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I have two questions about inhomogenous Fredholm integral equations of the first kind: $$f(x) = \int_a^b K(x,t) g(t) dt$$ where $f, K$ are known and $g$ is not.

  1. If a unique solution for $g$ exists, under what conditions is $g$ a probability density function -- i.e., a positive function that integrates to 1?

  2. For the regression equation $$r(x) \equiv E[Y | X=x] = \int_a^b E[Y | X=x, Z=z; \beta] g(z) dz,$$ solved by $g$ and $\beta$, must there exist another solution with $g’\neq g$ and $\beta’\neq\beta$? The regression function integrates over $Z$, which is not observed. I am particularly interested in cases of the form $$r(x) =\int_a^b \exp(-\exp(z + \beta x))\,g(z)dz$$ and using this to show that the parameter $\beta$ is not statistically identified.

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  • $\begingroup$ @MattF. sure, I've now done so $\endgroup$
    – Wilbur
    Mar 8, 2023 at 17:31
  • $\begingroup$ @MattF. -- to clarify, I'm interested in the case where there exists a $\beta, g$ that solves $r(x) = \int_a^b exp(-exp(z+\beta x)) g(z) dz$, but am not sure whether that $\beta, g$ solution is unique. $\endgroup$
    – Wilbur
    Mar 14, 2023 at 0:10
  • $\begingroup$ Right. I think the confusion might be due to me phrasing my original question as a general query about Fredholm integral equations. My general question is, $\endgroup$
    – Wilbur
    Mar 14, 2023 at 8:16
  • $\begingroup$ ... "Assume that, conditional on $K$ (i.e. conditional on $\beta$), there exists a unique $g$. When can we know that $g$ is a pdf?" Addressing the general question will help me prove the conjecture: "Assume that there exists a $\beta$ and a pdf $g$ that solves the equation. There exists a $\beta' \neq \beta$ and a pdf $g' \neq g$ that also solves the equation." $\endgroup$
    – Wilbur
    Mar 14, 2023 at 8:29
  • $\begingroup$ This is all in the post now, so I think the comments can be deleted. $\endgroup$
    – user44143
    Mar 14, 2023 at 13:54

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