Transformation of exponentials into piecewise constant hazard
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Let $X_1$ and $X_2$ follow independent exponential distributions with rates $lambda_1$ and $lambda_2$ respectively. Let $tau > 0$ be given and consider the hazard function $h(t)$ which is $lambda_1$ in $[0, tau[$ and $lambda_2$ in $[tau, infty[$. Find a function $F:mathbb{R}_+^2 to mathbb{R}_+$ so that $W = F(X_1, X_2)$ has hazard function $h(t)$.
I have no idea where to start. I found the density function corresponding to the hazard function, which is
$$
f(t) = begin{cases}
lambda_1 e^{-lambda_1 t} & text{if $t in [0, tau[$}\
lambda_2 e^{-lambda_2 (t-tau) + lambda_1 tau} & text{if $t in [tau, infty[$}
end{cases}
$$
It looks very similar to the exponential density but I'm having trouble finding the appropriate transformation. Any hints/ideas/solutions?
probability statistics
asked Nov 25 at 14:45
Lundborg
703414
add a comment |
up vote
0
down vote
favorite
Let $X_1$ and $X_2$ follow independent exponential distributions with rates $lambda_1$ and $lambda_2$ respectively. Let $tau > 0$ be given and consider the hazard function $h(t)$ which is $lambda_1$ in $[0, tau[$ and $lambda_2$ in $[tau, infty[$. Find a function $F:mathbb{R}_+^2 to mathbb{R}_+$ so that $W = F(X_1, X_2)$ has hazard function $h(t)$.
I have no idea where to start. I found the density function corresponding to the hazard function, which is
$$
f(t) = begin{cases}
lambda_1 e^{-lambda_1 t} & text{if $t in [0, tau[$}\
lambda_2 e^{-lambda_2 (t-tau) + lambda_1 tau} & text{if $t in [tau, infty[$}
end{cases}
$$
It looks very similar to the exponential density but I'm having trouble finding the appropriate transformation. Any hints/ideas/solutions?
probability statistics
asked Nov 25 at 14:45
Lundborg
703414
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $X_1$ and $X_2$ follow independent exponential distributions with rates $lambda_1$ and $lambda_2$ respectively. Let $tau > 0$ be given and consider the hazard function $h(t)$ which is $lambda_1$ in $[0, tau[$ and $lambda_2$ in $[tau, infty[$. Find a function $F:mathbb{R}_+^2 to mathbb{R}_+$ so that $W = F(X_1, X_2)$ has hazard function $h(t)$.
I have no idea where to start. I found the density function corresponding to the hazard function, which is
$$
f(t) = begin{cases}
lambda_1 e^{-lambda_1 t} & text{if $t in [0, tau[$}\
lambda_2 e^{-lambda_2 (t-tau) + lambda_1 tau} & text{if $t in [tau, infty[$}
end{cases}
$$
It looks very similar to the exponential density but I'm having trouble finding the appropriate transformation. Any hints/ideas/solutions?
probability statistics
asked Nov 25 at 14:45
Lundborg
703414
Let $X_1$ and $X_2$ follow independent exponential distributions with rates $lambda_1$ and $lambda_2$ respectively. Let $tau > 0$ be given and consider the hazard function $h(t)$ which is $lambda_1$ in $[0, tau[$ and $lambda_2$ in $[tau, infty[$. Find a function $F:mathbb{R}_+^2 to mathbb{R}_+$ so that $W = F(X_1, X_2)$ has hazard function $h(t)$.
I have no idea where to start. I found the density function corresponding to the hazard function, which is
$$
f(t) = begin{cases}
lambda_1 e^{-lambda_1 t} & text{if $t in [0, tau[$}\
lambda_2 e^{-lambda_2 (t-tau) + lambda_1 tau} & text{if $t in [tau, infty[$}
end{cases}
$$
It looks very similar to the exponential density but I'm having trouble finding the appropriate transformation. Any hints/ideas/solutions?
probability statistics
probability statistics
asked Nov 25 at 14:45
Lundborg
703414
asked Nov 25 at 14:45
Lundborg
703414
asked Nov 25 at 14:45
Lundborg
703414
asked Nov 25 at 14:45
Lundborg
703414
asked Nov 25 at 14:45
Lundborg
703414
703414
add a comment |
add a comment |
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