Questions tagged [asymptotics]

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

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32 views

is inverse of 1 + small op(1) equal to 1 + small op(1)?

One rule for small op and big Op is $$ (1+o_p(1))^{-1} = O_p(1) $$ (on page 13 of Vaart, A. W. van der. (1998). Asymptotic statistics. Cambridge University Press.) I am curious whether it is true to ...
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31 views

Bias of MLEs increases rather than decreases in n

In the context of conducting simulations to assess the performance of MLE point estimates for truncated data, I am encountering surprising settings in which the bias of MLEs is clearly non-monotonic ...
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346 views

MLE and non-normality

What is a non-trivial example of an identifiable model whose MLE is consistent, but the MLE's asymptotic distribution is not normal? Parametric setting and IID sample would be desirable.
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22 views

Optimal rate of convergence of nonparametric density estimators

Suppose that $X_1, X_2, \dots, X_n$ forms an independent and identically distributed sample from some $d$-dimensional probability distribution with unknown probability density function $f$. Let $x$ be ...
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14 views

Consistent Estimator for the Dispersion of a GLM

I am trying to figure out the proof for consistency of the estimators for an exponential dispersion family. The proof is well covered in the paper "Consistency and Asymptotic Normality of the ...
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22 views

"Second order correct", "first order asymptotics"

I keep seeing phrases like "first order asymptotic", "second order correct", "high order asymptotics" and I am honestly don't know how these terms are rigorously/...
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Are there small-sample cases where we can approximate the distribution of the Wald statistic better than with the Normal/Chi-squared distribution?

In my specific case, I'm referring to the parameter estimates of a binomial regression (regardless of whether we are estimating risk ratios or odds ratios). Can the problem of divergence of the Wald ...
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30 views

Does $|\hat{\theta}_n|-|\theta|=o_p(\alpha_n)$ implies $\hat{\theta}_n-\theta=o_p(\alpha_n)$

Does $|\hat{\theta}_n|-|\theta|=o_p(\alpha_n)$ implies $\hat{\theta}_n-\theta=o_p(\alpha_n)$? ($\alpha_n\rightarrow 0$ as $n\rightarrow \infty$)
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14 views

Rates of convergence with asymptotically negligibly noisy observations

Apologies in advance if this question is not completely well defined. Suppose that I am estimating a nonparametric model for a conditional expectation function $\mathbb E[Y_i | X_i]$ using some i.i.d. ...
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1answer
33 views

Are two weak instruments better than one weak instrument?

From my understanding, when using IV regression to eliminate confounding effects, we prefer to have a single strong instrument, over multiple weak instruments which can lead to bias. My question is, ...
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99 views

Mixed Model in a repeated measurement design and AUC

My goal is to predict the $Y_i=1$ for each subject $i$ given a set of explanatory variables $x_i$. Since I have repeated measurements for some subjects, I was told to use a mixed model strategy, i.e. ...
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35 views

How to calculate the expectation of the KDE using little-o?

This is possibly a duplicate of this question of mine, however, here I ask for clarification regarding an estimation that is done when calculating the expectation of the kernel density estimator (KDE) ...
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277 views

Delta method for Poisson ratio

Let $X_1,...,X_n$ be drawn from $Pois(\lambda)$ and $Y_1,...,Y_n$ from $Pois(\theta)$. I would like to find the asymptotic distribution of $$\frac{\overline X}{\overline X + \overline Y }$$ using ...
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67 views

Asymptotic Distribution of AUC

It is well known that the area under the curve ($AUC$) is equal to the Mann-Whitney U statistic (c.f. Why is ROC AUC equivalent to the probability that two randomly-selected samples are correctly ...
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47 views

Does $X_n=O_p(\alpha_n)$ implies $X_n^{-1}=O_p(1/\alpha_n)$

In which condition can I say that $X_n=O_p(\alpha_n)$ implies $X_n^{-1}=O_p(1/\alpha_n)$, since it holds for $X_n\sim N(0,1)$.(In this case, $X_n=O_p(1)$ and $\frac{1}{X_n}$ is a.s. finite and hence $\...
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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 ...
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Parameter estimation in linear regression

This is a regression problem. Let $r_i=f_iu_i+e_i,i=1,...,n$, where $u_i\sim^{i.i.d}N(0,1)$, and $e_i=O_p(\alpha_n),\alpha_n\rightarrow 0$ as $n\rightarrow 0$. When I take the log transformation of $...
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49 views

How many samples does one need to perform polynomial regression of degree $m$?

