Questions tagged [asymptotics]
Asymptotic theory studies the properties of estimators and test statistics when the sample size approaches infinity.
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questions
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Variance of $\frac{\sum{X_i}}n$, where $X_i$'s are i.i.d. Bernoulli random variables
It's abou Example 10.1.14 from Casella (2nd ed) For a random sample $X_1, \dots, X_n$, each having Bernoulli distribution ($P(X_i=1)=p$), we know $\mathrm{Var}_X=p(1-p)$.
It's said $\mathrm{Var}_p\hat{...
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1answer
24 views
What is the limit of the random forest (or bagging) estimator?
I am looking for a proof or intuition as to why the absolute limit of a random forest estimator is the expectancy of a single tree (see citation below), i.e:
$$
\hat{f}_{rf}(x) = \lim_{B \to \infty}[\...
2
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1answer
33 views
Asymptotic equivalence of Likelihood Ratio Staitistic and Wald Statistic
When we say the likelihood ratio statistic and the wald statistic of a set of binomial distributions are asymptotically equivalent, do we mean that the sampling distributions of the two statistics ...
2
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1answer
36 views
Randomly selecting integers with prescribed minimal distance and estimations
I am interested in sequences of $M$ distinct integers in $[[1,N]]$ (integers from $1$ to $N$) such that integers $I_m$ are separated by at least $\delta$ integers (taking into account the outer ...
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11 views
Intepretation of a sample size convergence assumption in a 2 sample Z test
Let us consider a two sample problem modeled with a fixed sample sizes $n_1, n_2$ (i.g without assuming that the group is chosen randomly).
If we want to proof asymptotic normality of a classic Z-...
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37 views
Asymptotic Distribution Using CLT
I have random variables $X_1, X_2, ... , X_n \sim \text{IID } f_X$ using the density function:
$$f_X(x) = \frac{2x}{\theta^2} \cdot \mathbb{I}(0 \leqslant x \leqslant \theta).$$
I have to use the ...
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0answers
15 views
Asymptotic distribution of mme of coefficient of variation
For random sample from unif(0,1) distribution, method of moments estimator for coefficient of variation is sample mean divided by sample standard deviation. Here, coefficient of variation theta is mu/...
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1answer
37 views
How to show two variables are asymptotically independent
Let $X_1,...,X_n$ be iid from $Exp(\theta)$ with density function $f(x) = \frac{1}{\theta}e^{-x/\theta}$. Show that $M_n = X_{n:n} - \theta \ln(n)$ and $T_n = nX_{1:n}$ are asmyptoically independent ...
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18 views
What is the rational behind bootstrapping a model?
At this point, I understand what bootstrapping is and how it works. What I would like to understand better is the exact properties of the method regarding its test error and, as a related question, ...
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0answers
32 views
Linear model with partially time-varying coefficients
Suppose we have a linear model with time-varying coefficients
$$
y_i = x_i' \beta_{t_i} + \epsilon_i, \; i = 1, 2, \cdots, n
$$
where the design points are $t_i = \frac{i}{n}$, and $\beta(t): [0,1] \...
3
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1answer
40 views
Prove that the MLE exists almost surely and is consistent
I need to show that given an i.i.d sample $X_1,\dots X_n$ arising from the model:
$$\{f(x,\theta)=\theta x^{\theta-1}exp\{-x^{\theta}\},x>0,\theta\in (0,\infty)\}$$
that the MLE exists with ...
4
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1answer
266 views
What is Asymptotic Independence
What does it mean if two random variables are asymptotically independent? And how would you prove that they are?
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19 views
The reason why a test is undersized?
Now I have a statistic $T_n$ for testing $H_0 \leftrightarrow H_1$, and I have proved that:
$$n T_n \rightarrow_d \chi_K^2$$
under $H_0$. Then an asymptotic $\chi^2$ test can be used, an asymptotic ...
