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Expected Cumulants and Correlations for Continuous Mixture Variables

Mixture distributions describe random variables that are drawn from more than one component distribution. Mixture distributions provide a useful way for describing heterogeneity in a population, especially when an outcome is a composite response from multiple sources. For a random variable \(Y\) from a finite mixture distribution with \(k\) components, the PDF can be described by:
\[\begin{equation} h_{Y}(y) = \sum_{i=1}^{k} \pi_{i} f_{Y_{i}}(y), \tag{.} \end{equation}\]

where \(\sum_{i=1}^{k} \pi_{i} = 1\). The \(\pi_{i}\) are mixing parameters which determine the weight of each component distribution \(f_{Y_{i}}(y)\) in the overall probability distribution. As long as each component has a valid PDF, the overall distribution \(h_{Y}(y)\) has a valid PDF. The main assumption is statistical independence between the process of randomly selecting the component distribution and the distributions themselves. Assume there is a random selection process that first generates the numbers \(1,\ ...,\ k\) with probabilities \(\pi_{1},\ ...,\ \pi_{k}\). After selecting number \(i\), where \(1 \leq i \leq k\), a random variable \(y\) is drawn from component distribution \(f_{Y_{i}}(y)\) (Davenport, Bezder, and Hathaway 1988; Schork, Allison, and Thiel 1996; Everitt 1996; Pearson 2011).

Continuous mixture variables are generated at the component level in SimCorrMix. The target correlation matrix rho in the simulation functions corrvar and corrvar2 is specified in terms of the correlations with components of continuous mixture variables. This allows the user to set the correlation between components of the same mixture variable to any desired value. If this correlation is set to \(0\), the intermediate correlation matrix Sigma may need to be converted to the nearest positive-definite matrix. This is done within the simulation functions by specifying use.nearPD = TRUE. Higham (2002)’s algorithm is executed with the Matrix::nearPD function (Bates and Maechler 2018). Otherwise, negative eigenvalues are replaced with \(0\).

1 Expected Cumulants of Continuous Mixture Variables

The components of the continuous mixture variables are created using either Fleishman (1978)’s third-order or Headrick (2002)’s fifth-order power method transformation (PMT) applied to standard normal variables. The PMT simulates continuous variables by matching standardized cumulants derived from central moments. Since some distributions may have large central moments, using standardized cumulants decreases the complexity involved in calculations. In view of this, let \(Y\) be a real-valued random variable with cumulative distribution function \(F\). Define the central moments, \({\mu}_{r}\), of \(Y\) as:
\[\begin{equation} {\mu}_{r} = {\mu}_{r}(Y) = E{[y-\mu]}^{r} = \int_{-\infty}^{+\infty}{[y-\mu]}^{r}dF(y). \tag{1.1} \end{equation}\]

Then the first six cumulants are given by (Kendall and Stuart 1977):

\[\begin{align} \begin{split} {\kappa}_{1} &= {\mu}_{1} = 0\ \ \ \ \ \ & &{\kappa}_{4} &= {\mu}_{4} - 3{{\mu}_{2}}^{2} \\ {\kappa}_{2} &= {\mu}_{2} & &{\kappa}_{5} &= {\mu}_{5} - 10{\mu}_{3}{\mu}_{2} \\ {\kappa}_{3} &= {\mu}_{3} & &{\kappa}_{6} &= {\mu}_{6} - 15{\mu}_{4}{\mu}_{2} - 10{{\mu}_{3}}^{2} + 30{{\mu}_{2}}^. \end{split} \tag{1.2} \end{align}\]

The cumulants are standardized by dividing \({\kappa}_{1}\) - \({\kappa}_{6}\) by \(\sqrt{{{\kappa}_{2}}^{r}} = ({\sigma}^{2})^{r/2} = {\sigma}^{r}\), where \({\sigma}^{2}\) is the variance of \(Y\) and \(r\) is the order of the cumulant:

\[\begin{align} \begin{split} 0 &= \frac{{\kappa}_{1}}{\sqrt{{{\kappa}_{2}}^{1}}} = \frac{{\mu}_{1}}{{\sigma}^{1}} &\ \ \ \ \ \ &{\gamma}_{2} &= \frac{{\kappa}_{4}}{\sqrt{{{\kappa}_{2}}^{4}}} = \frac{{\mu}_{4}}{{\sigma}^{4}} - 3 \\ 1 &= \frac{{\kappa}_{2}}{\sqrt{{{\kappa}_{2}}^{2}}} = \frac{{\mu}_{2}}{{\sigma}^{2}} &\ \ \ \ \ \ &{\gamma}_{3} &= \frac{{\kappa}_{5}}{\sqrt{{{\kappa}_{2}}^{5}}} = \frac{{\mu}_{5}}{{\sigma}^{5}} - 10{\gamma}_{1} \\ {\gamma}_{1} &= \frac{{\kappa}_{3}}{\sqrt{{{\kappa}_{2}}^{3}}} = \frac{{\mu}_{3}}{{\sigma}^{3}} &\ \ \ \ \ \ &{\gamma}_{4} &= \frac{{\kappa}_{6}}{\sqrt{{{\kappa}_{2}}^{6}}} = \frac{{\mu}_{6}}{{\sigma}^{6}} - 15{\gamma}_{2} - 10{{\gamma}_{1}}^{2} - 15. \end{split} \tag{1.3} \end{align}\]

The values \({\gamma}_{1}\) and \({\gamma}_{2}\) correspond to skewness and standardized kurtosis (so that the normal distribution has a value of 0, hereafter referred to as skurtosis). The corresponding sample values for the above can be obtained by replacing \({\mu}_{r}\) by \({m}_{r} = \sum_{j=1}^{n}{({x}_{j}-{m}_{1})}^{r}/n\) (Headrick 2002).

