Questions tagged [asymptotics]

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

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6 views

What is Gross Error Sensitivity and Asymptotic Variance in the context of Correlation Coefficient?

Why did the question arise? I was reading one of the versions of this paper. They have mentioned why did they use Kendall’s tau formula: We use the Kendall’s tau coefficient because it has low ...
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1answer
60 views

Asymptotic t test question- regression when you do not assume normality of errors

Say you are running a regression: $Y_i$= $X_i$$\beta$ + $\eta_i$ And we are not assuming normality of $\eta_i$. My understanding is that as long as your sample size is relatively large (and i know ...
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30 views

Asking for feedback on the application of a Central Limit Theorem

Let $\{X_{n,i}:1\leq i \leq d_n\}$ be a triangular array of mean zero random variables where $d_n$ is a positive increasing sequence ($d_n\leq n$). Under some conditions, a Central Limit Theorem ...
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35 views

What's the relationship between $\frac{1}{n}\sum_{i=1}f(X_{1i},X_{2i})$ and $\frac{1}{n^2}\sum_{i}\sum_{j}f(X_{1i},X_{2j})$?

Suppose $(X_1,X_2)$ is a bivariate random vector following distribution $G$. $f(x_1,x_2)$ is a known bivariate smooth function. Suppose we are interested in estimating $E[f(X_1,X_2)]$ using a random ...
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31 views

For arbitrary random variable $Z$, prove $P(\lvert Z1_{B^{c}}\lvert > \epsilon) \leq P(B^{c})$?

This question is asked to understand proof of Lemma 9.15 from Keener. For arbitrary random variable $Z$, show that $$P(\lvert Z1_{B^{c}} \lvert > \epsilon) \leq P(B^{c})$$ for event $B$ and ...
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How to simulate the supremum of a Gaussian Process

I have a problem where I need to estimate the quantiles of the supremum of a Gaussian Process certain point $t_0$ in time. This should be achieved by simulations. I have a centered Gaussian Process $...
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19 views

Is maximum-a-posteriori estimation consistent?

I am wondering if Maximum-a-Posteriori (MAP) estimates are consistent in the frequentist sense. When I am searching for this, usually what pops up is posterior consistency, for example in the sense ...
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59 views

Proving convergence in probability to zero, is this derivation correct? [closed]

I want to show that $A=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(\widehat{B}_{i}-B_{i})X_i$ converges in probability to 0, where $B_i=E(C_i|Z_i)$ and $C_i$ is i.i.d. binary and $Z_i$ is a discrete random ...
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33 views

Is kernel density estimator a linear transformation?

I am reading the book Nonparametric econometrics, I am thinking since the kernel density estimator is given as $$\hat{f}(x)=\frac {1}{nh}\sum_{i=1}^nK\left(\frac{X_i-x}{h}\right),$$ where $K(\cdot)$ ...
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32 views

What is the asymptotic distribution of $\cos(\bar{X})$ from IID standard normal data?

Suppose that $\bar{X}$ is the mean of $n$ independent standard normal random variables. I have seen it asserted that: $$n (\cos(\bar{X}) - 1) \overset{\text{dist}}{\longrightarrow} -\frac{1}{2} \...
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13 views

Some thoughts on finite sample properties of an estimator

I derived the mean squared errors (MSE) of a consistent estimator for two models: restricted and unrestricted. In addition, I showed that both has the same rate of convergence (that is, the asymptotic ...
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73 views

Asymptotic Distribution and Quantile (Inverse) Function

For a concrete example: Question. Suppose I have iid random variables $X_1, \dots, X_n$ with pdf $f(x) = \frac{1}{6|x|^{2/3}}$ for $x \in [-1, 0) \cup (0, 1]$. Find the asymptotic distribution of $...
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Proof clarification on almost sure convergence and Borel-Cantelli Lemma (update)

I believe it is easier if I print the proof below: Several other papers uses the same argument to bound $\lvert R_n(x)-E R_n(x) \rvert$ almost surely, so I believe it is correct. It should be ...
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1answer
27 views

How can I construct an asymptotic confidence interval using a specified pivotal quantity and the score test?

Let there be a random sample $X_1,...,X_n \sim Poison(\theta)$, where $\theta>0$ is unknown. Show that $P(\mathbf{X},\theta)=\frac{\bar{X}-\theta}{\sqrt{\bar{X}/n}}$ is asymptotically pivotal, then ...
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74 views

How can I find the asymptotic variance of the MLE of $\beta$ for $f_y(y|\beta,\mathbf{x})=\frac{\beta x}{1+\beta x}(\frac{1}{1+\beta x})^{y-1}$?

We have $f_Y(y|\theta)=\theta(1-\theta)^{y-1},y=1,2,...$, where $\theta=\frac{\beta x}{1+ \beta x}$ and $\beta >0$ is unknown. Given $(x_i,Y_i), i=1,...,n$, show that the asymptotic variance of $\...
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1answer
58 views

How can I obtain an asymptotic $1-\alpha$ confidence interval for $\tau$ given a hierarchical distribution?

Let $X \sim Gamma(\alpha,1)$ and $Y|X=x \sim Exp(\frac{1}{\theta x}), \alpha >1$ and $\theta >0$ are unknown. Let $\tau=E(Y)$. Suppose that based on the random sample $Y_1,...,Y_n$, we have ...
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2answers
46 views

How can I find the asymptotic relative efficiency of two quantities, estimating $\sigma$?