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26 views
Question about an implication relationship
Let $||\cdot||$ be the Euclidean norm. Suppose $X_1,X_2$ are two independent and identically distributed random variables, and $a_N(X_1,X_2)$ is a vector valued function that depends on factor $N$ and ...
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1answer
61 views
The logrank test statistic is equivalent to the score of a Cox regression. Is there an advantage of using a logrank test over a Cox regression?
I have understood the logrank test as a "safe" or "conservative" way to check for a difference between two survival curves. It is "safe" in the sense that it is a ...
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1answer
78 views
Does a version of the Delta Method exist for non-i.i.d. sequences?
I have a sequence of random variables that are non-independent, but usually identically distributed. I am wondering if a version of the Delta Method exists under the case when I only have that the ...
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1answer
56 views
(From van der Vaart's Asymptotic Statistics, page 161, U-statistic) Why we can always replace the function $h$ with a symmetric one?
I'm reading the following Chapter from van der Vaart's Asymptotic Statistics, Section 12.1 page 161 (see the screenshot below). For the $h$ function that it mentioned, I have two questions regarding ...
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1answer
30 views
Random Sampling: Weak and Strong Exogenity
$Y \ = \ X' \beta \ + \ e $
Where $Y = (y_1, ..., y_n)$ and $\beta = (\beta_0,..., \beta_k)$.
Why would Weak Exogenity under random sampling produce Strong Exogenity?
I know that weak exogenity is ...
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0answers
125 views
Proving the almost sure convergence of the Kolmogorov-Smirnov test statistic
Context: I have spent the last few weeks thinking about how the central limit theorem is enunciated: if we have a set of i.i.d. random variables $X_1, X_2, \ldots X_m$ then $\frac{\sum_{i=1}^mX_i - E[\...
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1answer
30 views
Asymptotic distribution after replacing quantities by consisent estimators
Suppose that we wish to estimate $T(\theta_1,\theta_2)$, a continuous function of several parameters. Suppose that we know the asymptotic distribution when $\theta_1$ is replaced by an estimator $\hat{...
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1answer
78 views
Exogenity: What does E(eX) really mean and why is it used?
What does it mean to talk about the expectation of the product of the error term and an independent variable? Like, why do we even need to mention $E(e_i X_{ik})$? What is it actually describing or ...
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1answer
42 views
asymptotic p values of standard normal distributed ? how to calculate them?
I do have an estimator which have an asymptotic standard normal distribution under the null hypothesis .How to calculate the asymptotic p values ?
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1answer
40 views
Does the Glivenko-Cantelli theorem work back and forth?
If we have a sample $X_1, X_2, \ldots, X_n \sim F$ then $\hspace{1mm}sup_x|F_n(x) -F(x)|\xrightarrow{a.s./p}0$.
Now, if I can come up with a theoretical cdf $F$ such that $\hspace{1mm}sup_x|F_n(x) -F(...
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46 views
Asymptotic normal distribution of MLE estimator using Fisher's information
Let $X_1, . . . , X_n$ be independent N(Āµ, Ļ$^2$
) random variables. Suppose that Āµ is known, Ļ is
unknown and that we want to estimate Ļ = log Ļ.
(a) Find the maximum likelihood estimator $\hat{\...
1
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1answer
23 views
Does Overall CTR obey asymptotic normal distribution?
Background
In business industry, I came across two different type of CTR metrics to measure our products. I want to perform hypothesis testing(AB test) for these metrics. However, I am not sure about ...
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votes
2answers
145 views
Is the MA($\infty$) process with i.i.d. noise strictly stationary?
I have a MA($\infty$) process defined by
$$
X_t = \sum_{k=0}^\infty \alpha_{k} \epsilon_{t-k}, \qquad t\in\mathbb{Z}
$$
where the sums converge a.s. and the $\epsilon_t$ are i.i.d. centered noise ...
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0answers
42 views
product of asymptotic standard normal distribution
Suppose $Z_n\xrightarrow{d} Z \sim N(0,I_p)$, why $Z_n^TZ_n\xrightarrow{d}\chi^2_p$?
