≤xY>y, and P(X>x, Y>y)PX>xY>y, respectively.

Now, based on the bivariate probability relation discussed in Chapter 3, we have the following:

Replacing the values of a, b, c, and d, we obtain the expression of parameter θ as follows:

θ=Hxy1−FXx−FYy+HxyFXx−HxyFYy−Hxy(6.1e)

Let u = F_X(x)u=FXx and v = F_Y(y)v=FYy. Equation (6.1e) may be written in the copula form by applying Sklar’s theorem as follows:

θ=Cuv1−u−v+Cuvu−Cuvv−Cuv(6.2)

Solving for C in Equation (6.2), we obtain the Plackett copula:

Cuvθ=1+θ−1u+v−1+θ−1u+v2−4θθ−1uv2θ−1;θ>0&θ≠1(6.3a)

Taking the partial derivatives with respect to u and v, its copula density function can be written as follows:

cuvθ=∂2Cuvθ∂u∂v=θ1+θ−1u+v−2uv1+θ−1u+v2−4θθ−1uv1.5(6.4)

Taking the partial derivative of equation (6.3a) with respect to u or v, the conditional probability distributions can be obtained as follows:

CV≤vU=u=PY≤yX=x=∂Cuvθ∂u=12+−1+u+v−uθ+vθ21+θ−1u+v2−4θθ−1uv(6.5)

CU≤uV=v=PX≤xY=y=∂Cuvθ∂v=12+−1+u+v+uθ−vθ21+θ−1u+v2−4θθ−1uv(6.6)

Example 6.1 Graph the Plackett copula function and its density function with θ = 20, θ = 1, and θ = 0.5θ=20,θ=1,andθ=0.5.

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.

6.1.2 Simulation of Bivariate Plackett Copula

Following the Rosenblatt transform (Rosenblatt, 1952), the random variable can be simulated as follows:

1. 
   

   1. Simulate two independent random variables (w₁, w₂)w1w2 from the uniform distribution U(0, 1)U01.

   
   
2. 
   

   2. Set u = w₁u=w1.

   
   
3. 
   

   3. Using Equation (6.5a) and set w₂ = C(v| u)w2=Cvu, i.e.,

   
   
   w2=∂Cuvθ∂u=12+−1+u+v−uθ+vθ21+θ−1u+v2−4θθ−1uv(6.7)
   
   

After some algebraic manipulation of Equation (6.7), vv can be solved as follows:

v=c−1−2w2d2b(6.8)

where

Example 6.2 Generate the random variables from the Plackett copula function.

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.

6.1.3 Parameter Estimation for Bivariate Plackett Copulas

As discussed in Section 3.6, the full ML, IFM, and semiparametric (pseudo-ML) methods may be applied to estimate the parameter numerically for the Plackett copula function. Here, without further discussion, we will give one example to illustrate the procedure of parameter estimation.

Example 6.3 Using the random variables (Table 6.2) and assuming (a) random variables X and Y are sampled from the normal distribution and gamma distribution, respectively, and (b) the joint distribution may be modeled using Plackett copula, estimate the parameters using full ML, IFM, and semiparametric methods.

Table 6.2. Sample data for Example 6.3.

No.	X	Y	No.	X	Y
1	11.276	5.049	26	12.793	12.942
2	19.570	12.015	27	16.772	4.140
3	10.864	3.691	28	12.215	4.522
4	14.517	9.233	29	24.909	7.689
5	17.512	6.862	30	17.580	12.331
6	14.312	5.343	31	17.200	7.060
7	17.785	12.689	32	10.621	5.583
8	9.457	8.182	33	10.310	19.026
9	13.290	8.531	34	8.957	3.648
10	15.470	31.129	35	18.735	7.534
11	18.392	20.848	36	11.536	7.519
12	9.411	8.567	37	16.264	10.727
13	18.883	15.874	38	21.382	21.947
14	11.749	12.142	39	19.153	11.813
15	14.173	10.224	40	17.355	7.988
16	14.044	6.223	41	17.877	12.159
17	13.032	7.594	42	14.799	9.622
18	18.374	14.827	43	11.457	11.147
19	17.979	14.283	44	18.601	14.626
20	7.656	4.639	45	11.636	4.732
21	14.642	10.039	46	11.427	6.263
22	19.871	16.856	47	15.067	11.378
23	7.769	17.575	48	16.328	14.778
24	12.870	7.763	49	21.471	29.678
25	14.119	6.964	50	15.327	9.639

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.