The standardized cumulants for a continuous mixture variable can be derived in terms of the standardized cumulants of its component distributions. Suppose the goal is to simulate a continuous mixture variable \(Y\) with PDF \(h_{Y}(y)\) that contains two component distributions \(Y_a\) and \(Y_b\) with mixing parameters \(\pi_a\) and \(\pi_b\):

\[\begin{equation} h_{Y}(y) = \pi_a f_{Y_a}(y) + \pi_b g_{Y_b}(y),\ y\ \in\ \bm{Y},\ \pi_a\ \in\ (0,\ 1),\ \pi_b\ \in\ (0,\ 1),\ \pi_a + \pi_b = 1. \tag{1.4} \end{equation}\] Here,
\[\begin{equation} \begin{split} Y_a &= \sigma_a Z_a' + \mu_a,\ Y_a \sim f_{Y_a}(y),\ y\ \in\ \bm{Y_a} \\ Y_b &= \sigma_b Z_b' + \mu_b,\ Y_b \sim g_{Y_b}(y),\ y\ \in\ \bm{Y_b} \end{split} \tag{1.5} \end{equation}\]

so that \(Y_a\) and \(Y_b\) have expected values \(\mu_a\) and \(\mu_b\) and variances \(\sigma_a^2\) and \(\sigma_b^2\). Assume the variables \(Z_a'\) and \(Z_b'\) are generated with zero mean and unit variance using Headrick’s fifth-order PMT given the specified values for skew (\(\gamma_{1_a}'\), \(\gamma_{1_b}'\)), skurtosis (\(\gamma_{2_a}'\), \(\gamma_{2_b}'\)), and standardized fifth (\(\gamma_{3_a}'\), \(\gamma_{3_b}'\)) and sixth (\(\gamma_{4_a}'\), \(\gamma_{4_b}'\)) cumulants:

\[\begin{equation} \begin{split} Z_a' &= c_{0_a} + c_{1_a}Z_a + c_{2_a}Z_a^2 + c_{3_a}Z_a^3 + c_{4_a}Z_a^4 + c_{5_a}Z_a^5,\ Z_a \sim N(0,\ 1) \\ Z_b' &= c_{0_b} + c_{1_b}Z_b + c_{2_b}Z_b^2 + c_{3_b}Z_b^3 + c_{4_b}Z_b^4 + c_{5_b}Z_b^5,\ Z_b \sim N(0,\ 1). \end{split} \tag{1.6} \end{equation}\]

The constants \(c_{0_a},\ ...,\ c_{5_a}\) and \(c_{0_b},\ ...,\ c_{5_b}\) are the solutions to the system of equations given in SimMultiCorrData::poly and calculated by SimMultiCorrData::find_constants. Similar results hold for Fleishman’s third-order PMT, where the constants \(c_{0_a},\ ...,\ c_{3_a}\) and \(c_{0_b},\ ...,\ c_{3_b}\) are the solutions to the system of equations given in SimMultiCorrData::fleish and \(c_{4_a} = c_{5_a} = c_{4_b} = c_{5_b} = 0\) (Fialkowski 2018).

The \(r^{th}\) expected value of \(Y\) can be expressed as:

\[\begin{equation} \begin{split} E_{h}[Y^r] &= \int y^r h_{Y}(y) dy = \pi_a \int y^r f_{Y_a}(y) dy + \pi_b \int y^r g_{Y_b}(y) dy \\ &= \pi_a E_{f}[Y_a^r] + \pi_b E_{g}[Y_b^r]. \end{split} \tag{1.7} \end{equation}\]

This expression can be used to derive expressions for the mean, variance, skew, skurtosis, and standardized fifth and sixth cumulants of \(Y\) in terms of the \(r^{th}\) expected values of \(Y_a\) and \(Y_b\).

1. Mean: Using \(r = 1\) yields:

\[\begin{equation} \begin{split} E_{h}[Y] &= \pi_a E_{f}[Y_a] + \pi_b E_{g}[Y_b] \\ &= \pi_a E_{f}[\sigma_a Z_a' + \mu_a] + \pi_b E_{g}[\sigma_b Z_b' + \mu_b] \\ &= \pi_a (\sigma_a E_{f}[Z_a'] + \mu_a) + \pi_b (\sigma_b E_{g}[Z_b'] + \mu_b). \end{split} \tag{1.8} \end{equation}\]

Since \(E_{f}[Z_a'] = E_{g}[Z_b'] = 0\), this becomes \(E_{h}[Y] = \pi_a \mu_a + \pi_b \mu_b\).

2. Variance: The variance of \(Y\) can be expressed by the relation \(Var_{h}[Y] = E_{h}[Y^2] - {(E_{h}[Y])}^2\). Using \(r = 2\) yields:

\[\begin{equation} \begin{split} E_{h}[Y^2] &= \pi_a E_{f}[Y_a^2] + \pi_b E_{g}[Y_b^2] \\ &= \pi_a E_{f}[{(\sigma_a Z_a' + \mu_a)}^2] + \pi_b E_{g}[{(\sigma_b Z_b' + \mu_b)}^2] \\ &= \pi_a E_{f}[\sigma_a^2 {Z_a'}^2 + 2\mu_a\sigma_aZ_a' + \mu_a^2] + \pi_b E_{g}[\sigma_b^2 {Z_b'}^2 + 2\mu_b\sigma_bZ_b' + \mu_b^2] \\ &= \pi_a (\sigma_a^2 E_{f}[{Z_a'}^2] + 2\mu_a\sigma_aE_{f}[Z_a'] + \mu_a^2) + \pi_b (\sigma_b^2 E_{g}[{Z_b'}^2] + 2\mu_b\sigma_bE_{g}[Z_b'] + \mu_b^2). \end{split} \tag{1.9} \end{equation}\]

Applying the variance relation to \(Z_a'\) and \(Z_b'\) gives:

\[\begin{equation} \begin{split} E_{f}[{Z_a'}^2] &= Var_{f}[Z_a'] + {(E_{f}[Z_a'])}^2 \\ E_{g}[{Z_b'}^2] &= Var_{g}[Z_b'] + {(E_{g}[Z_b'])}^2. \end{split} \tag{1.10} \end{equation}\] Since \(E_{f}[Z_a'] = E_{g}[Z_b'] = 0\) and \(Var_{f}[Z_a'] = Var_{g}[Z_b'] = 1\), \(E_{f}[{Z_a'}^2]\) and \(E_{g}[{Z_b'}^2]\) both equal \(1\).
Therefore, \(E_{h}[Y^2]\) simplifies to:
\[\begin{equation} E_{h}[Y^2] = \pi_a (\sigma_a^2 + \mu_a^2) + \pi_b (\sigma_b^2 + \mu_b^2), \tag{1.11} \end{equation}\] and the variance of \(Y\) is given by:
\[\begin{equation} Var[Y] = \pi_a (\sigma_a^2 + \mu_a^2) + \pi_b (\sigma_b^2 + \mu_b^2) - [\pi_a \mu_a + \pi_b \mu_b]^2. \tag{1.12} \end{equation}\]

