# Relation between minimum contrast estimate and minimum distance estimate?

What relation are between minimum contrast estimate and minimum distance estimate?

If I understand correctly, these two are different methods? or are they equivalent?

Thanks and regards!

Minimum distance estimate from Wikipedia

Let $\displaystyle X_1,\ldots,X_n$ be an independent and identically distributed (iid) random sample from a population with distribution $F(x;\theta)\colon \theta\in\Theta$ and $\Theta\subseteq\mathbb{R}^k (k\geq 1)$.

Let $\displaystyle F_n(x)$ be the empirical distribution function based on the sample.

Let $\hat{\theta}$ be an estimator for $\displaystyle \theta$. Then $F(x;\hat{\theta})$ is an estimator for $\displaystyle F(x;\theta)$.

Let $d[\cdot,\cdot]$ be a functional returning some measure of "distance" between the two arguments. The functional $\displaystyle d$ is also called the criterion function.

If there exists a $\hat{\theta}\in\Theta$ such that $d[F(x;\hat{\theta}),F_n(x)]=\inf\{d[F(x;\theta),F_n(x)]; \theta\in\Theta\},$ then $\hat{\theta}$ is called the minimum distance estimate of $\displaystyle \theta$.

Minimum contrast estimate from Bickel and Doksum's Mathematical Statistics Vol1 The minimum distance estimator is equivalent to the minimum contrast estimator, provided you pick $d$ and and $\rho$ such that $$d[F(x; \theta), \, F_n(x)] = \rho(X, \theta).$$
• I guess, technically, it depends on what kinds of functions d and $\rho$ are allowed – Stefan Wager Apr 26 '13 at 23:13