How can I display randomized estimator?

Question

Let $\textbf{X}=(X_1, X_2, \ldots, X_n)$ be a random sample from $f(x, \theta)$. I have Obtained a randomized estimator for unknown parameter $\theta$ as follows: $$ \delta(\textbf{X})=\begin{cases} T_1(\textbf{X})~~~~\text{with probability}~U(\textbf {X})\\ T_2(\textbf{X})~~~~\text{with probability}~1-U(\textbf {X})\\ \end{cases}, $$ where $U(\textbf{X})$ is a random variable and $ 0<U(\textbf{X})<1$. Is true statement "with probability$~U(\textbf {X})$"? I think probability is not a random variable. But I force to use notation $\textbf{X}$ for displaying randomized estimator. What is your idea?

Answer 1

This is a, mathematically terrible, description of the following setting.

One is given mathematical objects $X$, $T_1$, $T_2$, $U$ and $W$, some probabilistic and some deterministic, defined as follows:

• A random variable $X:(\Omega,\mathcal F)\to(\Xi,\mathcal X)$, which you denote by $\mathbf X$ in your question, probably with $\Xi=\mathbb R^n$ and $\mathcal X=\mathcal B(\mathbb R^n)$ but this point is irrelevant.
• Some deterministic real valued measurable functions $T_1$, $T_2$, and $U$, all defined on $(\Xi,\mathcal X)$, such that $0<U(\xi)<1$ for every $\xi$ in $\Xi$.
• A random variable $W$ on $(\Omega,\mathcal F)$, independent of $X$, uniformly distributed on $(0,1)$.

Then, one considers the event $$A=\{\omega\in\Omega\mid W(\omega)<U(X(\omega))\}$$ and one defines a new random variable $\Delta:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B(\mathbb R))$, as $$\Delta=\begin{cases}T_1(X)\,\quad\text{on}\quad A\\ T_2(X)\,\quad\text{on}\quad \Omega\setminus A\end{cases}$$ Thus, indeed, $$P(A\mid X)=U(X)$$ and $\Delta$ may be a randomized estimator of $\theta$ if $T_1(X)$ and $T_2(X)$ are, but the random object $\Delta$ is certainly not a deterministic function of $X$ alone.

Actually, $\Delta=\delta(X,W)$, where the deterministic measurable function $\delta$ is defined on $\Xi\times(0,1)$ by $$\delta(\xi,w)=\begin{cases}T_1(\xi)\,\quad\text{if}\quad w<U(\xi)\\ T_2(\xi)\,\quad\text{if}\quad w\geqslant U(\xi)\end{cases}$$