I encounter this problem when we get the asymptotic distribution of the maximum likelihood estimator (MLE). Suppose ...
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108 views
Is bootstrap problematic in small samples?
In "3 Things That Bother Me" (1988), Ed Leamer writes:
Bootstrap estimates of standard errors are based on the assumption that the observed sample is the same as the true distribution, ...
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1answer
42 views
Variance of sample moments - clarification on Serfling (1980)
In "Serfling, R. J. (1980). Approximation theorems of mathematical statistics", we read
In Theorem A, as one suspects, $k=1,2,...$, indicating the integer-moments, while $n$ is the sample ...
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1answer
67 views
If $\hat{a}=O_{p}(\sqrt{\frac{logn}{nh^b}}+h^c)$, what is $\hat{a}^2$ in terms of $O_{p}()$?
If $\hat{a}=O_{p}(\sqrt{\frac{logn}{nh^b}}+h^c)$, where $n$ is sample size, and $h$ is bandwidth that also depends on $n$. What is the order of $\hat{a}^2$ in terms of $O_{p}()$?
More specifically, ...
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16 views
Suppose $max\{a_i\}_{i=1}^{Rn}\overset{p}{\rightarrow} a_0$, where $a_i$ are i.i.d.r.v.. Are there any results on its rate of convergence?
Suppose $max\{a_i\}_{i=1}^{Rn}\overset{p}{\rightarrow} a_0$, where $a_i$ are i.i.d. random variables, $a_0$ is a constant and $R_n\rightarrow\infty$ as $n\rightarrow\infty$.
Are there any results on ...
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0answers
11 views
What are the available tools (results) that can be used to pin down the rate of convergence of an estimator besides CLT?
What are the available tools (results) that can be used to pin down the rate of convergence of an estimator besides CLT? It would be great if you could illustrate how to use the tools (results) with ...
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0answers
31 views
Evaluation of Limit involved in the proof of Asymptotic Unbiasedness
We know that $S^{2}$ is an unbiased estimator of $\sigma^{2}$ and $S$ is a biased estimator of $\sigma$. But if $n\rightarrow\infty$, then $S$ is an asymptotically unbiased estimator of $\sigma$. I ...
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0answers
16 views
Another 3-part question, this time on limiting distributions. Care to critique my work?
Let $X \stackrel{d}{\sim} Geometric(p)$ for $0 < p < 1$. E.g., $X$ has the pmf $f(x|p) = p(1-p)^{x-1}, x = 1, 2, ...$ with $E(X) = \frac{1}{p}$ and $Var(X) = \frac{1-p}{p^2}.$
a.) Find the limit ...
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0answers
23 views
Show that a sequence of random variables diverges to infinity in probability
I have sequences of real-valued random variables $\{X_T\}, \{Y_T\}$ and a sequence of real numbers $\{a_T\}$.
As $T\rightarrow\infty$, I know that
$$
a_T \rightarrow \infty
$$
and
$$
X_T \overset{d}{\...
2
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1answer
42 views
What does weak convergence mean for a stochastic process?
I am reading a paper in which stochastical processes $\{\mathcal{H}_T(u)\}_{u\in[0,1]}$ and $\{\mathcal{H}(u)\}_{u\in[0,1]} $ on [0,1] with $u$ as a time-index occur.
There is a theorem which states ...
3
votes
1answer
93 views
Asymptotic dist of an average involving OLS coefs?
Suppose that we have iid sample of size $n$. i.e., the random vector $(Y_{i}, X_{1i}, X_{2i}, X_{3i})$ is iid from $1,\ldots,n$. And suppose the following relationship is true:
$$
Y_i = \beta_0 + \...
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13 views
distribution marginal associations in linear model with random design
We have a vector $x \in \mathbb{R}^d$, and it's a random vector which has some underlying distribution $p(x)$. We are also given a fixed vector $\beta \in\mathbb{R}^d$. Suppose we have $n$ vectors i.i....
