# Questions tagged [asymptotics]

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

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### 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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### 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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### 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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### 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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### 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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### "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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### 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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### 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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### 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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### 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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### 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) ...
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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
$Y=X_i\beta_i+X_2\beta_2+\epsilon$ 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 \$...