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

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$p$-value for non-standard asymptotics

Suppose I have an asymptotic result like $$\sqrt{n}(T_n - \theta) \overset{D}{\to} \sum_{i=1}^k \lambda_i X_i$$ where $X_i$ are independent $\chi^2_1$. i.e. some test statistics $T_n$ is ...
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1answer
19 views

MLE asymptotic properties in non-regular families

I am working with asymptotic results about the MLEs and I know that if the family of distributions to whom the pdf of my sample belongs is exponential the regularity conditions for the asymptotic ...
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44 views

Convergence in distribution and degenerate random variable

Let $\{Y_n\}$ be a sequence of random variables with an associated sequence of CDFs $\{F_n\}$ given by : $$F_n(y) = \begin{cases} 0 & \textsf{for}&y <0 \\ (\frac{y}{\theta})^n & ...
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1answer
28 views

What is the asymptotic distribution of the variance of the error term (in MLE linear regression)

In most treatments of the MLE linear regression, the author focuses on the asymptotic normality of $\hat \beta_{MLE}$. To estimate $Var(\hat \beta_{MLE})$, which relies on $\sigma^2$, they also show ...
5
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81 views

What can go wrong using lagged terms as instrumental variables?

Can anybody give one example of when the set of all lagged $X$ can (or can't) be a good choice of IV's for $X_{t}$?
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20 views

Observed Fisher Info as an estimator of Expected Info

When I construct an asymptotic confidence interval for a parameter $\theta$, taken from a sample iid distributed with a generic pdf/pmf, I usually implement the mle $\hat\theta$ instead of $\theta$, ...
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1answer
29 views

Using asymptotic distribution under the null when the null hypothesis is false

This is a general question about hypothesis testing in statistics. A standard hypothesis test as I see it is based on 3 things: Stating a null hypothesis Computing a test statistic for the ...
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1answer
24 views

How do weak instruments violate full rank condition?

So I know that if the instruments $(Z_i)$ in 2SLS regression are weak i.e. $X_i$ and $Z_i$ are uncorrelated, then full rank condition for matrix $\Sigma_{ZX}=E[Z_iX_i]$ will fail to hold, but how can ...
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60 views

Is it better to use a MLE or a MME to build an asymptotic confidence interval for a real parameter $\theta$?

I thought that the answer was pretty straightforward given that the MLEs possess some strong asymptotic properties, i.e. normality, efficiency and consistency. But then, I have found that also MME ...
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21 views

Asymptotic normality for nonsmooth objective functions

Assume that $f ({\bf x}; \theta): \mathbb{R}^p \times \Theta \to \mathbb{R}$, where ${\bf x}$ is the vector of inputs (with some distribution) and $\theta$ is the vector of parameters. Also, assume ...
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17 views

Asymptotically exact confidence interval

I am trying to answer the following question: I have modeled it using a Poisson distribution, and from this calculated the Fisher information as $I(\lambda)= \frac{n}{\lambda}$. Then, when I ...
2
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1answer
53 views

Likelihood Ratio Test for the variance of a normal distribution

I've found that the asymptotic LR test is used in simple vs bilateral hypothesis test in which it is impossible to actually compute the rejection region, or better, in which we would need to find a ...
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1answer
23 views

To determine asymptotic distribution, when to ignore lower-order $O_p$ terms?

Let $A_{m,n} + B_{m,n}$ be such that: $A_{m,n} \overset{d}{\rightarrow} A$ as $m,n \rightarrow \infty$ $A_{m,n} = O_p(\sqrt{nm(n+m)})$ $B_{m,n} = O_p(\sqrt{mn})$. Under what conditions would the ...
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24 views

Negative population variable importance

The question is: Is it possible for the population variable importance to be negative ? The variable importance is defined below in the context it comes from: random forests. But no knowledge is ...
4
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1answer
46 views

Is finite and strictly positive variances all that is needed for the CLT Lindeberg condition?

I cannot understand what I am doing wrong here, so please somebody, point it out from me. The issue? I keep finding that the Lindeberg sufficient condition for the Central Limit Theorem for ...
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35 views
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109 views

Chebychev’s Weak Law of Large Numbers

This theorem is on Econometric Analysis (7th edition) by Greene (2012), Page 1071. It states that "If $x_i$, $i=1,2,...,n$ is a sample of observations such that $E(x_i)=\mu_i<\infty$ and ...
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31 views

Is Chi-Square test when divided by variance valid?

