Questions tagged [asymptotics]
Asymptotic theory studies the properties of estimators and test statistics when the sample size approaches infinity.
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1answer
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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0answers
8 views
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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0answers
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\}}$
...
3
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1answer
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 ...
3
votes
1answer
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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0answers
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 ...
2
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1answer
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 ...
4
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1answer
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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0answers
18 views
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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0answers
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$$
This seems pretty plausible.
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27 views
Consistency of variance estimator in OLS [duplicate]
Given the model,
$$
y_i = x_i'\beta + \epsilon_i \quad \epsilon_i \sim N(0, \sigma^2)
\quad iid \quad \forall i = 1, ..,n
$$
how can I prove that the estimator of the variance $\hat{\sigma}^2 = \...
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0answers
30 views
Tail behaviour of normal cdf?
Q: What is the tail behaviour of $\log \Phi(t)$ as $t \to \infty$?
Since $\Phi(t) \to 1$ as $t \to \infty$, we know that $\log \Phi(t)\to 0$, but I would like to know at what rate this function ...
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0answers
13 views
Convergence in distribution of the $2SLS$ estimator for some $\pi = \frac{h}{\sqrt{n}}$
Consider the following IV model.
$$y_{1,i} = y_{2,i} \beta + u_i$$
$$y_{2,i} = z_{i}' \pi + v_{2,i}$$
with $(z_i', u_i, v_{2,i})$ iid and $\begin{pmatrix}
u_i \\
v_{2,i}
\end{pmatrix} \sim N(0_{...
2
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1answer
36 views
What is a good aproximation in asymptotic normality?
I have a conceptual doubt. For example, suppose I have $X_i \stackrel{iid}{\sim} N(\theta^*,1)$ and I know that (I have the information) $\theta^{*}\geq 0$. So I have the Constrained Maximum ...
1
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1answer
33 views
Implications of i.i.d. sample
I have the following question:
I have managed to solve it but I wasn't sure if my reasoning was correct. So I can express the OLS estimator as
$\sqrt{n}(\hat{\beta} - \beta) = (\frac{1}{n}\sum_{=1}^{...
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votes
1answer
26 views
asymptotic behavior of OLS estimators
Conceptually when do estimated regression coefficients converge with the true values; is it
a) when the sample size of observations tends to infinity
or
b) when the regression with the same predictors ...
2
votes
2answers
62 views
Can statistical dependence arise only at the limit?
(This has been inspired by a comment exchange with @guy).
Assume we have two infinite sequences of random variables, $\{X_n\}$ and $\{Y_n\}$. Assume the RVs in $\{X_n\}$ are statistically ...
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0answers
45 views
Asymptotic distribution using the Delta method
Let $X_1, \ldots, X_n$ be i.i.d. normal random variables, where $X_i \sim N(\theta, \theta^2)$ with an unknown $\theta > 0$.
We could for example estimate $\theta$ using the average
$$\hat\theta_n ...
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votes
3answers
132 views
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 ...
4
votes
1answer
86 views
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-...
5
votes
1answer
77 views
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 ...
2
votes
2answers
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, \...
3
votes
0answers
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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0answers
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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0answers
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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0answers
26 views
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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vote
2answers
34 views
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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0answers
11 views
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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0answers
34 views
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
19 views
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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0answers
24 views
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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0answers
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:...
3
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1answer
82 views
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 \...
4
votes
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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1answer
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 ...
0
votes
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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0answers
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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2answers
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 $σ^2$....
1
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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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0answers
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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0answers
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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0answers
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 ...
2
votes
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 ...
1
vote
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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0answers
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 ...
3
votes
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 $X$.
...
0
votes
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 $y_t$ ...
