Z-estimators

Let $P$ be a probability measure on $(\mathcal{X},\mathcal{A})$, let $X_1,X_2,\dots$ be IID with law $P$ (Probability measures and probability spaces, Identical distribution and IID families), and let $(\mathrm{\Theta },d)$ be a metric candidate space with its Borel sigma-algebra (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric, The Borel sigma-algebra of a topological space). Fix a finite integer $q\ge 1$ and a norm on ${\mathbb{R}}^q$. For each $\theta \in \mathrm{\Theta }$, let ${\psi }_{\theta }:\mathcal{X}\to {\mathbb{R}}^q$ be measurable with $P\mathrm{\parallel }{\psi }_{\theta }\mathrm{\parallel }<\mathrm{\infty }$. Define ${\mathrm{\Psi }}_n(\theta )=P_n{\psi }_{\theta }=\frac{1}{n}{\sum }_{i=1}^n{\psi }_{\theta }(X_i),\mathrm{\Psi }(\theta )=P{\psi }_{\theta }\mathrm{.}$

A measurable estimator sequence ${\widehat{\theta }}_n$ into the Borel target $\mathrm{\Theta }$ is a Z-estimator if it is an exact or $o_P(1)$-approximate root: ${\mathrm{\Psi }}_n({\widehat{\theta }}_n)=0$ for every $n$, or $\mathrm{\parallel }{\mathrm{\Psi }}_n({\widehat{\theta }}_n)\mathrm{\parallel }=o_P(1)$. Measurability is as in Estimator sequences in IID experiments, and $o_P(1)$ has the convention of Stochastic order and convergence notation for finite-dimensional random vectors. Whenever a random candidate is substituted into an estimating equation, assume the evaluation is measurable.

Because $q\ge 1$ is finite, For $n\ge 1$ all norms on ${\mathbb{R}}^n$ are equivalent gives constants $c,C>0$ with $c\mathrm{\parallel }v{\mathrm{\parallel }}_2\le \mathrm{\parallel }v\mathrm{\parallel }\le C\mathrm{\parallel }v{\mathrm{\parallel }}_2$ for every $v\in {\mathbb{R}}^q$. Thus for every ${\mathbb{R}}^q$-valued sequence, convergence of the chosen-norm tails to zero is equivalent to the Euclidean $o_P(1)$ convention in Stochastic order and convergence notation for finite-dimensional random vectors.

This definition does not infer a Z-estimating equation from an M-criterion. Differentiating an M-criterion is justified only under additional differentiability and interior-point hypotheses; a nonsmooth criterion need not have a score equation.