# 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 Apr 26, 2013 at 23:13