Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [asymptotics]

Asymptotic theory studies the properties of estimators and test statistics when the sample size approaches infinity.

Asymptotic theory is concerned with the properties of estimators and test statistics in large samples which are assumed to tend towards infinity in size. This allows to obtain complicated estimators and tests which would not be available in small samples. Note that asymptotic theory is only an approximation with small samples, and it is not always a good approximation. Examples of asymptotic properties of estimators are consistency, regularity or their asymptotic distribution. Frequently used concepts in asymptotic theory include the weak and strong law of large numbers, the central limit theorem, certain classes of expansions (e.g. Taylor, Edgeworth, von Mises) and the Delta method.

831 questions
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What's the point of asymptotics?

I presume this is a rather stupid question, but I hope some of you can find a bit of time to entertain it. Looking at asymptotic behavior of estimators/test statistics etc means looking at their behavior as sample size approaches infinity. But if…
No Free Crunch Theorem
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What is the meaning of squared root n when we talk about asymptotic properties?

I just wonder what is the meaning or intuition behind $\sqrt{n}$ before $(\hat\beta-\beta)$ when we talk about asymptotic normality. Where does $\sqrt{n}$ come from? $\sqrt{n}(\hat\beta-\beta)\xrightarrow{ d }N(0,\sigma^2)$
jck21
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Asymptotic normality: do the following convergences hold?

Suppose we have a sequence $Z_n$ of estimators with mean $\mu_n$ and variance $\sigma_n^2$. Suppose we know that there exist constants $a_n$ and $b_n(>0)$ such that $\dfrac{Z_n-a_n}{b_n}\to\mathcal N(0,1)$ in distribution. Then, is it true that…
Landon Carter
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Convergence in distribution of a product of R.V.s?

If we have a sequence of variables $X_n$ that converges in distribution to $X$, and a sequence $Y_n$ that converges in distribution to $Y$, then does $X_nY_n$ converge in distribution to $XY$. Assume $X_n$ and $Y_n$ are independent. Weirdly, I…
user56834
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Big O almost surely

I came across the following: Let $Z_n$ be some sequence of random variables defined on a probability space $(\Omega, F, P)$ and suppose $$P(Z_n > n^{-1/2}(x+12*log(n))) \leq \exp(-x/6) $$ for all $x>0$. The author then claims that this implies $Z_n…
Joogs
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Proof clarification on almost sure convergence and Borel-Cantelli Lemma (update)

I believe it is easier if I print the proof below: Several other papers uses the same argument to bound $\lvert R_n(x)-E R_n(x) \rvert$ almost surely, so I believe it is correct. It should be clear that the aim of this result is to establish the…
Celine Harumi
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Question about asymptotic order

How does $\frac{n^{1-2/p}h_{n}^r}{\log(n)}\rightarrow \infty$, $n \rightarrow \infty$ and $h_{n}\rightarrow 0$ ($h_{n}$ is a function of $n$) imply $\frac{(\log(n))^{1/2}}{(nh_{n}^{r+2d})^{1/2}}\rightarrow 0$, where $\log(n)$ is the natural log of…
ExcitedSnail
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Convergence in distribution from ratio of the two densities

I want to prove $[X|y] \rightarrow \delta_0(X)$, as $y$ goes to $\infty$, i.e. the distribution of $[X|y]$ converges to degenerate distribution at zero for large enough $y$. I know the density $f(x|y)$ upto a multiplicative constant and also the…
VitalStatistix
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Obtain the asymptotic distribution of X/Y if X and Y are iid and independent of each other

. We just covered large sample theory and I'm going through the examples in our textbook. This one kinda confused me and I was hoping someone could help me understand why and how the book got, $W_i=\mu_X*Y_i - \mu_Y*X_i$ and how the book derived…
user42668
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Distribution of minimizer of sum of squared pairwise differences

Suppose I have iid random variables $X_1,\dots,X_n$ and some parametric function of $\theta,$ $f_{\theta},$ which is linear or nonlinear in $\theta$ (I am more interested in the nonlinear case, but if the linear case has been studied but the…
User0
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How to account for preliminary estimation of a quantile in an asymptotic variance?

Suppose I observe a random sample $X_1, ..., X_n$ from some continuous distribution $F$ and I want to estimate the parameter $\eta_q \equiv \mathbb{E}[g(q, X_i)]$ where $q$ is some specified quantile, e.g. the median, of the distribution of $X_i$…
inhuretnakht
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proposition whose proof I cannot find

Hi: I am reading a text that has the following proposition and it says that tbe proof is in Billingsley, 1979. I only have a later version of Billingsley and cannot find the proof in there. If anyone knows how to prove it or can tell me where I can…
mlofton
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Using Minkowski's inequality in a proof

Hi: I'm reading a proof in Halbert White's "asymptotic theory for econometricians" and I don't understand one of the steps. The theorem supposes the following: $E(X^2_{thi})^{1+\delta} < \triangle < \infty$ for some $\delta > 0 $ and all $h = 1,…
mlofton
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Are asymptotic distributions similar to limits?

If $X_1, X_2, X_3, ...$ are random variables and $(\bar{X_n}, \frac{1}{n} \sum_{1}^{n} X_i^2 - (\bar{X_n})^2)$ has some asymptotic distribution, will $(\bar{X_n}, \frac{1}{n-1} \sum_{1}^{n} X_i^2 - \frac{n}{n-1}(\bar{X_n})^2)$ have the same…
Noppawee Apichonpongpan
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What's the rate of $\log|1+\frac{O_p(M^{-1/2})}{fu}|$

What's the rate of $\log|1+\frac{O_p(M^{-1/2})}{fu}|$, where f is a real valued constant, u follows standard normal distribution and hence $u=O_p(1)$. So $\log|1+\frac{O_p(M^{-1/2})}{fu}|=O_p(?)$. I do not know how to deal with the absolute values.…
Patrick Star
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