3. Skew: Using Headrick (2002)’s expression, the skew of \(Y\) is given by \(\gamma_{1} = \frac{{\mu}_{3}}{{\sigma}^{3}} = \frac{{\mu}_{3}}{{(\sigma^2)}^{3/2}}\), where \(\sigma^2\) is the variance of \(Y\) and \({\mu}_{3} = E_{h}[Y^3]\). Using \(r = 3\) yields:

\[\begin{equation} \begin{split} E_{h}[Y^3] &= \pi_a E_{f}[Y_a^3] + \pi_b E_{g}[Y_b^3] \\ &= \pi_a E_{f}[{(\sigma_a Z_a' + \mu_a)}^3] + \pi_b E_{g}[{(\sigma_b Z_b' + \mu_b)}^3] \\ &= \pi_a E_{f}[\sigma_a^3 {Z_a'}^3 + 3\sigma_a^2\mu_a{Z_a'}^2 + 3\sigma_a\mu_a^2 Z_a' + \mu_a^3] \\ &\ \ \ \ + \pi_b E_{g}[\sigma_b^3 {Z_b'}^3 + 3\sigma_b^2\mu_b{Z_b'}^2 + 3\sigma_b\mu_b^2 Z_b' + \mu_b^3] \\ &= \pi_a ( \sigma_a^3 E_{f}[{Z_a'}^3] + 3\sigma_a^2\mu_aE_{f}[{Z_a'}^2] + 3\sigma_a\mu_a^2E_{f}[Z_a'] + \mu_a^3)\\ &\ \ \ \ + \pi_b (\sigma_b^3 E_{g}[{Z_b'}^3] + 3\sigma_b^2\mu_bE_{g}[{Z_b'}^2] + 3\sigma_b\mu_b^2 E_{g}[Z_b'] + \mu_b^3). \end{split} \tag{1.13} \end{equation}\]

Then \(E_{f}[{Z_a'}^3] = {\mu}_{3_a}'\) and \(E_{g}[{Z_b'}^3] = {\mu}_{3_b}'\) are given by:

\[\begin{equation} \begin{split} E_{f}[{Z_a'}^3] &= {(Var_{f}[Z_a'])}^{3/2} \gamma_{1_a}' = \gamma_{1_a}' \\ E_{g}[{Z_b'}^3] &= {(Var_{g}[Z_b'])}^{3/2} \gamma_{1_b}' = \gamma_{1_b}'. \end{split} \tag{1.14} \end{equation}\] Combining these with \(E_{f}[Z_a'] = E_{g}[Z_b'] = 0\) and \(E_{f}[{Z_a'}^2] = E_{g}[{Z_b'}^2] = 1\), \(E_{h}[Y^3]\) simplifies to:
\[\begin{equation} E_{h}[Y^3] = \pi_a ( \sigma_a^3 \gamma_{1_a}' + 3\sigma_a^2\mu_a + \mu_a^3) + \pi_b (\sigma_b^3 \gamma_{1_b}' + 3\sigma_b^2\mu_b + \mu_b^3). \tag{1.15} \end{equation}\] Therefore, the skew of \(Y\) is:
\[\begin{equation} \gamma_{1} = \frac{\pi_a ( \sigma_a^3 \gamma_{1_a}' + 3\sigma_a^2\mu_a + \mu_a^3) + \pi_b (\sigma_b^3 \gamma_{1_b}' + 3\sigma_b^2\mu_b + \mu_b^3)}{{(\pi_a (\sigma_a^2 + \mu_a^2) + \pi_b (\sigma_b^2 + \mu_b^2) - [\pi_a \mu_a + \pi_b \mu_b]^2)}^{3/2}}. \tag{1.16} \end{equation}\]

4. Skurtosis: Using Headrick (2002)’s expression, the standardized kurtosis of \(Y\) is given by \(\gamma_{2} = \frac{{\mu}_{4}}{{\sigma}^{4}} - 3 = \frac{{\mu}_{4}}{{(\sigma^2)}^2} - 3\), where \(\sigma^2\) is the variance of \(Y\) and \({\mu}_{4} = E_{h}[Y^4]\). Using \(r = 4\) yields:

\[\begin{equation} \begin{split} E_{h}[Y^4] &= \pi_a E_{f}[Y_{a}^4] + \pi_b E_{g}[Y_{b}^4] \\ &= \pi_a E_{f}[{(\sigma_{a} Z_{a}' + \mu_{a})}^4] + \pi_b E_{g}[{(\sigma_{b} Z_{b}' + \mu_{b})}^4] \\ &= \pi_a E_{f}[\sigma_{a}^4 {Z_{a}'}^4 + 4\sigma_{a}^3\mu_{a}{Z_{a}'}^3 + 6\sigma_{a}^2\mu_{a}^2{Z_{a}'}^2 + 4\sigma_{a}\mu_{a}^3 Z_{a}' + \mu_{a}^4] \\ &\ \ \ \ + \pi_b E_{g}[\sigma_{b}^4 {Z_{b}'}^4 + 4\sigma_{b}^3\mu_{b}{Z_{b}'}^3 + 6\sigma_{b}^2\mu_{b}^2{Z_{b}'}^2 + 4\sigma_{b}\mu_{b}^3 Z_{b}' + \mu_{b}^4] \\ &= \pi_a (\sigma_{a}^4 E_{f}[{Z_{a}'}^4] + 4\sigma_{a}^3\mu_{a}E_{f}[{Z_{a}'}^3] + 6\sigma_{a}^2\mu_{a}^2E_{f}[{Z_{a}'}^2] + 4\sigma_{a}\mu_{a}^3 E_{f}[Z_{a}'] + \mu_{a}^4) \\ &\ \ \ \ + \pi_b (\sigma_{b}^4 E_{g}[{Z_{b}'}^4] + 4\sigma_{b}^3\mu_{b}E_{g}[{Z_{b}'}^3] + 6\sigma_{b}^2\mu_{b}^2E_{g}[{Z_{b}'}^2] + 4\sigma_{b}\mu_{b}^3 E_{g}[Z_{b}'] + \mu_{b}^4). \end{split} \tag{1.17} \end{equation}\]

