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Questions tagged [asymptotics]

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

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30 views

Convergence of error term

I was deriving the ratio of a Laplace approximation with the true quantity and I got this: $$ \left(n\bar{x} + \alpha - \frac{1}{2}\right)\log\left(\frac{\bar{x}+(a-1)/n}{\bar{x}+\alpha/n}\right) + n\...
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The Cochran and Cox Approximation Pair T test Unequal Variance

Hi everyone I hope you are well. Maybe as you know according to Behners-Fisher problem (unequal variance case of samples) there are some kind of approximations. I have recently covered the ...
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17 views

Convergence in probability (asymptotic notation) result

Let $h=h_n$ be a sequence of numbers such that $h_n \rightarrow 0$ as $n \rightarrow \infty$, $\mu$ be a real constant and $f$ be some probability density function. I was wondering if the following ...
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16 views

The limit distribution of Wilcoxon signed rank statistic?

An alternative representation of the Wilcoxon signed rank statistic $V$ is $V=\sum_{i\le j}\mathbb{I}_{\{X_i+X_j>0\}}=\sum_i\mathbb{I}_{\{X_i>0\}}+\sum_{i<j}\mathbb{I}_{\{X_i+X_j>0\}}$ ...
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49 views

AIC for increasing sample size

I am using AIC as a model selection criteria in one of my projects. However, since AIC isn't dependent on the number of points sampled, for large n the log likelihood term rapidly outscales the ...
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104 views

What “nice” property of a confidence set is being violated

Suppose that I have $X_{i} \overset{i.i.d.}{\sim} P$ with $E[X_{i}]=\mu$ and $V[X_{i}^{2}] = \sigma^{2}<\infty$. Then by the central limit theorem I know that: \begin{align} \sqrt{n} (\bar{X}_{n} ...
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19 views

Behavior of likelihood far from peak

In the neighborhood of the maximum likelihood point the log-likelihood function is often fruitfully expanded as a quadratic function of the parameters. Are there any general results about the shape ...
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63 views

How to derive the asymptotic distribution of t-statistic?

Let ${X_n}$ be an IID sample such that ${X_i} \sim N(\mu,\sigma^2)$. When both $\mu$ and $\sigma$ are unknown, we construct $t(\hat{\mu},s)=\dfrac{\sqrt{n}(\hat{\mu}-\mu)}{s}$, where $s$ is the sample ...
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66 views

Convergence of scaled $L_1$ distance between two sorted random vectors with same limiting distribution

Let $ X=(X_1,\dots,X_n) $ and $ Y=(Y_1,\dots,Y_n) $ where the RVs $ X_1,\dots X_n, Y_1,\dots Y_n $ are independent and have the same limiting distribution (assume for simplicity that all moments exist)...
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Type 2 error in t-test on time series

I have an AR(1) time series with $1>\phi>0$. If I naively use t-test to check $H_0:\mu=0$ and it does not reject the null, then can I accept the result? I think yes because for a time-series ...
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18 views

Asymptotic probability that chi-squared distribution exceeds its mean

Let $\chi_k$ be the chi-squared distribution with $k$ degrees of freedom. $\chi_k$ has mean $k$. Is it true that $$\lim_{k \to \infty} \Pr[\chi_k \geq k] = 1/2$$ This seems pretty plausible.
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Consistency of variance estimator in OLS [duplicate]

Given the model, $$ y_i = x_i'\beta + \epsilon_i \quad \epsilon_i \sim N(0, \sigma^2) \quad iid \quad \forall i = 1, ..,n $$ how can I prove that the estimator of the variance $\hat{\sigma}^2 = \...
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30 views

Tail behaviour of normal cdf?

Q: What is the tail behaviour of $\log \Phi(t)$ as $t \to \infty$? Since $\Phi(t) \to 1$ as $t \to \infty$, we know that $\log \Phi(t)\to 0$, but I would like to know at what rate this function ...
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Convergence in distribution of the $2SLS$ estimator for some $\pi = \frac{h}{\sqrt{n}}$

Consider the following IV model. $$y_{1,i} = y_{2,i} \beta + u_i$$ $$y_{2,i} = z_{i}' \pi + v_{2,i}$$ with $(z_i', u_i, v_{2,i})$ iid and $\begin{pmatrix} u_i \\ v_{2,i} \end{pmatrix} \sim N(0_{...
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What is a good aproximation in asymptotic normality?

I have a conceptual doubt. For example, suppose I have $X_i \stackrel{iid}{\sim} N(\theta^*,1)$ and I know that (I have the information) $\theta^{*}\geq 0$. So I have the Constrained Maximum ...
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1answer
33 views

Implications of i.i.d. sample

I have the following question: I have managed to solve it but I wasn't sure if my reasoning was correct. So I can express the OLS estimator as $\sqrt{n}(\hat{\beta} - \beta) = (\frac{1}{n}\sum_{=1}^{...
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26 views

asymptotic behavior of OLS estimators

Conceptually when do estimated regression coefficients converge with the true values; is it a) when the sample size of observations tends to infinity or b) when the regression with the same predictors ...
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62 views

Can statistical dependence arise only at the limit?