The Pearson chi-square test is $$ \sum_{i=1}^k \dfrac{(X_i - \text{E}[X_i])^2}{\text{E}[X_i]}$$ A good link as to why it is divided by the expected value (rather than variance) is ...
3
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100 views

Law of large numbers and convergence

Suppose $X{\sim}N(\mu_1,\sigma_1^2)$ and $Y{\sim}N(\mu_2,\sigma_2^2)$, $x_1,...x_n$ and $y_1,...,y_n$ are i.i.d samples from X and Y, respectively. Consider the estimator ...
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105 views

How to formally prove two formulas are asymptotically equivalent?

I have two formulas about variance estimation of linear combination of order statistics given order statistics $X_{1}\leqslant X_{2} \leqslant ...\leqslant X_{n} $ from a random sample of arbitrary ...
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258 views

Mathematical definition of Infill Asymptotics

I am writing a paper that uses infill asymptotics and one of my reviewers has asked me to please provide a rigorous mathematical definition of what infill asymptotics is (i.e., with math symbols and ...
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26 views

how to find asymptotic joint distribution of two linear combination of order statistics?

Suppose I have n order statistics from some unknown continuous distribution funciton F(x), $X_{1}\leqslant X_{2}\leqslant...\leqslant X_{n}$. And I have two linear combination of these order ...
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56 views

Why do statisticians prove asymptotic normality?

In many statistics papers, authors suggest a new data analysis methodology and prove its properties such as consistency or asymptotic normality. I think it's a kind of tradition or custom. I ...
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113 views

Asymptotic normality of order statistic of heavy tailed distributions

Background: I have a sample which I want to model with a heavy tailed distribution. I have some extreme values, such that the spread of the observations are relatively large. My idea was to model this ...
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23 views

Does the asymptotic distribution of sample median follow from statistical functionals?

I know that if $F_n$ is the empirical distribution function and $F$ is the true distribution function, then, with $T$ being any statistical functional (satisfying the von-Mises derivative conditions), ...
3
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54 views

Gaussian Process Infill Asymptotics

I had asked a question recently about what happens to the predictive variance of a Gaussian process as you let $n\rightarrow\infty$ and have realized what that the name of these type of asymptotic ...
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1answer
24 views

Distribution of weighted Cox regression coefficients

I know that under typical conditions, the coefficients of a Cox model, the coefficients are asymptotically normally distributed. I plan to weight my Cox model by inverse probability of treatment ...
6
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1answer
102 views

Fisher information for $\rho$ in a bivariate normal distribution

I have seen many times people using the Delta method in order to find the asymptotic distribution of $r$, the sample correlation coefficient, for bivariate normal data. This distribution is given by ...
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1answer
32 views

Asymptotics of a quadratic form with growing vector / matrix dimensions

Let $\ {\bf x}_n=\big\{x_1,x_2,...,x_n\big\}$ be a vector of random variables with ${\bf m}_n=\big\{\mu_1,\mu_2,...,\mu_n\big\}$ and $\ n^{1/2}({\bf x}_n-{\bf m}_n)\rightarrow N(0,\Omega_n)$ Assuming ...
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1k views

Latin Hypercube Sampling Asymptotics

I am trying to construct a proof for a problem I am working on and one of the assumptions that I am making is that the set of points I am sampling from is dense over the entire space. Practically, I ...
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21 views

Understanding Borel-Cantelli in an “asymptotic” setting

Let $X_1(p),...,X_j(p),\dots$ be a sequence of random variables that depend on a parameter $1\le p \le \infty$. Suppose that the $n$-th variable satisfies $Pr [X_n>t] < c_n/p^n$ for some ...
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Asymptotic Distribution of Sample Quantile

My question is related to the proof below. Why can we say that $F^{-1}(Y_n(x))=X_{[np]}$? I would like to understand better the reason for that. Any help would be appreciated. I've also asked ...
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Proof of the asymptotic normality of the sample estimator $\rho_k$ and of the Bartlett formula

I was reading the two above proofs from Brockwell and Davis (1991), Time series theory and methods Theorems 7.2.1 and 7.2.2, but there's a lot of formalism that I don't get very well... Can somebody ...
3
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1answer
107 views