Then \(E_{f}[{Z_{a}'}^4] = {\mu}_{4_{a}}'\) and \(E_{g}[{Z_{b}'}^4] = {\mu}_{4_{b}}'\) are given by:

\[\begin{equation} \begin{split} E_{f}[{Z_{a}'}^4] &= {(Var_{f}[Z_{a}'])}^2 (\gamma_{2_{a}}' + 3) = \gamma_{2_{a}}' + 3 \\ E_{g}[{Z_{b}'}^4] &= {(Var_{g}[Z_{b}'])}^2 (\gamma_{2_{b}}' + 3) = \gamma_{2_{b}}' + 3. \end{split} \tag{1.18} \end{equation}\]

Since \(E_{f}[Z_{a}'] = E_{g}[Z_{b}'] = 0\) and \(E_{f}[{Z_{a}'}^2] = E_{g}[{Z_{b}'}^2] = 1\), \(E_{h}[Y^4]\) simplifies to:

\[\begin{equation} \begin{split} E_{h}[Y^4] &= \pi_a (\sigma_{a}^4(\gamma_{2_{a}}' + 3) + 4\sigma_{a}^3\mu_{a}\gamma_{1_{a}}' + 6\sigma_{a}^2\mu_{a}^2 + \mu_{a}^4) \\ &\ \ \ \ + \pi_b (\sigma_{b}^4(\gamma_{2_{b}}' + 3) + 4\sigma_{b}^3\mu_{b}\gamma_{1_{b}}' + 6\sigma_{b}^2\mu_{b}^2 + \mu_{b}^4). \end{split} \tag{1.19} \end{equation}\]

Therefore, the skurtosis of \(Y\) is:

\[\begin{equation} \begin{split} \gamma_{2} &= \frac{\pi_a (\sigma_{a}^4(\gamma_{2_{a}}' + 3) + 4\sigma_{a}^3\mu_{a}\gamma_{1_{a}}' + 6\sigma_{a}^2\mu_{a}^2 + \mu_{a}^4)}{{(\pi_a (\sigma_a^2 + \mu_a^2) + \pi_b (\sigma_b^2 + \mu_b^2) - [\pi_a \mu_a + \pi_b \mu_b]^2)}^2} \\ \\ &\ \ \ \ + \frac{\pi_b (\sigma_{b}^4(\gamma_{2_{b}}' + 3) + 4\sigma_{b}^3\mu_{b}\gamma_{1_{b}}' + 6\sigma_{b}^2\mu_{b}^2 + \mu_{b}^4)}{{(\pi_a (\sigma_a^2 + \mu_a^2) + \pi_b (\sigma_b^2 + \mu_b^2) - [\pi_a \mu_a + \pi_b \mu_b]^2)}^2}. \end{split} \tag{1.20} \end{equation}\]

5. Fifth Cumulant: Using Headrick (2002)’s expression, the standardized fifth cumulant of \(Y\) is given by \(\gamma_{3} = \frac{{\mu}_{5}}{{\sigma}^{5}} - 10{\gamma}_{1} = \frac{{\mu}_{5}}{{(\sigma^2)}^{5/2}} - 10{\gamma}_{1}\), where \(\sigma^2\) is the variance of \(Y\) and \({\mu}_{5} = E_{h}[Y^5]\). Using \(r = 5\) yields:

\[\begin{equation} \begin{split} E_{h}[Y^5] &= \pi_a E_{f}[Y_{a}^5] + \pi_b E_{g}[Y_{b}^5] \\ &= \pi_a E_{f}[{(\sigma_{a} Z_{a}' + \mu_{a})}^5] + \pi_b E_{g}[{(\sigma_{b} Z_{b}' + \mu_{b})}^5] \\ &= \pi_a E_{f}[\sigma_{a}^5 {Z_{a}'}^5 + 5\sigma_{a}^4\mu_{a}{Z_{a}'}^4 + 10\sigma_{a}^3\mu_{a}^2{Z_{a}'}^3 + 10\sigma_{a}^2\mu_{a}^3{Z_{a}'}^2 + 5\sigma_{a}\mu_{a}^4 Z_{a}' + \mu_{a}^5] \\ &\ \ \ \ + \pi_b E_{g}[\sigma_{b}^5 {Z_{b}'}^5 + 5\sigma_{b}^4\mu_{b}{Z_{b}'}^4 + 10\sigma_{b}^3\mu_{b}^2{Z_{b}'}^3 + 10\sigma_{b}^2\mu_{b}^3{Z_{b}'}^2 + 5\sigma_{b}\mu_{b}^4 Z_{b}' + \mu_{b}^5] \\ &= \pi_a (\sigma_{a}^5 E_{f}[{Z_{a}'}^5] + 5\sigma_{a}^4\mu_{a}E_{f}[{Z_{a}'}^4] + 10\sigma_{a}^3\mu_{a}^2E_{f}[{Z_{a}'}^3] + 10\sigma_{a}^2\mu_{a}^3E_{f}[{Z_{a}'}^2] + 5\sigma_{a}\mu_{a}^4 E_{f}[Z_{a}'] + \mu_{a}^5) \\ &\ \ \ \ + \pi_b (\sigma_{b}^5 E_{g}[{Z_{b}'}^5] + 5\sigma_{b}^4\mu_{b}E_{g}[{Z_{b}'}^4] + 10\sigma_{b}^3\mu_{b}^2E_{g}[{Z_{b}'}^3] + 10\sigma_{b}^2\mu_{b}^3E_{g}[{Z_{b}'}^2] + 5\sigma_{b}\mu_{b}^4 E_{g}[Z_{b}'] + \mu_{b}^5). \end{split} \tag{1.21} \end{equation}\]

Then \(E_{f}[{Z_{a}'}^5] = {\mu}_{5_{a}}'\) and \(E_{g}[{Z_{b}'}^5] = {\mu}_{5_{b}}'\) are given by:

\[\begin{equation} \begin{split} E_{f}[{Z_{a}'}^5] &= {(Var_{f}[Z_{a}'])}^{5/2} (\gamma_{3_{a}}' + 10\gamma_{1_{a}}') = \gamma_{3_{a}}' + 10\gamma_{1_{a}}' \\ E_{g}[{Z_{b}'}^5] &= {(Var_{g}[Z_{b}'])}^{5/2} (\gamma_{3_{b}}' + 10\gamma_{1_{b}}') = \gamma_{3_{b}}' + 10\gamma_{1_{b}}'. \end{split} \tag{1.22} \end{equation}\]