(This has been inspired by a comment exchange with @guy). Assume we have two infinite sequences of random variables, $\{X_n\}$ and $\{Y_n\}$. Assume the RVs in $\{X_n\}$ are statistically ...
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45 views

Asymptotic distribution using the Delta method

Let $X_1, \ldots, X_n$ be i.i.d. normal random variables, where $X_i \sim N(\theta, \theta^2)$ with an unknown $\theta > 0$. We could for example estimate $\theta$ using the average $$\hat\theta_n ...
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When does $X_n\stackrel{d}{\rightarrow}X$ and $Y_n\stackrel{d}{\rightarrow}Y$ imply $X_n+Y_n\stackrel{d}{\rightarrow}X+Y$?

The question: $X_n\stackrel{d}{\rightarrow}X$ and $Y_n\stackrel{d}{\rightarrow}Y \stackrel{?}{\implies} X_n+Y_n\stackrel{d}{\rightarrow}X+Y$ I know that this does not hold in general; Slutsky's ...
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How is it possible for both the likelihood and log-likelihood to be asymptotically normal?

I was trying to understand asymptotic normality of the posterior better, and came across a confusing point. So let's say we have a likelihood, $L(\theta | X) = \Pi_{i=1}^n p(X_i | \theta)$, so the log-...
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distribution for scaled Maximum of n independent Weibulls for $n \to \infty$

Assume that $X_1, X_2,...\sim Weibull(\lambda, k) \quad iid.$, i.e. $F(X_1\leq x) = 1-e^{-(\lambda x)^k}$ define $M_n:= \max\{X_1, ..., X_n\}$ and $\tilde{M}_n:=\frac{M_n-b_n}{a_n}$ according to ...
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41 views

Asymptotic Expectation of Ratio of Sample Averages

I have two random variables: $X$ and $Y$. I know that: \begin{equation} E[X]=E[Y]=\mu>0 \end{equation} I know that variance of both can be bounded: \begin{equation} \operatorname{Var}[X]<k, \...
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60 views

Terminology: “Central Limit Theorem” for Delta Method

This is a question about when is it appropriate to call an asymptotic normality statement, the "Central Limit Theorem" (CLT). More specifically, suppose I have $X_1, X_2, \dots X_n$ iid from a ...
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26 views

why is method of moments estimates asymptotically normal

I have noticed that a lot of statistics textbooks contain lengthy discussions and detailed proofs on showing that MLE estimates are asymptotically normal (under regularity conditions). On the other ...
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35 views

Approximating standard error that contains a parameter, by replacing the parameter with its estimate

I am a bit confused about the following step I have seen in the stats literature which seems to me a bit circular. Say you are approximating the standard error of the MoM estimate of an exponential ...
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Limit behavior of “weighted” Poisson Binomial distribution

Given $X_n \sim \operatorname{Binomial}(n, p_n)$ it is known by the Poisson Limit Theorem that as $n \to \infty$ that $$X_n \to \operatorname{Poisson}(np_n).$$ This can be generalized to hold for ...
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Question about expectation in OLS?

Consider the linear model $$y_i = x_i^T\beta + \epsilon_i.$$ In ordinary least squares it is assumed that the errors satisfy $E[\epsilon_i]=0$. This implies that that $\dfrac{X^T\epsilon}{n} \to 0$ ...
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Sandwich standard errors versus typical standard error estimation

Question: do sandwich estimators of the standard errors equal the typical estimation of the standard error IF the data was generated with constant variance in the residuals and as the sample size ...
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Incorrect computation in Knight and Fu (2000)?

I'm currently reading Knight and Fu's 2000 paper on the asymptotics of "Bridge" estimators with a particular focus on LASSO as a special case. In the proof of theorem 2, they make the claim that under ...
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1answer
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Feature selection by lasso in two samples compared to one joint sample

Let's say you have two sets of features $X_1$ and $X_2$ together with a response variable $Y$. I wonder whether the two following procedures are identical asymptotically (or in finite samples) in ...
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Is there an asymptotic distribution expansion that does not go negative when truncated?

I am looking into a problem where I have a distribution that converges to the normal distribution as its parameters become large. I am looking at the rate of this convergence, and seeking to describe ...
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36 views

Parametric Bootstrap Central limit theorem non i i d

I am having paired data with missing values in a single arm. I am willing to use parametric bootstrap with specific quadratic tests to test the hypothesis of equality of means. My model is as follows:...
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1answer
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Is the distribution of the logarithm of the mean of Bernoulli random variables ($\log \overline X$) still asymptotically normal?