Product and sum of big $O_p$ random variables

I seems that people often use the following properties $O_p(a_n)O_p(b_n) = O_p(a_nb_n)$ and $O_p(a_n)+O_p(b_n) = O_p(a_n+b_n)$. I'm wondering, if these are true for any sequences $a_n,b_n$. The reason ...
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52 views

notation for point estimation in “All of Statistics” by Larry Wasserman

I'm currently reading All of Statistics by Larry Wasserman. The question is about the Delta Method, page 160 in the pdf (I have a hard copy, there it's page 133). The authors defines $\hat{\theta}$ or ...
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22 views

Asymptotics of the MLE: a different flavor of proof? [Reference request]

I'm currently trying to understand more about the properties of the maximum likelihood estimator. It's known that, in the large data-limit, the MLE becomes an unbiased estimator with almost Gaussian ...
3
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1answer
111 views

Fisher consistency versus “standard” consistency

My question relates two types of consistency. In particular, how does the Fisher consistency differ from standard notions of consistency, such as convergence in probability to the generative ...
5
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1answer
99 views

Is it possible to construct a hypothesis test for the existence of a mean of a symmetric distribution?

In practice, we often assume that the process we are examining has a mean, and so statistics involving averages are defined. As pure conjecture, I was wondering if one can construct a hypothesis test ...
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Asymptotic variance of “log odds ratio”?

Can anyone state the asymptotic variance of "log odds ratio" for the PRODUCT BINOMIAL case in a 2 X 2 contingency table ?
2
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1answer
50 views

Convergence of Bernoulli sampling procedure

Let $X$ be a variable which has density $f_{X}$ and mean $E[X]=\mu$ Define $B$, a Bernoulli variable, which has a probability $P(b_{i}=1)=p_{i}=\frac{1}{N^\gamma}$ Consider the following ...
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23 views

What is the intuition behind asymptotic normality of a parameter vector

I'm studying statistical inference and am trying to understand what it means for the estimate of a parameter vector to be asymptotically normal. I can understand the univariate version of this but ...
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49 views

Conjecturing asymptotic normality for a sum of dependent random variables

I am hunting for the asymptotic distribution of a scaled partial sum of pair-wise equi-correlated, identically distributed continuous random variables $$W =n^{-\delta}\sum_{i=1}^nY_i(n),\;\; ...
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26 views

Theoretical properties of Gaussian Process Emulator

I am studying Guassian Prcess Emulator (GPE) to approximate computationally expensive computer models. Basically, we suppose the computer model, or simulator, is denoted by $f(x)$, where $x$ is the ...
5
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1answer
85 views

Intuition for difference in asymptotic distribution

I am looking for some intuition on the following: Assume a sequence of random variables $\{X_n\}$ for which we know that the following hold: $$X_n \xrightarrow{p} c \neq 0$$ $$Z_n \equiv \sqrt{n}(X_n ...
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1answer
39 views

Why are the Wald standard errors often very poor estimates of the uncertainty of variances?

As from this post as @Ben Bolker pointed out that : .. note that these (as often pointed out by Doug Bates) the Wald standard errors are often very poor estimates of the uncertainty of variances, ...
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Testing the variance component in a mixed effects model

Say $y=X\beta+ Zu +\epsilon$ is our mixed effects model where $u=(u_1,..,u_r)$ and $u_{j} \stackrel{i.i.d.}{\sim} N(0, \sigma^2_{a})$ for $j=1,...,r$ and $\epsilon=(\epsilon_1,...,\epsilon_n)$ are ...
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How to compare orders of magnitude?

In Fan and Li's paper "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties", they provided a proof to Theorem 1. The very last part of the proof is as follows. Some ...
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1answer
42 views

Cross-validating the tbats/bats function in forecast

Is there a way to cross validate the tbats/bats function in the forecast package in R? I have been trying to get CV weighted parameters which then I can pass to a function for revised estimates. ...
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Does a “pruned” i.i.d. multivariate sample behave as the i.i.d. sample?

Let $z_1,\cdots,z_n$ be $n$ points drawn i.i.d. from $\mathbb{C}N_p(0,\Sigma_n)$. The distribution of the covariance $S_n=\frac1n\sum_{i=1}^n z_i z_i^*$ is well known in the limit as $n,p(n)\to\infty$ ...
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1answer
122 views

How to prove test statistic has $\chi^2$ distribution using minimum chi-square estimator?

This is for iid case where the test statistic is sum of $(observed - expected)^2/expected$. It is supposedly shown on page 5 of Fisher (1924): ...