- #1
dragon_chen
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I am pondering over how to solve the following (seemingly nonstandard) integral equation.
Let h(t) be a known function which is non-negative, strictly increasing and satisfies that h(t) → 0 as t→-∞ and h(t)→1 as t→∞. Indeed, h(t) can be viewed as a cumulative distribution function for a continuous random variable.
Let K be a given positive integer. The problem is to find the function f(x) satisfying that
h(t) = ∫(F(t+x)^K)f(x)dx
where the integral is taken from -∞ to ∞, F is a functional of f and satisfies F(y)=∫f(x)dx where the integral is taken from -∞ to y. The function f to be solved is positive with support on the whole real line and is integrated to 1 over its support. Indeed, f is viewed as a density of a continuous random variable whose cumulative distribution function is F. There is a location constraint on the solution f that should satisfy that ∫xf(x)dx=0. Namely, the expectation of the random variable with density f should be zero.
The integral equation seems nonstandard since the kernel F(t+x) is unknown though it depends only on f.
Could anyone suggest how to deal with this problem ?
Thank you very much !
Let h(t) be a known function which is non-negative, strictly increasing and satisfies that h(t) → 0 as t→-∞ and h(t)→1 as t→∞. Indeed, h(t) can be viewed as a cumulative distribution function for a continuous random variable.
Let K be a given positive integer. The problem is to find the function f(x) satisfying that
h(t) = ∫(F(t+x)^K)f(x)dx
where the integral is taken from -∞ to ∞, F is a functional of f and satisfies F(y)=∫f(x)dx where the integral is taken from -∞ to y. The function f to be solved is positive with support on the whole real line and is integrated to 1 over its support. Indeed, f is viewed as a density of a continuous random variable whose cumulative distribution function is F. There is a location constraint on the solution f that should satisfy that ∫xf(x)dx=0. Namely, the expectation of the random variable with density f should be zero.
The integral equation seems nonstandard since the kernel F(t+x) is unknown though it depends only on f.
Could anyone suggest how to deal with this problem ?
Thank you very much !