Since \(E_{f}[Z_{a}'] = E_{g}[Z_{b}'] = 0\) and \(E_{f}[{Z_{a}'}^2] =\) \(E_{g}[{Z_{b}'}^2] = 1\), \(E_{h}[Y^5]\) simplifies to:

\[\begin{equation} \begin{split} E_{h}[Y^5] &= \pi_a (\sigma_{a}^5(\gamma_{3_{a}}' + 10\gamma_{1_{a}}') + 5\sigma_{a}^4\mu_{a}(\gamma_{2_{a}}' + 3) + 10\sigma_{a}^3\mu_{a}^2\gamma_{1_{a}}' + 10\sigma_{a}^2\mu_{a}^3 + \mu_{a}^5) \\ &\ \ \ \ + \pi_b (\sigma_{b}^5(\gamma_{3_{b}}' + 10\gamma_{1_{b}}') + 5\sigma_{b}^4\mu_{b}(\gamma_{2_{b}}' + 3) + 10\sigma_{b}^3\mu_{b}^2\gamma_{1_{b}}' + 10\sigma_{b}^2\mu_{b}^3 + \mu_{b}^5). \end{split} \tag{1.23} \end{equation}\]

Therefore, the standardized fifth cumulant of \(Y\) is:

\[\begin{equation} \begin{split} \gamma_{3} &= \frac{\pi_a (\sigma_{a}^5(\gamma_{3_{a}}' + 10\gamma_{1_{a}}') + 5\sigma_{a}^4\mu_{a}(\gamma_{2_{a}}' + 3) + 10\sigma_{a}^3\mu_{a}^2\gamma_{1_{a}}' + 10\sigma_{a}^2\mu_{a}^3 + \mu_{a}^5)}{{(\pi_a (\sigma_a^2 + \mu_a^2) + \pi_b (\sigma_b^2 + \mu_b^2) - [\pi_a \mu_a + \pi_b \mu_b]^2)}^{5/2}} \\ \\ &\ \ \ \ + \frac{\pi_b (\sigma_{b}^5(\gamma_{3_{b}}' + 10\gamma_{1_{b}}') + 5\sigma_{b}^4\mu_{b}(\gamma_{2_{b}}' + 3) + 10\sigma_{b}^3\mu_{b}^2\gamma_{1_{b}}' + 10\sigma_{b}^2\mu_{b}^3 + \mu_{b}^5)}{{(\pi_a (\sigma_a^2 + \mu_a^2) + \pi_b (\sigma_b^2 + \mu_b^2) - [\pi_a \mu_a + \pi_b \mu_b]^2)}^{5/2}} - 10{\gamma}_{1}. \end{split} \tag{1.24} \end{equation}\]

6. Sixth Cumulant: Using Headrick (2002)’s expression, the standardized sixth cumulant of \(Y\) is given by \(\gamma_{4} = \frac{{\mu}_{6}}{{\sigma}^{6}} - 15{\gamma}_{2} - 10{{\gamma}_{1}}^{2} - 15 = \frac{{\mu}_{6}}{{(\sigma^2)}^3} - 15{\gamma}_{2} - 10{{\gamma}_{1}}^{2} - 15\), where \(\sigma^2\) is the variance of \(Y\) and \({\mu}_{6} = E_{h}[Y^6]\). Using \(r = 6\) yields:

\[\begin{equation} \begin{split} E_{h}[Y^6] &= \pi_a E_{f}[Y_{a}^6] + \pi_b E_{g}[Y_{b}^6] \\ &= \pi_a E_{f}[{(\sigma_{a} Z_{a}' + \mu_{a})}^6] + \pi_b E_{g}[{(\sigma_{b} Z_{b}' + \mu_{b})}^6] \\ &= \pi_a E_{f}[\sigma_{a}^6 {Z_{a}'}^6 + 6\sigma_{a}^5\mu_{a}{Z_{a}'}^5 + 15\sigma_{a}^4\mu_{a}^2{Z_{a}'}^4 + 20\sigma_{a}^3\mu_{a}^3{Z_{a}'}^3 + 15\sigma_{a}^2\mu_{a}^4{Z_{a}'}^2 + 6\sigma_{a}\mu_{a}^5 Z_{a}' + \mu_{a}^6] \\ &\ \ \ \ + \pi_b E_{g}[\sigma_{b}^6 {Z_{b}'}^6 + 6\sigma_{b}^5\mu_{b}{Z_{b}'}^5 + 15\sigma_{b}^4\mu_{b}^2{Z_{b}'}^4 + 20\sigma_{b}^3\mu_{b}^3{Z_{b}'}^3 + 15\sigma_{b}^2\mu_{b}^4{Z_{b}'}^2 + 6\sigma_{b}\mu_{b}^5 Z_{b}' + \mu_{b}^6] \\ &= \pi_a (\sigma_{a}^6 E_{f}[{Z_{a}'}^6] + 6\sigma_{a}^5\mu_{a}E_{f}[{Z_{a}'}^5] + 15\sigma_{a}^4\mu_{a}^2E_{f}[{Z_{a}'}^4] + 20\sigma_{a}^3\mu_{a}^3E_{f}[{Z_{a}'}^3] + 15\sigma_{a}^2\mu_{a}^4E_{f}[{Z_{a}'}^2] \\ &\ \ \ \ + 6\sigma_{a}\mu_{a}^5 E_{f}[Z_{a}'] + \mu_{a}^6) \\ &\ \ \ \ + \pi_b (\sigma_{b}^6 E_{g}[{Z_{b}'}^6] + 6\sigma_{b}^5\mu_{b}E_{g}[{Z_{b}'}^5] + 15\sigma_{b}^4\mu_{b}^2E_{g}[{Z_{b}'}^4] + 20\sigma_{b}^3\mu_{b}^3E_{g}[{Z_{b}'}^3] + 15\sigma_{b}^2\mu_{b}^4E_{g}[{Z_{b}'}^2] \\ &\ \ \ \ + 6\sigma_{b}\mu_{b}^5 E_{g}[Z_{b}'] + \mu_{b}^6). \end{split} \tag{1.25} \end{equation}\]