Let $\overline X$ be the mean of a Bernoulli random variable (r.v.) $$\overline X = \frac{1}{n}\sum_{i=1}^{n} X_i$$ where $X_i \in \{0, 1\}$. So based on Central Limit Theoreom, $$\overline X \sim \...
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1answer
70 views

Does convergence in distribution imply asymptotic stationarity?

Let ${\bf \tilde{x}}_1, {\bf \tilde{x}}_2, \ldots$ be a (possibly non-stationary) stochastic sequence of $d$-dimensional random vectors that converges in distribution. Does it immediately follow that ...
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40 views

In large sample theory does $\hat{\theta} \sim N(\theta, I^{-1}(\hat{\theta}))$ refer to only diagonal entries of $Cov(\hat{\theta})$?

In large sample theory does $\hat{\theta} \sim N(\theta, I^{-1}(\hat{\theta})$) implicitly mean that only diagonal entries of $Cov(\hat{\theta})=I^{-1}(\hat{\theta})$?
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23 views

Best estimator for the variance of the empirical median

Let $X_i$ be $n=2k+1$ IID continuous random variables with distribution function $F$ and quantile function $F^{-1}$. Let $m$ be their empirical median. It is well known that: If the $X_i$ were ...
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1answer
27 views

jackknife estimator with central limit theorem

Let $\hat{\theta}_n$ be an estimator of the parameter $\theta$ from the sample $\Omega_n$ of $n$ observations, satisfying that $\sqrt{n} (\hat{\theta}_n-\theta) \overset{d}{\longrightarrow} \mathcal{N}...
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17 views

Confidence interval with an unknown constant bias

Assume that we have an estimator $T_n$ of the parameter $\theta$ where $n$ is the sample size and there exists an unknown constant $C$ such that $\sqrt{n}(T_n-\theta) - C \overset{d}{\longrightarrow} ...
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141 views

Deriving an expression for a confidence interval for σ^2 using the asymptotic distribution of √n(σ̂^2−σ^2)

We have We have $X1,…,Xn i.i.d N(μ,σ^2) $where $μ$ is known and $σ^2$ isn't known. $σ̂^2=(\frac{1}{n})∑(X_i−μ)^2$. First of all what I did, I derived an equitailed 95% confidence interval for $σ^2$....
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1answer
190 views

Proving Asymptotic distribution of $\sqrt n( \widehat\sigma^2 -\sigma^2)$

I am looking at trying to derive an expression for the asymptotic distribution. We have $X_1,\ldots, X_n$ i.i.d $N(\mu, σ^2)$. So we have defined $\hat \sigma^2 = \frac 1n \sum_{i=1}^n(X_i-\mu)^2$. (...
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1answer
197 views

Asymptotic distribution of sample variance via multivariate delta method

I was trying to get the asymptotic distribution of sample variance using multivariate delta method and without normality assumption. So I defined the random vector $ z = \left( \begin{matrix} X \\...
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20 views

Examples of convergence in distribution using CDF directly [duplicate]

I find very interesting the example that if we let $Q_n$ be the maximum of n i.i.d. with distribution $U[0,\alpha]$, then $n(Q_n - \alpha)$ converges to an exponential distribution. See e.g. here for ...
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57 views

Asymptotic properties for LASSO in logistic regression

I am trying to find a paper or book that gives the asymptotic properties for the LASSO for logistic regression. Anyone has any suggestions please?
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124 views

Why without regularization, the asymptotic nature of logistic regression would keep driving loss towards 0 in high dimensions?

While understanding the Logistic regression, I didn't completely get the behavior of its asymptotic nature which says: Without regularization, the asymptotic nature of logistic regression i.e (it ...
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1answer
43 views

Asymptotic Normality of Posterior distribtuion

I am trying to understand asymptotic Normality of posterior distributuions. Specifically, what are the (regularity) conditions that allow us to use this approximation? The problem is that the ...
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1answer
72 views

Establishing convergence in probability from a related convergence in distribution

Is it true that $\sqrt n (\hat{\theta}-\theta) \ \rightarrow_d \ N(0,\sigma^2)$ implies $\text{plim} \ \hat{\theta} = \theta $? If so, how can I prove this? Attempted proof: My proof is like ...
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36 views

Perform a wald's test to check the statistical significance of biasedness of MLE

I have done a simulation to see that MLEs are asymptotically unbiased. I want to know whether I can perform a wald test here to check the statistical significance of the difference between the mean of ...
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2answers
299 views

Convergence to a Uniform Distribution

$\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} $ Show that if $P(X_n = i/n)=1/n$ for every $i = 1,...,n$, then $X_n$ converges in distribution to a uniformly distributed random variable $X$. ...
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1answer
117 views

Using the asymptotic normal approximation to derive confidence intervals for a binomial distribution

Let $y_1, ... y_T$ be a random sample of T = 100 observations on iid Bernoulli distributed random variables $Y_t$ that represent individual decision making where $y_t$ = 1 with probability θ and $y_t$ ...