📂Mathematical Statistics

Regularity Conditions in Mathematical Statistics

Overview

In subjects that utilize mathematics, the term Regularity Conditions usually refers to conditions that allow for a wide range of applications and make theoretical developments more comfortable. In mathematical statistics, they are as follows.

Assumptions ¹

Consider a random variable $X$ with probability density function $f \left( x ; \theta \right)$ for a parameter $\theta \in \Theta$. The random sample $X_{1} , \cdots , X_{n}$ drawn iid from the same distribution as $X$ has the same probability density function $f(x ; \theta)$ and realizations $\mathbf{x} := \left( x_{1} , \cdots , x_{n} \right)$. The following function $L$ is called the Likelihood Function. $$ L ( \theta ; \mathbf{x} ) := \prod_{k=1}^{n} f \left( x_{k} ; \theta \right) $$ Finally, let’s say $\theta_{0}$ is the true value of $\theta$.

• (R0): The probability density function $f$ is injective with respect to $\theta$. In formula, it satisfies the following. $$ \theta \ne \theta’ \implies f \left( x_{k} ; \theta \right) \ne f \left( x_{k} ; \theta’ \right) $$
• (R1): The probability density function $f$ has the same support for all $\theta$.
• (R2): The true value $\theta_{0}$ is an interior point of $\Omega$.
• (R3): The probability density function $f$ is twice differentiable with respect to $\theta$.
• (R4): The integral $\int f (x; \theta) dx$ is twice differentiable with respect to $\theta$, with the differentiation being interchangeable with the integral sign.
• (R5): The probability density function $f$ is thrice differentiable with respect to $\theta$. Moreover, for all $\theta \in \Theta$, there exists constants $c> 0$ and a function $M(x)$ satisfying $E_{\theta_{0}} \left[ M ( X ) \right] < \infty$ and the following. $$ \left| {{ \partial^{3} } \over { \partial \theta ^{3} }} \log f (x ; \theta) \right| \le M (x) \qquad , \forall x \in \mathcal{S}_{X} , \forall \theta \in \left( \theta_{0} - c , \theta_{0} + c \right) $$