Suppose $(X_i, Y_i)$, $i = 1,\dots, n$ are random variables such that $$X_i\sim N(0,1)$$ $$Y_i = f(X_i) + \epsilon_i$$ where the $\epsilon_i$ are i.i.d. standard Gaussian and $f(x)=\sum_{k = 0}^\infty ...
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1answer
87 views

How is the asymptotic justification of the "linearization by influence function method" for surveys established?

The survey R package recently adopted the "linearization by influence function" method of estimating covariances between domain estimates. The central ...
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51 views

Fisher vs. Asymptotic Consistency - Example using a single observation as the population mean estimator

I am learning about Fisher Consistency and came across this section of a Wikipedia article (https://en.wikipedia.org/wiki/Fisher_consistency#Relationship_to_asymptotic_consistency_and_unbiasedness) ...
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62 views

Asymptotic covariance matrix of $\bar{\pmb x}$

In a text I'm reading it says that we define $$ \begin{align} \bar{\pmb x}= \begin{bmatrix}\bar x_1 \\ \bar x_2 \end{bmatrix} \end{align} $$ And then immediately says the asymptotic covariance matrix ...
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Estimating rate of convergence from MSE

Given that I have calculated values for the MSE for differing values of $n$ and the estimator $\hat{\theta}$. Is it possible to calculate the rate of convergence, $O_p(n^{-r})$, of this estimator by ...
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26 views

Variable converge to known distribution, then what about its likelihood function

If we only know $X$ converge to normal distribution, then is it possible to obtain its likelihood function ? I know it is impossible to get the exact likelihood function. However, can we claim that ...
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91 views

Smoothed CDF to calculate asymptotic normality

If we have the following estimator: $\hat{F_Z}(z)=\frac{1}{N}\sum_{i=1}^N1\{Z_i\leq z\}$. The CDF of $Z$ is defined as $F_Z(z)=Pr(Z\leq z)$. $Z_1, ..., Z_N$ is i.i.d. data. What would be the steps to ...
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Consistency and Asymptotic Normality in Generalized Linear Mixed Models

I am looking for a general, and perhaps classical, reference with a proof of the consistency and asymptotic normality of marginal maximum likelihood estimators of the parameters of generalized linear ...
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23 views

limit behaviour of a quadratic form

I'm reading a book on portfolio optimization and risk management, and I wanna clarify what the author wants to say. Let $\mathbf{X}=[X_1,...,X_n]$ be a random vector with mean $\mathbf{\mu}=E\{\mathbf{...
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388 views

Asymptotic Normality and Consistency

I have difficulties understanding the concept of asymptotic normality and consistency. Take an estimator of a parameter which is consistent and asymptotically normally distributed. Because it is ...
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1answer
164 views

Asymptotic bias of LASSO vs. none of SCAD

I am reading a paper which says that LASSO is asymptotically biased while SCAD is not. I take asymptotic (un)biasedness to concern the slope estimators from LASSO and SCAD as the sample size goes to ...
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18 views

About the validity of an expansion result

Suppose I have an expansion result: $sup_{p\in P,x\in X}|G_N(p,x)-g_N(p,x)|=O_p(a_N)$. Suppose now I have some (nonparametric) estimator for $p$ denoted by $\widehat{p}=\widehat{p}(x)\in P$ for any $x\...
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123 views

Likelihood function as number of observations increases

If we have $n$ iid observations from some $X \sim p(\cdot|\theta)$, what happens to the likelihood function $p(x_1,\dots,x_n|\theta)$ as $n\rightarrow \infty$? I plotted the product of several $\...
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1answer
41 views

What, if any, asymptotic arguments are used in moving between various statements of the central limit theorem?

What, if any, asymptotic arguments are used in moving between the various statements of the central limit theorem (e.g. in terms of sample means compared with standardised sample means)? Context. My ...
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54 views

Reference for regularity conditions for asymptotic of MLE

I wonder if there is a complete list of regularity conditions for MLE asymptotic normality. I read this post and found a list of 6 conditions but the answer does not include any reference. I read the ...
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2answers
35 views

Clarification on $m$ and $n$ I'n the $m$-out-of-$n$ bootstrap

I've asked questions on the $m$-out-of-$n$ bootstrap here before. Responses have been quite valuable, but one key aspect still sparks some confusion from non-statisticians. Chernick (2007, 2011) ...
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125 views

Asymptotic distribution of $ \sqrt{n} (θ_n - θ)$ then CI of $θ$

Suppose we have $X_1,...,X_n$ iid with distribution: $f(x)=xe^{−(\frac{x^2 − θ^2}{2})},x≥θ, θ > 0$ By calculated the median of $f(x)$, $X_{\frac{1}{2}}$ equals $ \sqrt{θ^2 + \log4}$ we obtain an ...
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261 views

