10 votes

Convergence rate of a nonparametric estimator

Yes, in some cases the rate can be much faster for specific functions than in the general case. Consider the model $y_i = \theta^*_i + \epsilon_i$ where $\epsilon_i \sim N(0, \sigma^2)$ and $\theta^*...
angryavian's user avatar
  • 2,238
7 votes

Does the Law of Large Numbers work better for some Distributions?

The answer to the first question is yes and no and yes First, yes: the variance of $X$ matters. For example, if the variance of $X$ (call it $\sigma^2$) is finite, $\bar X_n-\mu_n$ has variance $\...
Thomas Lumley's user avatar
2 votes

Convergence of squared sample average

Two points: $E[\bar y^2]=E[\bar y]^2+\mathrm{var}[\bar y]$. The first term of this is the zero that you have, the second term is non-zero, as @whuber suggested This is exactly the sort of question ...
Thomas Lumley's user avatar
2 votes
Accepted

Asymptotic normality implies consistency

Your choice of $Y_N$ and $X_N$ are good, but you applied the theorem incorrectly. It does not make sense to take a limit in $N$, and end up with some result that depends on $N$ like $\mathcal{N}(0, a^...
angryavian's user avatar
  • 2,238

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