Table 6.3. Cumulative probability computed using the fitted normal and gamma distributions and Weibull probability plotting-position formula.

X	FMLE	IFM	Empirical	Y	FMLE	IFM	Empirical
19.570	0.871	0.882	0.900	12.015	0.638	0.633	0.640
10.864	0.129	0.141	0.160	3.691	0.047	0.045	0.040
14.517	0.427	0.449	0.460	9.233	0.434	0.428	0.460
17.512	0.724	0.742	0.680	6.862	0.242	0.237	0.220
14.312	0.406	0.428	0.440	5.343	0.132	0.129	0.140
17.785	0.747	0.764	0.720	12.689	0.680	0.675	0.720
9.457	0.067	0.075	0.100	8.182	0.348	0.343	0.400
13.290	0.308	0.328	0.360	8.531	0.377	0.371	0.420
15.470	0.526	0.548	0.560	31.129	0.996	0.996	0.980
18.392	0.795	0.810	0.800	20.848	0.946	0.944	0.920
9.411	0.065	0.073	0.080	8.567	0.380	0.374	0.440
18.883	0.829	0.843	0.860	15.874	0.830	0.827	0.840
11.749	0.183	0.199	0.260	12.142	0.646	0.641	0.660
14.173	0.392	0.414	0.420	10.224	0.512	0.506	0.540
14.044	0.380	0.401	0.380	6.223	0.193	0.189	0.180
13.032	0.284	0.304	0.340	7.594	0.300	0.295	0.320
18.374	0.794	0.809	0.780	14.827	0.789	0.785	0.820
17.979	0.763	0.780	0.760	14.283	0.764	0.760	0.760
7.656	0.025	0.028	0.020	4.639	0.091	0.088	0.100
14.642	0.440	0.462	0.480	10.039	0.498	0.492	0.520
19.871	0.887	0.897	0.920	16.856	0.863	0.860	0.860
7.769	0.026	0.030	0.040	17.575	0.883	0.881	0.880
12.870	0.270	0.289	0.320	7.763	0.314	0.309	0.360
14.119	0.387	0.409	0.400	6.964	0.250	0.245	0.240
12.793	0.264	0.282	0.300	12.942	0.694	0.690	0.740
16.772	0.656	0.676	0.620	4.140	0.066	0.064	0.060
12.215	0.217	0.234	0.280	4.522	0.085	0.082	0.080
24.909	0.994	0.995	0.980	7.689	0.308	0.303	0.340
17.580	0.730	0.748	0.700	12.331	0.658	0.653	0.700
17.200	0.696	0.715	0.640	7.060	0.257	0.253	0.260
10.621	0.116	0.127	0.140	5.583	0.148	0.145	0.160
10.310	0.101	0.111	0.120	19.026	0.916	0.914	0.900
8.957	0.052	0.058	0.060	3.648	0.045	0.044	0.020
18.735	0.819	0.833	0.840	7.534	0.295	0.290	0.300
11.536	0.169	0.184	0.220	7.519	0.294	0.289	0.280
16.264	0.607	0.628	0.580	10.727	0.549	0.544	0.560
21.382	0.945	0.951	0.940	21.947	0.959	0.958	0.940
19.153	0.847	0.859	0.880	11.813	0.625	0.620	0.620
17.355	0.710	0.729	0.660	7.988	0.332	0.327	0.380
17.877	0.755	0.772	0.740	12.159	0.647	0.642	0.680
14.799	0.456	0.478	0.500	9.622	0.465	0.459	0.480
11.457	0.164	0.178	0.200	11.147	0.579	0.574	0.580
18.601	0.810	0.824	0.820	14.626	0.780	0.776	0.780
11.636	0.175	0.191	0.240	4.732	0.096	0.093	0.120
11.427	0.162	0.176	0.180	6.263	0.196	0.192	0.200
15.067	0.484	0.506	0.520	11.378	0.596	0.590	0.600
16.328	0.613	0.634	0.600	14.778	0.787	0.783	0.800
21.471	0.948	0.953	0.960	29.678	0.995	0.994	0.960
15.327	0.511	0.533	0.540	9.639	0.466	0.461	0.500

Table 6.4. Estimated parameters using the preceding three methods.

