Non-monotonic transformation of Uniform distribution and derivative
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Let $Xsim Uniform[0,1]$ and $z:[0,1]rightarrow[0,1]$ be a non-monotonic function (but with very nice features such as continuity, differentiability, etc).
If a function $alpha$ is defined to be $$alpha(t)=P[z(X)leq t],$$ Is there anyway that we can get a closed form representaiton of $alpha'(t)$?
transformation uniform-distribution
asked Nov 14 at 7:16
Greenteamaniac
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up vote
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favorite
Let $Xsim Uniform[0,1]$ and $z:[0,1]rightarrow[0,1]$ be a non-monotonic function (but with very nice features such as continuity, differentiability, etc).
If a function $alpha$ is defined to be $$alpha(t)=P[z(X)leq t],$$ Is there anyway that we can get a closed form representaiton of $alpha'(t)$?
transformation uniform-distribution
asked Nov 14 at 7:16
Greenteamaniac
134
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $Xsim Uniform[0,1]$ and $z:[0,1]rightarrow[0,1]$ be a non-monotonic function (but with very nice features such as continuity, differentiability, etc).
If a function $alpha$ is defined to be $$alpha(t)=P[z(X)leq t],$$ Is there anyway that we can get a closed form representaiton of $alpha'(t)$?
transformation uniform-distribution
asked Nov 14 at 7:16
Greenteamaniac
134
Let $Xsim Uniform[0,1]$ and $z:[0,1]rightarrow[0,1]$ be a non-monotonic function (but with very nice features such as continuity, differentiability, etc).
If a function $alpha$ is defined to be $$alpha(t)=P[z(X)leq t],$$ Is there anyway that we can get a closed form representaiton of $alpha'(t)$?
transformation uniform-distribution
transformation uniform-distribution
asked Nov 14 at 7:16
Greenteamaniac
134
asked Nov 14 at 7:16
Greenteamaniac
134
asked Nov 14 at 7:16
Greenteamaniac
134
asked Nov 14 at 7:16
Greenteamaniac
134
asked Nov 14 at 7:16
Greenteamaniac
134
134
add a comment |
add a comment |
1 Answer
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Yes, there is:
$$alpha'(t) = mathcal I(t in [0, 1]) sum_{x_i: z(x_i) = t} left| frac{dz}{dt}(x_i) right|^{-1}$$
Here, $mathcal I$ is the indicator function that takes value $1$ if $t$ is within the range $[0, 1]$ and the sum is over all values of $x_i$ such that $z(x_i) = t$.
The way one arrives at this answer - if you look at Wikipedia's change of variables page, there's an equation for the density of random variables under non-monotonic transformations. Essentially, we add up density contributions from all $x$ that can map to $t$ and given the piecewise monotonicity, we can apply the usual Jacobian adjustment on each monotonic segment of $z$.
answered 21 hours ago
Aditya Ravuri
566
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Yes, there is:
$$alpha'(t) = mathcal I(t in [0, 1]) sum_{x_i: z(x_i) = t} left| frac{dz}{dt}(x_i) right|^{-1}$$
Here, $mathcal I$ is the indicator function that takes value $1$ if $t$ is within the range $[0, 1]$ and the sum is over all values of $x_i$ such that $z(x_i) = t$.
The way one arrives at this answer - if you look at Wikipedia's change of variables page, there's an equation for the density of random variables under non-monotonic transformations. Essentially, we add up density contributions from all $x$ that can map to $t$ and given the piecewise monotonicity, we can apply the usual Jacobian adjustment on each monotonic segment of $z$.
answered 21 hours ago
Aditya Ravuri
566
add a comment |
up vote
0
down vote
Yes, there is:
$$alpha'(t) = mathcal I(t in [0, 1]) sum_{x_i: z(x_i) = t} left| frac{dz}{dt}(x_i) right|^{-1}$$
Here, $mathcal I$ is the indicator function that takes value $1$ if $t$ is within the range $[0, 1]$ and the sum is over all values of $x_i$ such that $z(x_i) = t$.
The way one arrives at this answer - if you look at Wikipedia's change of variables page, there's an equation for the density of random variables under non-monotonic transformations. Essentially, we add up density contributions from all $x$ that can map to $t$ and given the piecewise monotonicity, we can apply the usual Jacobian adjustment on each monotonic segment of $z$.
answered 21 hours ago
Aditya Ravuri
566
add a comment |
up vote
0
down vote
up vote
0
down vote
Yes, there is:
$$alpha'(t) = mathcal I(t in [0, 1]) sum_{x_i: z(x_i) = t} left| frac{dz}{dt}(x_i) right|^{-1}$$
Here, $mathcal I$ is the indicator function that takes value $1$ if $t$ is within the range $[0, 1]$ and the sum is over all values of $x_i$ such that $z(x_i) = t$.
The way one arrives at this answer - if you look at Wikipedia's change of variables page, there's an equation for the density of random variables under non-monotonic transformations. Essentially, we add up density contributions from all $x$ that can map to $t$ and given the piecewise monotonicity, we can apply the usual Jacobian adjustment on each monotonic segment of $z$.
answered 21 hours ago
Aditya Ravuri
566
Yes, there is:
$$alpha'(t) = mathcal I(t in [0, 1]) sum_{x_i: z(x_i) = t} left| frac{dz}{dt}(x_i) right|^{-1}$$
Here, $mathcal I$ is the indicator function that takes value $1$ if $t$ is within the range $[0, 1]$ and the sum is over all values of $x_i$ such that $z(x_i) = t$.
The way one arrives at this answer - if you look at Wikipedia's change of variables page, there's an equation for the density of random variables under non-monotonic transformations. Essentially, we add up density contributions from all $x$ that can map to $t$ and given the piecewise monotonicity, we can apply the usual Jacobian adjustment on each monotonic segment of $z$.
answered 21 hours ago
Aditya Ravuri
566
answered 21 hours ago
Aditya Ravuri
566
answered 21 hours ago
Aditya Ravuri
566
answered 21 hours ago
Aditya Ravuri
566
566
add a comment |
add a comment |
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