I am currently doing research that requires me to understand dependence modeling. As a first step, I am reading An Introduction to Copulas. I am, stuck on the first example problem which I have re-written below and would like some well earned wisdom to help unstick me.

Example 1

Let $Y$ and $Z$ be two IID random variables each with CDF, $F()$. Let $X_1 = \min(Y, Z)$ and $X_2 = \max(Y, Z)$ with marginals $F_1$ and $F_2$ respectively. Use Sklar's Theorem to find an expression that satisfies $C(F_1(X_1)$, $F_2(X_2)) = F(x_1, x_2)$.

To solve this we need to start by finding $F(x_1,x_2)$, and then find the marginal distributions F_1(X_1), $F_2(X_2)$. After playing with it, and using the hint given in the notes, I understand(ish) how they arrived at the following:

1. $F(x_1, x_2) = 2F(\min(x_1,x_2))F(x_2) - F(\min(x_1,x_2))$

2. $F_1(X_1) = 2F(x_1) - F(x_1)$

3. $F_2(X_2) = F(x_2)^2$

Now is where it gets hard for me. The copula derived is

$C(u_1, u_2) = 2 \min(1 - \sqrt{1 - u_1}, \sqrt{u_2})\sqrt{u_2} - \min(1 - \sqrt{1 - u_1}, \sqrt{u_2})^2 = F(x_1, x_2)$

I understand how equation (3) was used to get $\sqrt{u_2} = F(x_2)$. But, I am having trouble with how to derive $F(\min(x_1,x_2)) = \min(1 - \sqrt{1 - u_2}, \sqrt{u_2})$.

Any help would be appreciated and let me know if there is anything unclear about my question!

I am currently doing research that requires me to understand dependence modeling. As a first step, I am reading An Introduction to Copulas. I am, stuck on the first example problem which I have re-written below and would like some well earned wisdom to help unstick me.

Example 1

Let $Y$ and $Z$ be two IID random variables each with CDF, $F()$. Let $X_1 = \min(Y, Z)$ and $X_2 = \max(Y, Z)$ with marginals $F_1$ and $F_2$ respectively. Use Sklar's Theorem to find an expression that satisfies $C(F_1(X_1)$, $F_2(X_2)) = F(x_1, x_2)$.

To solve this we need to start by finding $F(x_1,x_2)$, and then find the marginal distributions F_1(X_1), $F_2(X_2)$. After playing with it, and using the hint given in the notes, I understand(ish) how they arrived at the following:

1. $F(x_1, x_2) = 2F(\min(x_1,x_2))F(x_2) - F(\min(x_1,x_2))$

2. $F_1(X_1) = 2F(x_1) - F(x_1)^2$

3. $F_2(X_2) = F(x_2)^2$

Now is where it gets hard for me. The copula derived is

$C(u_1, u_2) = 2 \min(1 - \sqrt{1 - u_1}, \sqrt{u_2})\sqrt{u_2} - \min(1 - \sqrt{1 - u_1}, \sqrt{u_2})^2 = F(x_1, x_2)$

I understand how equation (3) was used to get $\sqrt{u_2} = F(x_2)$. But, I am having trouble with how to derive $F(\min(x_1,x_2)) = \min(1 - \sqrt{1 - u_2}, \sqrt{u_2})$.

Any help would be appreciated and let me know if there is anything unclear about my question!

I am currently doing research that requires me to understand dependence modeling. As a first step, I am reading An Introduction to Copulas. I am, stuck on the first example problem which I have re-written below and would like some well earned wisdom to help unstick me.

Example 1

Let Y and Z be two IID random variables each with CDF, F(). Let X₁ := min(Y, Z) and X₂ := max(Y, Z) with marginals F₁ and F₂ respectively. Use Sklar's Theorem to find an expression that satisfies C(F₁(X₁), F₂(X₂)) = F(x₁, x₂).

To solve this we need to start by finding F(x₁, x₂), and then find the marginal distributions F₁(X₁), F₂(X₂). After playing with it, and using the hint given in the notes, I understand(ish) how they arrived at the following:

(1) $F(x_1, x_2) = 2F(min(x_1,x_2))F(x_2) - F(min(x_1,x_2))$

(2) $F_1(X_1) = 2F(x_1) - F(x_1)$

(3) $F_2(X_2) = F(x_2)^2$

Now is where it get's hard for me. The copula derived is

$C(u_1, u_2) = 2 min(1 - \sqrt{1 - u_1}, \sqrt{u_2})\sqrt{u_2} - min(1 - \sqrt{1 - u_1}, \sqrt{u_2})^2 = F(x_1, x_2)$

I understand how equation (3) was used to get $\sqrt{u_2} = F(x_2)$. But, I am having trouble with how to derive $F(min(x_1,x_2)) = min(1 - \sqrt{1 - u_2}, \sqrt{u_2})$.

Any help would be appreciated and let me know if there is anything unclear about my question!

I am currently doing research that requires me to understand dependence modeling. As a first step, I am reading An Introduction to Copulas. I am, stuck on the first example problem which I have re-written below and would like some well earned wisdom to help unstick me.

Example 1

Let $Y$ and $Z$ be two IID random variables each with CDF, $F()$. Let $X_1 = \min(Y, Z)$ and $X_2 = \max(Y, Z)$ with marginals $F_1$ and $F_2$ respectively. Use Sklar's Theorem to find an expression that satisfies $C(F_1(X_1)$, $F_2(X_2)) = F(x_1, x_2)$.

To solve this we need to start by finding $F(x_1,x_2)$, and then find the marginal distributions F_1(X_1), $F_2(X_2)$. After playing with it, and using the hint given in the notes, I understand(ish) how they arrived at the following:

1. $F(x_1, x_2) = 2F(\min(x_1,x_2))F(x_2) - F(\min(x_1,x_2))$

2. $F_1(X_1) = 2F(x_1) - F(x_1)$

3. $F_2(X_2) = F(x_2)^2$

Now is where it gets hard for me. The copula derived is

$C(u_1, u_2) = 2 \min(1 - \sqrt{1 - u_1}, \sqrt{u_2})\sqrt{u_2} - \min(1 - \sqrt{1 - u_1}, \sqrt{u_2})^2 = F(x_1, x_2)$

I understand how equation (3) was used to get $\sqrt{u_2} = F(x_2)$. But, I am having trouble with how to derive $F(\min(x_1,x_2)) = \min(1 - \sqrt{1 - u_2}, \sqrt{u_2})$.

Any help would be appreciated and let me know if there is anything unclear about my question!