Method	Univariate		Copula
	X~normal	Y~Gamma	θθ	LL
Full ML	(15.224, 3.846)	(4.039, 0.369)	7.500	–275.327
IFM	(15.011, 3.851)	(4.069, 0.369)	7.167	8.106
Semiparametric	—	—	7.759	8.464

Figure 6.3 Comparison of frequency and the fitted probability distributions using IFM and Full MLE.

Figure 6.4 Comparison of observations with simulated random variables with three estimation methods.

Example 6.4 Using the sample data and the parameters estimated with the IFM method in Example 6.3, compute the joint return period and conditional return period of

T(X>19 ∩ Y>21), T(X>19| Y>21), T(X>19| Y = 21)

TX>19∩Y>21,TX>19Y>21,TX>19Y=21.

Solution: Applying the parameters estimated for the marginal distributions listed in Table 6.4 for the IFM method, we have

F_X(X ≤ 19; μ = 15.011, σ = 3.852)≈0.850 (Normal distribution)

FXX≤19μ=15.011σ=3.852≈0.850(Normaldistribution)

F_Y(Y ≤ 21; α = 4.069, β = 2.712)≈0.946 (Gamma distribution)

FYY≤21α=4.069β=2.712≈0.946(Gammadistribution)

1. 
   

   i.  T(X>19 ∩ Y>21)TX>19∩Y>21

   
   

   In this case, we are evaluating the recurrence interval if both X and Y exceed the value given in the preceding. Applying Equation (3.127) for the “and” case, we have the following:

   
   
   F(X>19  ∩​  Y>21)=1−FX(X≤19)−FY(Y≤21)+F(X≤19,Y≤21)=1−FX(X≤19)−FY(Y≤21)+Cplackett(FX(X≤19),FY(Y≤21);θ)=1−0.850−0.946+Cplackett(0.850,0.946;7.167)=1−0.850−0.946+0.824=0.028
   
   
   TX>19∩Y>21=1FX>19∩Y>21=10.028=36.10time units
   
   
   
   
2. 
   

   ii. T(X>19| Y>21)TX>19Y>21

   
   

   In this case, we are evaluating the recurrence interval of X > 19 under the condition of Y > 21:

   
   
   F(X>19|Y>21)=F(X>19∩​Y>21)F(Y>21)=1−FX(X≤19)−FY(Y≤21)+Cplackett(FX(X≤19),FY(Y≤21);θ)1−F(Y≤21)=1−0.850−0.946+0.8241−0.946=0.0280.054=0.516
   
   
   
   

   Applying Equation (2.91), we have the following:

   
   
   TX>19Y>21=11−FYy≤2111−FXx≤19−FYY≤21+CplackettFXFYθ=10.053710.0277≈672time units
   
   
   
   
3. 
   

   iii. T(X>19| Y = 21)TX>19Y=21

In this case, we are evaluating the recurrence interval of X > 19 under the condition that Y is exactly equal to 21:

F(X>19|Y=21)=1−F(X≤19|Y=21)=1−∂C(FX(X≤19),FY(Y≤21);θ)∂FY(Y≤21)=1−0.528=0.472T(X>19|Y=21)=1F(X>19|Y=21)=10.472≈2.12(time units)

Comparing the joint return period with the two conditional return periods we calculated, it is seen that the recurrence interval is longest for the conditional return period of (X>19| Y>21).X>19Y>21.

6.2.1 Definition of Cross-Product Ratio for the Trivariate Plackett Copula

For the given (u, v, wu,v,w), there are three compatible bivariate copulas: C_UV, C_VWCUV,CVW, and C_UWCUW. Analogous to the bivariate case, the trivariate constant cross-product ratio θ_UVWθUVW can be defined following Kao and Govindaraju (2008), Song and Singh (2010) as

θUVW=P000P011P101P110P111P100P010P001(6.9)

where

P000=CUVWuvwP100=CVWvw−CUVWuvwP010=CUWuw−CUVWuvwP001=CUVuv−CUVWuvwP110=w−CUVuw−CVWvw+CUVWuvwP101=v−CUVuv−CVWvw+CUVWuvwP011=u−CUVuv−CVWuw+CUVWuvwP111=1−u−v−w+CUVuv+CVWvw+CUWuw−CUVWuvw(6.10)

