Suppose $(x_{i})_{i=1}^n$ are i.i.d. sample. I want to construct a consistent estimator of $\mu=\mathbb{E}[\mathbb{E}[f(X_1,X_2)|X_1]^2]$, where $f(x_1,x_2)\ne f(x_2,x_1)$. I use $X_1$ for random, $x_1$ for realization.
My attempt: $\hat{\mu} = \frac{1}{n} \sum_{i=1}^n[\frac{1}{n-1}\sum_{j=1,j\ne i}^n f(x_i,x_j)]^2$
reasoning: $\frac{1}{n-1}\sum_{j=1,j\ne i}^n f(x_i,x_j)\xrightarrow{p}\mathbb{E}[f(x_i,X_j)|x_i]$ by the law of large number. Then, $\hat{\mu} = \frac{1}{n}\sum_{i}\mathbb{E}[f(x_i,X_j)|x_i]^2 +o_p(1)\xrightarrow{p}\mathbb{E}[\mathbb{E}[f(X_i,X_j)|X_i]^2].$
I am sort of sure that I am wrong because of $U$-statistic. But I don't know which part goes wrong.