Then \(E_{f}[{Z_{a}'}^6] = {\mu}_{6_{a}}'\) and \(E_{g}[{Z_{b}'}^6] = {\mu}_{6_{b}}'\) are given by:

\[\begin{equation} \begin{split} E_{f}[{Z_{a}'}^6] &= {(Var_{f}[Z_{a}'])}^3 (\gamma_{4_{a}}' + 15\gamma_{2_{a}}' + 10{\gamma_{1_{a}}'}^2 + 15) = \gamma_{4_{a}}' + 15\gamma_{2_{a}}' + 10{\gamma_{1_{a}}'}^2 + 15 \\ E_{g}[{Z_{b}'}^6] &= {(Var_{g}[Z_{b}'])}^3 (\gamma_{4_{b}}' + 15\gamma_{2_{b}}' + 10{\gamma_{1_{b}}'}^2 + 15) = \gamma_{4_{b}}' + 15\gamma_{2_{b}}' + 10{\gamma_{1_{b}}'}^2 + 15. \end{split} \tag{1.26} \end{equation}\]

Since \(E_{f}[Z_{a}'] = E_{g}[Z_{b}'] = 0\) and \(E_{f}[{Z_{a}'}^2] = E_{g}[{Z_{b}'}^2] = 1\), \(E_{h}[Y^6]\) simplifies to:

\[\begin{equation} \begin{split} E_{h}[Y^6] &= \pi_a (\sigma_{a}^6(\gamma_{4_{a}}' + 15\gamma_{2_{a}}' + 10{\gamma_{1_{a}}'}^2 + 15) + 6\sigma_{a}^5\mu_{a}(\gamma_{3_{a}}' + 10\gamma_{1_{a}}') + 15\sigma_{a}^4\mu_{a}^2(\gamma_{2_{a}}' + 3) + 20\sigma_{a}^3\mu_{a}^3\gamma_{1_{a}}' \\ &\ \ \ \ + 15\sigma_{a}^2\mu_{a}^4 + \mu_{a}^6) \\ &\ \ \ \ + \pi_b (\sigma_{b}^6(\gamma_{4_{b}}' + 15\gamma_{2_{b}}' + 10{\gamma_{1_{b}}'}^2 + 15) + 6\sigma_{b}^5\mu_{b}(\gamma_{3_{b}}' + 10\gamma_{1_{b}}') + 15\sigma_{b}^4\mu_{b}^2(\gamma_{2_{b}}' + 3) + 20\sigma_{b}^3\mu_{b}^3\gamma_{1_{b}}' \\ &\ \ \ \ + 15\sigma_{b}^2\mu_{b}^4 + \mu_{b}^6). \end{split} \tag{1.27} \end{equation}\]

Therefore, the standardized sixth cumulant of \(Y\) is:

\[\begin{equation} \begin{split} \gamma_{4} &= \frac{\pi_a (\sigma_{a}^6(\gamma_{4_{a}}' + 15\gamma_{2_{a}}' + 10{\gamma_{1_{a}}'}^2 + 15) + 6\sigma_{a}^5\mu_{a}(\gamma_{3_{a}}' + 10\gamma_{1_{a}}') + 15\sigma_{a}^4\mu_{a}^2(\gamma_{2_{a}}' + 3) + 20\sigma_{a}^3\mu_{a}^3\gamma_{1_{a}}' + 15\sigma_{a}^2\mu_{a}^4 + \mu_{a}^6)}{{(\pi_a (\sigma_a^2 + \mu_a^2) + \pi_b (\sigma_b^2 + \mu_b^2) - [\pi_a \mu_a + \pi_b \mu_b]^2)}^3} \\ \\ &\ \ \ \ + \frac{\pi_b (\sigma_{b}^6(\gamma_{4_{b}}' + 15\gamma_{2_{b}}' + 10{\gamma_{1_{b}}'}^2 + 15) + 6\sigma_{b}^5\mu_{b}(\gamma_{3_{b}}' + 10\gamma_{1_{b}}') + 15\sigma_{b}^4\mu_{b}^2(\gamma_{2_{b}}' + 3) + 20\sigma_{b}^3\mu_{b}^3\gamma_{1_{b}}' + 15\sigma_{b}^2\mu_{b}^4 + \mu_{b}^6)}{{(\pi_a (\sigma_a^2 + \mu_a^2) + \pi_b (\sigma_b^2 + \mu_b^2) - [\pi_a \mu_a + \pi_b \mu_b]^2)}^3} \\ &\ \ \ \ - 15{\gamma}_{2} - 10{{\gamma}_{1}}^{2} - 15. \end{split} \tag{1.28} \end{equation}\]

Extension to more than two component distributions:

If the desired mixture distribution \(Y\) contains more than two component distributions, the expected values of \(Y\) are again expressed as sums of the expected values of the component distributions, with weights equal to the associated mixing parameters. For example, assume \(Y\) contains \(k\) component distributions \(Y_{1},\ ...,\ Y_{k}\) with mixing parameters given by \(\pi_{1},\ ...,\ \pi_{k}\), where \(\sum_{i=1}^{k} \pi_{i} = 1\). The component distributions are described by the following parameters: means \(\mu_{1},\ ...,\ \mu_{k}\), variances \(\sigma_{1}^2,\ ...,\ \sigma_{k}^2\), skews \(\gamma_{1_{1}}',\ ...,\ \gamma_{1_{k}}'\), skurtoses \(\gamma_{2_{1}}',\ ...,\ \gamma_{2_{k}}'\), fifth cumulants \(\gamma_{3_{1}}',\ ...,\ \gamma_{3_{k}}'\), and sixth cumulants \(\gamma_{4_{1}}',\ ...,\ \gamma_{4_{k}}'\). Then the \(r^{th}\) expected value of \(Y\) can be expressed as:
\[\begin{equation} E_{h}[Y^r] = \int y^r h_{Y}(y) dy = \sum_{i=1}^{k} \pi_{i} \int y^r f_{Y_{i}}(y) dy = \sum_{i=1}^{k} \pi_{i} E_{f_{i}}[Y_{i}^r]. \tag{1.29} \end{equation}\]

Therefore, a method similar to that above can be used to derive the system of equations defining the mean, variance, skew, skurtosis, and standardized fifth and sixth cumulants of \(Y\). These equations are used within the function calc_mixmoments to determine the values for a mixture variable. Some code has been modified from the SimMultiCorrData package (Fialkowski 2018).