Here C_UV, C_VWCUV,CVW, and C_UWCUW are bivariate Plackett copulas with dependence parameters θ_UV, θ_VWθUV,θVW, and θ_UWθUW, for the given θ_UVWθUVW. Denoting z = C_UVW(u, v, w)z=CUVWuvw, one can compute C_UVW(u, v, w)CUVWuvw as follows:

where

a1=CVWvw,a2=CUWuw,a3=CUVuva4=1−u−v−w+CUVuv+CVWvw+CUWuwb1=CUWuw+CVWvw−wb2=CUVuv+CVWvw−vb3=CUWuw+CUVuv−u(6.12)

For the given θ_UV, θ_VW, θ_UWθUV,θVW,θUW, and θ_UVWθUVW, the corresponding trivariate Plackett copula may be obtained from Equations (6.11) and (6.12). For C_UVW(u, v, w)CUVWuvw to be a valid three-copula, the following conditions needs to be satisfied:

1. 
   

   1. Since each component in Equation (6.11) is a probability measure, we have the following:

   
   
   
   

   C_UVW(u, v, w) ∈ [b, a], b =  max (0, b₁, b₂, b₃); a =  min (a₁, a₂, a₃, a₄)

   CUVWuvw∈ba,b=max0b1b2b3;a=mina1a2a3a4(6.13)
   
   
   
   
2. 
   

   2. Equation (6.13) is the Fréchet–Hoeffding bounds for trivairate joint distributions with the known bivariate joint distributions (Joe, 1997).

   
   
3. 
   

   3. As discussed in Section 3.1.2 of Chapter 3, Equations (3.23)–(3.26) need to be satisfied.

   
   
4. 
   

   4. The copula density is CUVWuvw=∂3CUVWuvw∂u∂v∂w≥0. Following Kao and Govidaraju (2008), the derivation of the density function will be discussed in Section 6.2.2.

With the fulfillment of the preceding four conditions, for the given cross-product ratio parameters θ_UV, θ_VW, θ_UWθUV,θVW,θUW, and θ_UVWθUVW, z = C_UVW(u, v, w)z=CUVWuvw may be computed numerically with the following steps:

1. 
   

   1. Compute C_UV, C_VW, and C_UW using Equation (6.3).

   
   
2. 
   

   2. To compute C_UVWCUVW, Equation (5.11) can be rewritten as follows:

   
   
   θUVW−1z4+−θUVWa1+a2+a3+a4+(b1+b2+b3]2+θUVWa1a2+a1+a2a3+a4+a3a4−b1b2+b3b1+b2z2+−θUVWa1a2a3+a4+a3a4a1+a2+b1b2b3z+θUVWa1a2a3a4=0(6.14)
   Let f(z) represent the left side of Equation (6.14). We may use Newton’s iterative method to compute z numerically as follows:
   
   zn+1=zn−fznf’zn(6.15)
   where f^‘(z)f′z is the first derivative of f(z)fz with respect to z; z_nzn and z_n + 1zn+1 are the nth and (n+1)th iteratively computed values of z.
   

6.2.2 Derivation of Density Function of the Trivariate Plackett Copula

Following Kao and Govindaraju (2008), the density function of trivariate Plackett copula may be derived in the following manner:

1. 
   

   1. Solve C_UVWCUVW using given parameter θ_UVWθUVW and known bivariate copulas from Equation (6.11) or equivalently Equation (6.14).

   
   
2. 
   

   2. Compute first-order derivatives of ∂CUV∂u,∂CUV∂v,∂CUW∂u,∂CUW∂w,∂CVW∂v, and ∂CVW∂w from the corresponding known bivariate copulas. Similar to the vine copula discussed in Chapter 4, these bivariate copulas are not required to belong to the Plackett copula family, and each may belong to a different copula family.

   
   
3. 
   

