Variance of mixture of normals

There are several statistical methods to deal competing risk in survival analysis. Mixture model also can be used when there was competing risk in survival analysis. The following paper introduced how to use mixture model to deal with competing risk in survival analysis.

Larson, M. G., & Dinse, G. E. (1985). A mixture model for the regression analysis of competing risks data. Journal of the royal statistical society: series C (Applied Statistics), 34(3), 201-211.

Here I am trying to replicate the paper results, first, let us review what is a mixture distribution using normal distrbutions.

Let \(I_{1-\epsilon}\) be a discrete random variable defined by

\[\begin{equation} I_{1-\epsilon} = \begin{cases} 1 & \text{with probability }1-\epsilon\\ 0 & \text{with probability } \epsilon \end{cases} \end{equation}\]

Here, \(E(I_{1-\epsilon})=1-\epsilon\) and \(Var(I_{1-\epsilon})=(1-\epsilon)\epsilon\)

suppose \(Z\) is the standard normal distribution, and \(Z\) and \(I_{1-\epsilon}\) are independent. A mixture nomal distribution can be written as

\[W=ZI_{1-\epsilon}+\sigma_cZ(1-I_{1-\epsilon})\] It is quite easy to show that \(E(W)=0\)

We show how to calculate the variance of \(W\) here

Remember when \(X\) and \(Y\) are two independent randome variables we have

\({Var}(XY) = E(X^2Y^2) − (E(XY))^2=E(X^2)E(Y^2)-[E(X)]^2[E(Y)]^2\\= [Var(X)+E(X)^2][Var(Y)+E(Y)^2]-[E(X)]^2[E(Y)]^2\\={\rm Var}(X){\rm Var}(Y)+{\rm Var}(X)(E(Y))^2+{ Var}(Y)(E(X))^2\)

Therefore,

\(Var(W)=Var(ZI_{1-\epsilon})+\sigma_c^2Var(Z(1-I_{1-\epsilon}))\\=\left\{Var(Z)Var(I_{1-\epsilon})+Var(Z)[E(I_{1-\epsilon})]^2+Var(I_{1-\epsilon})[E(Z)]^2\right\}+\left\{\sigma_c^2[Var(Z)Var(1-I_{1-\epsilon})+Var(Z)[E(1-I_{1-\epsilon})]^2+Var(1-I_{1-\epsilon})[E(Z)]^2 \right\}\\=(1-\epsilon)\epsilon+(1-\epsilon)^2+0+\sigma_c^2[(1-\epsilon)\epsilon+1*(1-(1-\epsilon))^2+0]\\=1-\epsilon+\sigma_c^2(\epsilon-\epsilon^2+\epsilon^2)\\=1+\epsilon(\sigma_c^2-1)\)

For above derivations we will use that \(Var(a\pm bx)=b^2Var(X)\)

The cdf of the\(W\) is \[F_W(w)=\Phi(w)(1-\epsilon)+\Phi(w/\sigma_c)\epsilon\]

The pdf of the \(W\) is

\[f_W(w)=\phi(w)(1-\epsilon)+\phi(w/\sigma_c)\frac{\epsilon}{\sigma_c}\]

Note, from CDF to pdf we will use he second theorem of calculus and chain rule, i.e

\[\frac{d}{dx}\int_{g(x)}^{f(x)}h(t)\,dt=h(f(x))\cdot f'(x)-h(g(x))\cdot g'(x)\]