# Differentiability (Hessian) of $\int \log F$ when $\int \log f$ is differentiable?

For a specific probability density function $$f$$ with support on $${\mathbb R}$$, which is not differentiable everywhere, I have proven that the Hessian matrix of $$g(\theta) = \int \log f(x;\theta)d H(x),$$ exists for all $$\theta \in {\mathbb R}^p$$, where $$H$$ is another distribution function. Let $$F(x;\theta) = \int_{-\infty}^x f(t;\theta)dt$$ be the distribution function. I want to check if the Hessian of $$G(\theta) = \int \log F(x;\theta)dH(x),$$ also exists.

Is there a direct method of showing this? This is, some general result I can appeal to? If it wasn't for the logarithm, I could use exchange the integral and derivative symbols, for instance.

• How can you derivate $g(\theta)$ if $f$ is not differentiable in $\theta$? – Liviu Nicolaescu May 8 at 20:05