   3. Compute the first-order derivatives of P₀₀₀, P₀₁₀, P₁₀₀, P₀₁₁, P₁₁₀, P₁₀₁, P₀₁₁P000,P010,P100,P011,P110,P101,P011, and P₁₁₁P111 with respect to u, v, w, respectively, as follows:

   
   
   ∂P000∂u=−∂P100∂u=∂CUVW∂u∂P010∂u=−∂P110∂u=−∂CUVW∂u+∂CUW∂u∂P001∂u=−∂P101∂u=−∂CUVW∂u+∂CUV∂u∂P011∂u=−∂P111∂u=∂CUVW∂u−∂CUV∂u−∂CUW∂u+1(6.16)
   
   
   ∂P000∂v=−∂P010∂v=∂CUVW∂v∂P100∂v=−∂P110∂v=−∂CUVW∂v+∂CVW∂v∂P001∂v=−∂P011∂v=−∂CUVW∂v+∂CUV∂v∂P101∂v=−∂P111∂v=∂CUVW∂v−∂CUV∂v−∂CVW∂v+1(6.17)
   
   
   ∂P000∂w=−∂P001∂w=∂CUVW∂w∂P100∂w=−∂P101∂w=−∂CUVW∂w+∂CVW∂w∂P010∂w=−∂P011∂w=−∂CUVW∂w+∂CUW∂w∂P110∂w=−∂P111∂w=∂CUVW∂w−∂CUW∂w−∂CVW∂w+1(6.18)
   
   
   
   
4. 
   

   4. Compute ∂CUVW∂u,∂CUVW∂v,∂CUVW∂w as follows:

   
   
   ∂P000∂uP011P101P110+P000∂P011∂uP101P110+P000P011∂P101∂uP110+P000P011P101∂P110∂u–θUVW(∂P111∂uP100P010P001+P111∂P100∂uP010P001+P111P100∂P010∂uP001+P111P110P101∂P110∂u)=0(6.19)
   
   
   ∂P000∂vP011P101P110+P000∂P011∂vP101P110+P000P011∂P101∂vP110+P000P011P101∂P110∂v–θUVW(∂P111∂vP100P010P001+P111∂P100∂vP010P001+P111P100∂P010∂vP001+P111P110P101∂P110∂v)=0(6.20)
   
   
   ∂P000∂wP011P101P110+P000∂P011∂wP101P110+P000P011∂P101∂wP110+P000P011P101∂P110∂w–θUVW(∂P111∂wP100P010P001+P111∂P100∂wP010P001+P111P100∂P010∂wP001+P111P110P101∂P110∂w)=0(6.21)
   
   
   
   
5. 
   

   5. Compute the bivariate density function of c_UV, c_VW, c_UWcUV,cVW,cUW.

   
   
6. 
   

   6. Compute the second-order derivative of P₀₀₀, P₀₁₀, P₁₀₀, P₀₁₁, P₁₁₀, P₁₀₁, P₀₁₁P000,P010,P100,P011,P110,P101,P011, and P₁₁₁P111 with respect to u, v, w, respectively, as follows:

   
   
   ∂2P000∂u∂v=−∂2P100∂u∂v=−∂2P010∂u∂v=∂2P110∂u∂v=∂2CUVW∂u∂v∂2P001∂u∂v=−∂2P101∂u∂v=−∂2P011∂u∂v=∂2P111∂u∂v=−∂2CUVW∂u∂v+∂2CUV∂u∂v(6.22)
   
   
   ∂2P000∂u∂w=−∂2P100∂u∂w=−∂2P001∂u∂w=∂2P101∂u∂w=∂2CUVW∂u∂w∂2P010∂u∂w=−∂2P110∂u∂w=−∂2P011∂u∂w=∂2P111∂u∂v=−∂2CUVW∂u∂w+∂2CUW∂u∂w(6.23)
   
   
   ∂2P000∂v∂w=−∂2P010∂v∂w=−∂2P001∂v∂w=∂2P011∂v∂w=∂2CUVW∂v∂w∂2P100∂v∂w=−∂2P110∂v∂w=−∂2P101∂v∂w=∂2P111∂v∂w=−∂2CUVW∂v∂w+∂2CVW∂v∂w(6.24)
   
   
   
   
7. 
   