Basic properties of the kernel density estimator

This is a question from a mathematical statistics textbook, used at the first and most basic mathematical statistics course for undergraduate students. This exercise follows the chapter on ...
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1answer
45 views

Delta Method around zero is a N(0, 0)

I have this problem: $\sqrt N \hat{\theta} \sim N(0, V)$ where $E(\hat{\theta}) = \theta_{0} = 0$. I must find the asymthotic distribution of $\frac{N}{V}\hat{\theta}^{2}$ but if I use the Delta ...
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1answer
103 views

Asymptotic distribution of OLS standard errors

Consider the linear regression model: $y_i = \beta + u_i$. We might write this as: $$Y = X\beta + U\text{ with }Y = \begin{pmatrix}y_1 \\ y_2 \\ \vdots \\ y_n\end{pmatrix}, X = \begin{pmatrix}1 \\ 1 \\...
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24 views

Standard Error of ERGM Coefficients

I am trying to calculate the standard error of ERGM coefficients, which is estimated by MCMC sample. For an ERGM $P(y;\eta) = \exp[\eta^\top g(y) - \psi(\eta)]$, denote $\eta$ as the true parameter, $\...
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144 views

Deriving the limiting distribution of the Hodges-Le Cam estimator in Bickel and Doksum (2015)

I am trying to better understand the Hodges-Le Cam estimator, and am having difficulty rendering explicit some of the asymptotic arguments in the derivation of the estimator's limiting distribution. I ...
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1answer
174 views

Asymptotic normality of MLE

We know under regularity conditions the MLE is asymptotically normal. Usually, it is said that in practice it's hard to check these assumptions. However, I wondered whether we can claim that these ...
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32 views

Asymptotic Variances of MLE with Orthogonality Constraints

I have a model that is parameterized by orthogonal matrix $\boldsymbol \Gamma$ and other unconstrained parameters $\theta$ that are closely related to $\boldsymbol \Gamma$ within the model. ...
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1answer
87 views

Asymptotic chi-squared distribution of likelihood ratio statistic in regression problem

There is a famous result, going back to Wilks (1938) "The large-sample distribution of the likelihood ratio for testing composite hypotheses" (Ann. Math. Stat., 9, 60-62) that states that ...
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38 views

Asymptotic efficiency of estimators of autoregressive models

Are OLS or MLE estimators of autoregressive model asymptotically efficient if errors are i.i.d? Consider the case of an AR(1) model $$x_t=\alpha x_{t-1} + \epsilon_t$$ with $\epsilon_t$ ~ $i.i.d. N(0,\...
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1answer
30 views

Normal approximation and Hoeffding bound

Hoeffding bound for any $\epsilon>0$ is: $$P_F(|\bar{X}_n-\mu(F)|\geq \epsilon)\leq 2 \exp\{-\frac{n\epsilon^2}{2}\}=h(\sqrt{n}\epsilon)$$ wherever $|X|<1$. Now I want to have a comparison ...
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69 views

Convergence of uniformely distributed random variables on a sphere

I am reading "Asymptotic Statistics" by A.W van der Vaart and I am stuck with an exercise of chapter 2. Here is the question : for each $n \in \mathbb{N}$, let $U_n$ be uniformly distributed ...
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1answer
50 views

Why approximate delta-method Variance isn't multiplied by $\frac{1}{n}$?

I'm reading Casella-Berger chapter 10, where they introduce asymptotic evaluations. I don't seem quite to understand how the factor $\sqrt{n}$ works when we are using asymptotic evaluations in order ...
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23 views

Local Data Generating Process in Semiparametric Statistics

I am a bit confused about the LDGP assumption that is mentioned in books on semiparametric statistics. For example, in Semiparametric Theory and Missing Data by Tsiatis, the LDGP is defined as follows:...
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1answer
18 views

OLS Consistency for Count Data?

Why would we choose Poisson / NB regression (GLM) over OLS for fitting count data? Is there a way to show that OLS estimator would lost it consistency and asymptotic normality for count data? I'm ...
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44 views

relation between wald statistic and f statistic

$Y=X_i\beta_i+X_2\beta_2+\epsilon$ Y and $\epsilon$ are n*1 matrix $X_1$ is $n*k_1$ matrix $X_2$ is $n*k_2$ matrix $\beta_1$ is $k_1*1$ matrix $\beta_2$ is $k_2*1$ matrix F is f statistic for testing $...
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28 views

Different regularity conditions for finite population CLT

I am having trouble understanding the different regularity conditions for different versions of the finite population central limit theorem. I would greatly appreciate any help or insight anyone has. ...

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