2 Approximate Correlations for Continuous Mixture Variables:

Even though the correlations for the continuous mixture variables are set at the component level, we can approximate the resulting correlations for the mixture variables. The example from the Overall Workflow for Generation of Correlated Data vignette is used for demonstration.

Assume \(M1\) and \(M2\) are two continuous mixture variables. Let \(M1\) have \(k_{M1}\) components with mixing probabilities \(\alpha_1, ..., \alpha_{k_{M1}}\). The standard deviations of the components are \(\sigma_{M1_1}, \sigma_{M1_2}, ..., \sigma_{M1_{k_{M1}}}\). Let \(M2\) have \(k_{M2}\) components with mixing probabilities \(\beta_1, ..., \beta_{k_{M2}}\). The standard deviations of the components are \(\sigma_{M2_1}, \sigma_{M2_2}, ..., \sigma_{M2_{k_{M2}}}\).

2.1 Correlation between continuous mixture variables M1 and M2:

The correlation between the mixture variables \(M1\) and \(M2\) is given by:
\[\begin{equation} Cor(M1, M2) = \frac{E[M1M2] - E[M1]E[M2]}{\sigma_{M1} \sigma_{M2}}. \tag{2.1} \end{equation}\]

Equation (2.1) requires the expected value of the product of \(M1\) and \(M2\). Since \(M1\) and \(M2\) may contain any desired number of components and these components may have any continuous distribution, there is no general way to determine this expected value. Therefore, it will be approximated by expressing \(M1\) and \(M2\) as sums of their component variables:

\[\begin{equation} Cor(M1, M2) = \frac{E[(\sum_{i = 1}^{k_{M1}} \alpha_i M1_i) (\sum_{j = 1}^{k_{M2}} \beta_j M2_j)] - E[\sum_{i = 1}^{k_{M1}} \alpha_i M1_i]E[\sum_{j = 1}^{k_{M2}} \beta_j M2_j]}{\sigma_{M1} \sigma_{M2}}, \tag{2.2} \end{equation}\] where
\[\begin{equation} \begin{split} E[(\sum_{i = 1}^{k_{M1}} \alpha_i M1_i) (\sum_{j = 1}^{k_{M2}} \beta_j M2_j)] &= E[\alpha_1 M1_1 \beta_1 M2_1 + \alpha_1 M1_1 \beta_2 M2_2 + ... + \alpha_{k_{M1}} M1_{k_{M1}} \beta_{k_{M2}} M2_{k_{M2}}] \\ &= \alpha_1 \beta_1 E[M1_1 M2_1] + \alpha_1 \beta_2 E[M1_1 M2_2] + ... + \alpha_{k_{M1}} \beta_{k_{M2}} E[M1_{k_{M1}} M2_{k_{M2}}]. \end{split} \tag{2.3} \end{equation}\] Using the general correlation equation, for \(1 \leq i \leq k_{M1}\) and \(1 \leq j \leq k_{M2}\):
\[\begin{equation} E[M1_i M2_j] = \sigma_{M1_i} \sigma_{M2_j} Cor(M1_i, M2_j) + E[M1_i] E[M2_j], \tag{2.4} \end{equation}\]

so that we can rewrite \(Cor(M1, M2)\) as:

\[\begin{equation} \begin{split} Cor(M1, M2) &= \frac{\alpha_1 \beta_1 (\sigma_{M1_1} \sigma_{M2_1} Cor(M1_1, M2_1) + E[M1_1] E[M2_1])}{\sigma_{M1} \sigma_{M2}} \\ &\ \ \ \ + ... + \frac{\alpha_{k_{M1}} \beta_{k_{M2}} (\sigma_{M1_{k_{M1}}} \sigma_{M2_{k_{M2}}} Cor(M1_{k_{M1}}, M2_{k_{M2}}) + E[M1_{k_{M1}}] E[M2_{k_{M2}}])}{\sigma_{M1} \sigma_{M2}} \\ &- \frac{\alpha_1 \beta_1 E[M1_1] E[M2_1] + ... + \alpha_{k_{M1}} \beta_{k_{M2}} E[M1_{k_{M1}}] E[M2_{k_{M2}}]}{\sigma_{M1} \sigma_{M2}} \\ &= \frac{\sum_{i = 1}^{k_{M1}} \alpha_i \sigma_{M1_i} \sum_{j = 1}^{k_{M2}} \beta_j \sigma_{M2_j} Cor(M1_i, M2_j)}{\sigma_{M1} \sigma_{M2}}. \end{split} \tag{2.5} \end{equation}\]

For this example:

\[\begin{equation} \begin{split} Cor(M1, M2) = &\frac{\alpha_1 \sigma_{M1_1} [\beta_1 \sigma_{M2_1} Cor(M1_1, M2_1) + \beta_2 \sigma_{M2_2} Cor(M1_1, M2_2) + \beta_3 \sigma_{M2_3} Cor(M1_1, M2_3)]}{\sigma_{M1} \sigma_{M2}} \\ &+ \frac{\alpha_2 \sigma_{M1_2} [\beta_1 \sigma_{M2_1} Cor(M1_2, M2_1) + \beta_2 \sigma_{M2_2} Cor(M1_2, M2_2) + \beta_3 \sigma_{M2_3} Cor(M1_2, M2_3)]}{\sigma_{M1} \sigma_{M2}} \\ & \\ = &\frac{0.4 * 1 * [0.3 * \sqrt{\pi^2/3} * 0.35 + 0.2 * \sqrt{8} * 0.35 + 0.5 * \sqrt{6/196.625} * 0.35]}{2.2 * 2.170860} \\ &+ \frac{0.6 * 1 * [0.3 * \sqrt{\pi^2/3} * 0.35 + 0.2 * \sqrt{8} * 0.35 + 0.5 * \sqrt{6/196.625} * 0.35]}{2.2 * 2.170860} \end{split} \tag{2.6} \end{equation}\]

library("SimCorrMix")
L <- calc_theory("Logistic", c(0, 1))
C <- calc_theory("Chisq", 4)
B <- calc_theory("Beta", c(4, 1.5))
mix_pis <- list(c(0.4, 0.6), c(0.3, 0.2, 0.5))
mix_mus <- list(c(-2, 2), c(L[1], C[1], B[1]))
mix_sigmas <- list(c(1, 1), c(L[2], C[2], B[2]))
p_M11M21 <- p_M11M22 <- p_M11M23 <- 0.35
p_M12M21 <- p_M12M22 <- p_M12M23 <- 0.35
p_M1M2 <- matrix(c(p_M11M21, p_M11M22, p_M11M23, p_M12M21, p_M12M22, p_M12M23), 
  2, 3, byrow = TRUE)
rhoM1M2 <- rho_M1M2(mix_pis, mix_mus, mix_sigmas, p_M1M2)

The correlation between \(M1\) and \(M2\) is approximated as 0.0877342.