   7. Compute ∂2CUVW∂u∂v,∂2CUVW∂v∂w,∂2CUVW∂u∂w. As an example, ∂2CUVW∂u∂v may be computed by applying ∂∂v to Equation (6.19) as follows:

   
   
   ∂2P000∂u∂vP011P101P110+∂P000∂u∂P011∂vP101P110+∂P000∂uP011∂P101∂vP110+∂P000∂vP011P101∂P110∂v+∂P000∂v∂P011∂uP101P110+P000∂2P011∂u∂vP101P110+P000∂P011∂u∂P101∂vP110+P000∂P011∂uP101∂P110∂v+∂P000∂vP011∂P101∂uP110+P000∂P011∂v∂P101∂uP110+P000P011∂2P101∂u∂vP110+P000P011P101∂u∂P110∂v+∂P000∂vP011P101∂P110∂u+P000∂P011∂vP101∂P110∂u+P000P011∂P101∂v∂P110∂u+P000P011P101∂2P110∂u∂v−θUVW(∂2P111∂u∂vP100P010P001+∂P111∂u∂P100∂vP010P001+∂P111∂uP100∂P010∂vP001+∂P111∂uP100P010∂P001∂v+∂P111∂v∂P100∂uP010P001+P111∂2P100∂u∂vP010P001+P111∂P100∂u∂P010∂vP001+P111∂P100∂uP010∂P001∂v+∂P111∂vP100∂P010∂uP001+P111∂P100∂v∂P010∂uP001+P111P100∂2P010∂u∂vP001+P111P100∂P010∂u∂P001∂v+∂P111∂vP100P010∂P001∂u+P111∂P100∂vP010∂P001∂u+P111P100∂P010∂v∂P001∂u+P111P100P010∂2P001∂u∂v)(6.25)
   
   
   
   

   Similarly, applying ∂∂w,∂∂u, we can obtain ∂2CUVW∂v∂w,∂2CUVW∂u∂w from Equations (6.20) and (6.21), respectively.

   
   
8. 
   

   8. Compute the probability density function ∂3CUVW∂u∂v∂w for the trivariate Plackett copula.

   
   

   Applying ∂∂w to Equation (6.22), we have the following:

   
   
   ∂3P000∂u∂v∂w=−∂3P100∂u∂v∂w=−∂3P010∂u∂v∂w=∂3P110∂u∂v∂w=∂3CUVW∂u∂v∂w∂3P001∂u∂v∂w=−∂3P101∂u∂v∂w=−∂3P011∂u∂v∂w=∂3P111∂u∂v∂w=−∂3CUVW∂u∂v∂w(6.26)
   
   
   
   

   Applying ∂∂w to Equation (6.25), we obtain a new third-order derivative equation (we omit the derivative here). Substituting Equation (6.26) into the new equation derived for the third-order derivative, we have the density function as a function of P₀₀₀, P₀₁₁, P₁₀₁, P₁₁₀, P₁₁₁, P₀₁₀, P₀₁₀, P₀₀₁P000,P011,P101,P110,P111,P010,P010,P001.

Example 6.5 Express the PDF of trivariate Plackette copula with the following information:θ_UVW = 20; θ_UV = 15; θ_UW = 1.3; θ_VW = 1.4; u = 0.5; v = 0.975; w = 0.975θUVW=20;θUV=15;θUW=1.3;θVW=1.4;u=0.5;v=0.975;w=0.975

Solution: Applying the equations derived for the trivariate Plackett copula, we can compute the trivariate Plackett copula density function by following these procedure and steps:

1. 
   

   1. Compute the bivariate Plackett copula for the paired variables with Equation (6.3); using bivariate variable (u, v)uv as an example, we have the following:

   
   
   a3=CUV=CUV(0.5,0.975;15)        =[1+(15−1)(0.5+0.975)]−[1+(15−1)(0.5+0.975)]2−4(15)(15−1)(0.5)(0.975)2(15−1)=0.498
   
   
   
   

   Similarly, we have the following:

   
   
   
   

   a₂ = C_UW = C_UW(0.5, 0.975; 1.3) = 0.489

   a2=CUW=CUW0.50.9751.3=0.489
   
   
   
   

   a₁ = C_VW = C_VW(0.975, 0.975; 1.4) = 0.951

   a1=CVW=CVW0.9750.9751.4=0.951
   
   
   
   
2. 
   

