Regularity Conditions

Assumptions

• $(R0)$ : The cdfs are distinct; i.e., $\theta \ne {\theta }^{\prime }⟹F(x_i;\theta )\ne F(x_i;{\theta }^{\prime })$
• $(R1)$ : The pdfs have common support for all $\theta$
• $(R2)$ : The point ${\theta }_0$ is an interior point in $\Omega$
• $(R3)$ : The pdf $f(x;\theta )$ is twice differentiable as a function of $\theta$.
• $(R4)$ : The integral $\int f(x;\theta )$ can be differentiated twice under the integral sign as a function of $\theta$.
• $(R5)$ : The pdf $f(x;\theta )$ is three times differentiable as a function of $\theta$. Further, for all $\theta \in \Omega$, there exist a constant $c$ and a function $M(x)$ such that $\left|\frac{{\partial }^3}{\partial {\theta }^3}\log f(x;\theta )\right|\le M(x)$ with $E_{{\theta }_0}[M(X)]<\infty$, for all ${\theta }_0-c<\theta <{\theta }_0+c$ and all $x$ in the support of $X$.