2.2 Correlation between continuous mixture variable M1 or M2 and other random variable Y:

Here \(Y\) can be an ordinal, a continuous non-mixture, or a regular or zero-inflated Poisson or Negative Binomial variable. The correlation between the mixture variable \(M1\) and \(Y\) is given by:
\[\begin{equation} Cor(M1, Y) = \frac{E[M1Y] - E[M1]E[Y]}{\sigma_{M1} \sigma_Y}. \tag{2.7} \end{equation}\]

Equation (2.7) requires the expected value of the product of \(M1\) and \(Y\). Since \(M1\) may contain any desired number of components and these components may have any continuous distribution, there is no general way to determine this expected value. Therefore, it will again be approximated by expressing \(M1\) as a sum of its component variables:

\[\begin{equation} Cor(M1, Y) = \frac{E[(\sum_{i = 1}^{k_{M1}} \alpha_i M1_i) Y] - E[\sum_{i = 1}^{k_{M1}} \alpha_i M1_i]E[Y]}{\sigma_{M1} \sigma_Y}, \tag{2.8} \end{equation}\] where
\[\begin{equation} \begin{split} E[(\sum_{i = 1}^{k_{M1}} \alpha_i M1_i) Y] &= E[\alpha_1 M1_1 Y + \alpha_2 M1_2 Y + ... + \alpha_{k_{M1}} M1_{k_{M1}} Y] \\ &= \alpha_1 E[M1_1 Y] + \alpha_2 E[M1_2 Y] + ... + \alpha_{k_{M1}} E[M1_{k_{M1}} Y]. \end{split} \tag{2.9} \end{equation}\] Using the general correlation equation, for \(1 \leq i \leq k_{M1}\):
\[\begin{equation} E[M1_i Y] = \sigma_{M1_i} \sigma_Y Cor(M1_i, Y) + E[M1_i] E[Y], \tag{2.10} \end{equation}\]

so that we can rewrite \(Cor(M1, Y)\) as:

\[\begin{equation} \begin{split} Cor(M1, Y) &= \frac{\alpha_1 (\sigma_{M1_1} \sigma_Y Cor(M1_1, Y) + E[M1_1] E[Y])}{\sigma_{M1} \sigma_Y} \\ &\ \ \ \ \ + ... + \frac{\alpha_{k_{M1}} (\sigma_{M1_{k_{M1}}} \sigma_Y Cor(M1_{k_{M1}}, Y) + E[M1_{k_{M1}}] E[Y])}{\sigma_{M1} Y} \\ &\ \ \ \ \ - \frac{\alpha_1 E[M1_1] E[Y] + ... + \alpha_{k_{M1}} E[M1_{k_{M1}}] E[Y]}{\sigma_{M1} \sigma_Y} \\ &= \frac{\sum_{i = 1}^{k_{M1}} \alpha_i \sigma_{M1_i} Cor(M1_i, Y)}{\sigma_{M1}}. \end{split} \tag{2.11} \end{equation}\] Similarly,
\[\begin{equation} Cor(M2, Y) = \frac{\sum_{j = 1}^{k_{M2}} \beta_j \sigma_{M2_j} Cor(M2_j, Y)}{\sigma_{M2}}. \tag{2.12} \end{equation}\]

For this example, \(Y\) can be \(O1\), \(C1\), \(C2\), \(P1\), or \(NB1\). Let \(Y = C1\). Then we have:

\[\begin{equation} \begin{split} Cor(M1, C1) = &\frac{\alpha_1 \sigma_{M1_1} Cor(M1_1, C1) + \alpha_2 \sigma_{M1_2} Cor(M1_2, C1)}{\sigma_{M1}} \\ = &\frac{0.4 * 1 * 0.35 + 0.6 * 1 * 0.35}{2.2} \end{split} \tag{2.13} \end{equation}\]

p_M11C1 <- p_M12C1 <- 0.35
p_M1C1 <- c(p_M11C1, p_M12C1)
rho_M1C1 <- rho_M1Y(mix_pis[[1]], mix_mus[[1]], mix_sigmas[[1]], p_M1C1)

The correlation between \(M1\) and \(C1\) is approximated as 0.1590909. Since \(O1\), \(C2\), \(P1\), and \(NB1\) have the same target pairwise correlations with \(M1_1\) and \(M1_2\) as \(C1\), their correlations with \(M1\) are also approximated as 0.1590909.

Similarly,
\[\begin{equation} \begin{split} Cor(M2, C1) = &\frac{\beta_1 \sigma_{M2_1} Cor(M2_1, C1) + \beta_2 \sigma_{M2_2} Cor(M2_2, C1) + \beta_3 \sigma_{M2_3} Cor(M2_3, C1)}{\sigma_{M2}} \\ = &\frac{0.3 * \sqrt{\pi^2/3} * 0.35 + 0.2 * \sqrt{8} * 0.35 + 0.5 * \sqrt{6/196.625} * 0.35}{2.170860} \end{split} \tag{2.14} \end{equation}\]

p_M21C1 <- p_M22C1 <- p_M23C1 <- 0.35
p_M2C1 <- c(p_M21C1, p_M22C1, p_M23C1)
rho_M2C1 <- rho_M1Y(mix_pis[[2]], mix_mus[[2]], mix_sigmas[[2]], p_M2C1)

The correlation between \(M2\) and \(C1\) is approximated as 0.1930151. Since \(O1\), \(C2\), \(P1\), and \(NB1\) have the same target pairwise correlations with \(M2_1\), \(M2_2\), and \(M2_3\) as \(C1\), their correlations with \(M2\) are also approximated as 0.1930151.

References

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