   2. Compute the trivariate Plackett copula value using Equation (6.14), and solve it numerically as follows: C_UVW(0.5, 0.975, 0.975; [15, 1.4, 1.3, 20]) = 0.488CUVW0.50.9750.975151.41.320=0.488

   
   

   where the remaining a^‘sa′s and b^‘sb′s needed in Equation (6.12) are computed as follows:

   
   
   
   

   a₄ = 1 − 0.5 − 0.975 − 0.975 + 0.498 + 0.589 + 0.951 = 0.488

   a4=1−0.5−0.975−0.975+0.498+0.589+0.951=0.488
   
   
   
   

   b₁ = 0.489 + 0.951 − 0.975 = 0.465

   b1=0.489+0.951−0.975=0.465
   
   
   
   

   b₂ = 0.498 + 0.951 − 0.975 = 0.474

   b2=0.498+0.951−0.975=0.474
   
   
   
   

   b₃ = 0.498 + 0.489 − 0.5 = 0.487

   b3=0.498+0.489−0.5=0.487
   
   
   
   
3. 
   

   3. Compute the derivatives needed to compute the trivariate Plackett density:

   
   
   
   

   P₀₀₀ = C_UVW = 0.488, P₁₀₀ = C_VW − C_UVW = 0.463,

   P000=CUVW=0.488,P100=CVW−CUVW=0.463,
   
   
   
   

   P₀₁₀ = C_UW − C_UVW = 7.903 × 10⁻⁴,  P₀₀₁ = C_UV − C_UVW = 0.0074,

   P010=CUW−CUVW=7.903×10−4,P001=CUV−CUVW=0.0074,
   
   
   
   

   P₁₁₀ = w − C_UW − C_VW + C_UVW =  − 0.0234,  P₁₀₁ = v − C_UV − C_VW + C_UVW = 0.017

   P110=w−CUW−CVW+CUVW=−0.0234,P101=v−CUV−CVW+CUVW=0.017
   
   
   
   

   P₀₁₁ = u − C_UV − C_UW + C_UVW = 0.004

   P011=u−CUV−CUW+CUVW=0.004
   
   
   
   

   P₁₁₁ = 1 − u − v − w + C_UV + C_VW + C_UW − C_UVW =  − 0.003

   P111=1−u−v−w+CUV+CVW+CUW−CUVW=−0.003
   
   
   
   

   The rest computation will need to apply the numerical method (i.e., Newton’s method). Here we will only lists the final results:

   
   
   ∂2CUV∂u∂v=0.594;∂2CUW∂u∂w=0.985;∂2CVW∂v∂w=1.348
   
   
   ∂P000∂u=0.015;∂P111∂v=0.697;∂P111∂w=1.9461
   
   
   ∂CUVW∂u=0.946;∂CUVW∂v=−0.555;∂CUVW∂w=1.542
   
   
   ∂2CUVW∂u∂v=9.101;∂2CUVW∂u∂w=29.094;∂2CUVW∂v∂w=458.057
   
   

Finally, we have the trivariate Plackett copula density as follows:

cUVW=∂3CUVW∂u∂v∂w=8.412.

6.2.3 Estimation of Cross-Product Ratio (Copula Parameter) for the Trivariate Plackett Copula

Following the same procedure for the bivariate Plackett copula, the parameter for the trivariate Plackett copula may be estimated. For a trivariate sample of X = {x_i1, x_i2, x_i3, i = 1, …n}X=xi1xi2xi3i=1…n with u_i1 = F₁(x_i1), u_i2 = F₂(x_i2), u_i3 = F3(x_i3),ui1=F1xi1,ui2=F2xi2,ui3=F3xi3, we can then write the pseudo-MLE as follows:

LθUVW=∑i=1nlogcUVWui1ui2ui3θUVW(6.27)

Taking the derivative with respect to θ_UVWθUVW and setting the derivative equal to 0, we have the following:

1n∂L∂θUVW=1n∑i=1n∂logcUVWui1ui2ui3θUVW∂θUVW=0(6.27a)

As shown in the previous section, the trivariate Plackett copula does not have an analytical form of the trivariate Plackett copula density function, and the parameter may be optimized by the numerical scheme (e.g., central differencing). Compared to the bivariate case, the parameter estimation of the trivariate Plackett copula is more tedious. It holds true, compared to the asymmetric Archimedean, vine, and meta-elliptical copulas.