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1

It depends on what you count as 'digestible', but I like the version in van der Vaart's Asymptotic Statistics, which develops consistency of the bootstrap from the delta method in both the finite-dimensional and infinite-dimensional case.

0

Why would we choose Poisson / NB regression (GLM) over OLS for fitting count data? This can be more to do with modelling the relationship between the variance and the mean. Recall that in Poisson regression, the variance is equal to the mean. The negative binomial relaxes this slightly by allowing the variance to be quadratic in the mean.

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I have a very straight forward answer to this from a machine learning perspective. So given $\theta$ is mean square consistent, a reformulation is (Bias - Variance tradeoff): $$\mathbb{E}\left[(\theta-\mu)^2\right] \to 0 \Leftrightarrow Var\left[\theta\right] + Bias\left(\theta,\mu\right)^2 \to 0$$ as $n \to \infty$. As both ...

0

Let the model be, $y=X\beta+e$ where $e_i\sim i.i.d N(0,\sigma^2<\infty)$ and $i=1,2,\ldots,T$. Then by central limit theorem: $$\sqrt T (\beta-\hat\beta)\overset{d}{\rightarrow} N(0,(X'X)^{-1} \sigma^2)$$ This in turn implies that $$\sqrt T (\beta-\hat\beta) = O_P(1)$$ The above equation defines "$\hat{\beta}$ is $\sqrt{n}$-consistent". What ...

2

Maximum likelihood estimators in logistic regression are one example. The MLE is infinite with non-zero probability, which stays non-zero for all $n$, though it decreases exponentially with $n$. $E[(W_n-\theta)^2]$ is infinite, and does not converge to zero. If you don't like estimators that can be non-finite, take $W_n$ as mean of a sample of size $n$ ...

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