# Tag Info

Accepted

### Why does the continuity correction (say, the normal approximation to the binomial distribution) work?

In fact it doesn't always "work" (in the sense of always improving the approximation of the binomial cdf by the normal at any $x$). If the binomial $p$ is 0.5 I think it always helps, except ...
• 283k
Accepted

### Intuitive understanding of the difference between consistent and asymptotically unbiased

Asymptotic unbiasedness $\impliedby$ consistency + bounded variance Consider an estimator $\hat{\theta}_n$ for a parameter $\theta$. Asymptotic unbiasedness means that the bias of the estimator goes ...
• 125k

### Intuitive understanding of the difference between consistent and asymptotically unbiased

They are related ideas, but an asymptotically unbiased estimator doesn't have to be consistent. For example, imagine an i.i.d. sample of size $n$ ($X_1, X_2, ..., X_n$) from some distribution with ...
• 283k
Accepted

### What's the point of asymptotics?

The first reason we look at the asymptotics of estimators is that we want to check that our estimator is sensible. One aspect of this investigation is that we expect a sensible estimator will ...
• 125k

• 21.1k

### Consistency of lasso

You'll be disappointed to find that the consistency that matters the most with lasso is the consistency about which predictors are chosen. If you simulate two moderately large datasets and perform ...
• 92.1k

### Generalized CLT for any operation

There can only be such a theorem if $g$ is well-behaved in some sense: in particular, it should only depend on the set of values provided, not the order. Here is a big class of such functions, to ...
• 1,780

### What's the convergence rate in the context of convergence in probability?

I would argue that the most widely accepted definition of a convergence rate uses the "big-Oh" and "small-oh" notation. That is, convergence in probability is written as $z_n-z=o_p(1)$, while a rate ...
• 33.3k
Accepted

### Asymptotically Normally Distributed

But when we say "an estimator is asymptotically normally distributed", what does it mean? Using similar language to your first sentence, when we say an estimator is asymptotically normally ...
• 283k

### Does asymptotically unbiased mean convergence in probability?

Let $Y_i \sim \mbox{i.i.d. } N(\mu,1)$. Then $\hat\theta_n=Y_n$ is an unbiased (and hence asymptotically unbiased) estimator of $\mu$. However, it does not converge in probability, and thus is not ...
• 31.6k
Accepted

### Does $X\stackrel{d}\to X_1$ and $Y\stackrel{d}\to Y_1$ imply $X+Y\stackrel{d}\to X_1+Y_1$?

What $X+Y$ and $X_1+Y_1$ converge to depends upon the joint distributions of $(X,Y)$ and $(X_1,Y_1)$; what $X, X_1, Y, Y_1$ converge to individually depends upon the marginal distributions. You can ...
• 38.7k

### What is Asymptotic Independence

Assume you have a probability measure $P$. Two events $A$ and $B$ are independent iff $P(A\cap B) = P(A)P(B)$. Two random variables $X$ and $Y$ are independent iff for any measurable sets $A$ and $B$, ...
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Accepted

### What are "poor finite sample properties"?

In the context of hypothesis tests, poor finite sample properties usually mean that the actual rejection rate of the test differs from the nominal one. Recall that the nominal one is the level at ...
• 33.3k
The reason $e =\lim\limits_{n \to \infty}\left(1+\frac1n\right)^{n}=\frac1{\lim\limits_{n \to \infty}\left(1-\frac1n\right)^{n}}$ appears is precisely because you are taking a particular limit, though ...