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1answer
41 views
Why is this true?
suppose $T$ is a binary variable and $x$ is a continuous scalar, and $g(x)=E[T|x]$ is the conditional expectation of $T$. Suppose I estimate $g(x)$ using kernel regression $\widehat{g}(x)=\frac{\sum_{...
0
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1answer
20 views
Asymptotic equivalence and Kolmogorov-Smirnov Distance
Suppose we have two sequences of random variables $\{X_k\}$ and $\{Y_k\}$, both converging to the same distribution, say $N(0,V)$, for some covariance matrix $V$. Does this imply that
$$\sup_{u}\left|...
1
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1answer
29 views
Do these two random variables have the same asymptotic distribution?
Let $\{X_k\}$ be a sequence of dependent random variables with mean 0. Define $\bar{Y}_k = \frac{1}{\sqrt k}\sum_{i=1}^k X_i$.
Let $\{W_k\}$ be a sequence of i.i.d. random variables with mean 1 and ...
0
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1answer
72 views
How to calculate the asymptotic p-value for a test on a Poisson i.i.d random variable?
Disclaimer: This is a question I couldn't solve. I feel it is ethical to state this here so you can evaluate how much of your solution will share through either statements or hints.
Let $X_{1},ā¦,X_n $ ...
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0answers
22 views
Consistent estimate of an interval
Suppose I have an interval $[a,b]$, where $a$ is known and $b$ is unknown. Suppose I have a consistent estimator for $b$ denoted as $\widehat{b}$ so that $\widehat{b}=b+o_{p}(1)$. My question: is the ...
3
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2answers
133 views
Convergence in distribution of sum of random variables
Let $\{x_{1,n}\}_{n\in\mathbb{N}},...,\{x_{k,n}\}_{n\in\mathbb{N}}$ be random sequences of zero mean random variables satisfying
$$x_{1,n}\overset{d}{\to} N(0,\sigma^2_1),\cdots, x_{k,n}\overset{d}{\...
2
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1answer
43 views
Convergence Question
Suppose that $X_m \in \mathbb{R}^d$ and $W_m \in \mathbb{R}^{d \times d}$ be a sequence of random variables such that the following asymptotic statements are true
\begin{equation*}
\begin{aligned}
...
1
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1answer
36 views
Find MLE and CI for $\psi = \mathbb{P}_\theta(X_1>0)$ when $X_1,…,X_n \sim \text{IID } \mathcal{N}(\theta,1)$
Question: Let $X_1,\cdots,X_n \sim \text{IID }\mathcal{N}(\theta,1)$ where $\theta\in\mathbb{R}$ is unknown and let $\psi = \mathbb{P}_\theta(X_1>0)$. Find the maximum likelihood estimator $\hat{\...
1
vote
1answer
35 views
Linear regression with one generated regressor
Suppose I have the regression model: $Y_i=T^{\top}_{i}\beta_0+e_{i}$ with $E(e_i|X_i)=0$, where we have two regressors $X_i,\ E(D|X_{i})$ so that $T^{\top}_{i}=[X_i,\ E(D|X_{i})]$. $X_{i}$ is a ...
1
vote
1answer
87 views
MLE Asymptotic Normality regularity conditions
I had this lecture of mathematical statistics about asymptotic normality of MLE. In order to prove this, a series of regularity conditions were stated, and the identifiability condition was among them....
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0answers
19 views
Is this proof of convergence in probability to zero correct?
I want to show that $A=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(\widehat{B}_{i}-B_{i})X_i$ converges in probability to 0, where $B_i=E(C_i|Z_i)$ and $C_i$ is i.i.d. binary and $Z_i$ is a discrete random ...
0
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0answers
26 views
What is Gross Error Sensitivity and Asymptotic Variance in the Context of Correlation Coefficient?
Why did the question arise?
I was reading one of the versions of this paper. They have mentioned why did they use Kendallās tau formula:
We use the Kendallās tau coefficient because it has low ...