Let $X_1,...,X_n$ be a random sample from $N(0,\sigma^2)$, where $\sigma>0$ is unknown. We try to estimate $\sigma$ using $T_1=\sqrt{\frac{\pi}{2}}\frac{1}{n}\sum^n_{i=1}|X_i|$ and $T_2=\sqrt{\frac{...
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1answer
55 views

Showing Lyapunov (Lindeberg) condition holds for sum of independent bernoulli distribution with poisson tail probability

I'm trying to find an asymptotic distribution of the follwing random variable $$Z_n=\sum_{i=0}^n Y_i$$ where $Y_i = I[T_i<t]$ with $T_i \text{~} Gamma(i, \lambda)$. Here $t$ is a fixed number. ...
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Limit of multivariate hypergeometric distribution for infinite colours/categories?

The multivariate hypergeometric distribution gives the probability of drawing, without replacement and in any order, $n_1$ balls with colour (or of category) #1, $n_2$ balls with colour #2, ... $n_c$ ...
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25 views

What is the difference between a non-central limit theorem and the usual central limit theorems?

I'm reading a paper where the authors prove the following theorem. They then say that this constitutes a non-central limit theorem for the variables in question. Since I have never heard this term (...
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55 views

Proof of theorem on Poisson distribution [duplicate]

Can someone help prove this theorem? Many thanks! If $p\to0$ and $n\to\infty$ in such a way that $\lim np = \lambda > 0$, then for $k=0, 1,\dots$: $$\lim_{n\to\infty}\binom nkp^k (1-p)^{n-k}=\...
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Asymptotics of Marginal Likelihood

I'm working with Bayes factors, and I want to develop some intuition for the result $$ \frac{m_1(\mathbf{X})}{p_n(\mathbf{X}|\hat\theta_n)}\xrightarrow{p}\frac{\pi_1(\theta_0)\sqrt{2\pi}}{\sqrt{\...
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1answer
54 views

Formula for difference in order statistics [closed]

Is there a specific formula one can use to compute the differences in order statistics, say $x_i - x_{i-1}$ when the underlying distribution of $x$ is standard normal? Also what is the asymptotic ...
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58 views

Prove $\lim_{n\to\infty} X_{n} = \lim_{n\to\infty} Y_{n}$ implies that $\lim_{n\to\infty} E[X_{n}] =\lim_{n\to\infty} E[Y_{n}]$

Prove $\lim_{n\to\infty} X_{n} = \lim_{n\to\infty} Y_{n}$ implies that $\lim_{n\to\infty} E[X_{n}] =\lim_{n\to\infty} E[Y_{n}]$ given $X_{n}, Y_{n}$ are increasing sequences of positive integrable ...
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7 views

How to proof the asymptotic properties of the penalized spline estimator using asymptotic notations?

Please could someone proof how the Average Mean Squared Error of penalized spline estimator is given as \begin{eqnarray} AMSE(\hat{l} )=\...
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1answer
56 views

Weibull's MLE consistency and asymptotic normality

Let X = $(X_1, \dots, X_n)$ be a sample from Weibull distribution $W(\alpha, \beta)$ with fixed and known $\alpha$. Find MLE of parametric function $g(\beta) = \beta^{\alpha}$. Check if bias is equal ...
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21 views

Does convergence in probability imply $\sqrt{n}$-consistency?

Consider a linear model $y=X\beta+\varepsilon$, and take $\tilde\beta$ as an estimator of the population parameter $\beta$. If $\left\|\tilde\beta-\beta\right\|_2\rightarrow0$ in probability, does ...
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2answers
86 views

Question about asymptotic order

How does $\frac{n^{1-2/p}h_{n}^r}{\log(n)}\rightarrow \infty$, $n \rightarrow \infty$ and $h_{n}\rightarrow 0$ ($h_{n}$ is a function of $n$) imply $\frac{(\log(n))^{1/2}}{(nh_{n}^{r+2d})^{1/2}}\...
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28 views

Variance of $k$th order statistic of normal vector [duplicate]

Let $Z \sim \mathcal{N}(0, I)$. Let $Z_{(k)}$ be the $k$th order statistic of $Z$. Is it true that $\text{Var}(Z_{(k)}) \to 0$ as $n\to \infty$ for $1 \leq k \leq n$? Any estimate on the rate? What ...
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28 views

Almost sure convergence of the minimum eigenvalue of a sample covariance matrix

I was wondering if someone could provide a reference to the following result. Consider the $p\times p $ sample matrix $$\frac{1}{n} \sum_{i=1}^n x_i x_i',$$ where $x_i$ are i.i.d. $p\times 1$ random ...
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33 views

Distribution of quadratic forms in mixed model

I have a question related to the distribution or asymptotic distribution of quadratic forms that arise in the linear mixed model. Suppose, $$Y=X\beta + H\delta + \epsilon$$ where $\epsilon \sim N_n(0,...
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41 views

The distribution of bias of MLE and the asymptotic distribution of $n(\widehat{\beta}_n - \beta)$

$Y_i = \beta X_i + \epsilon_i$, $i=1,...,n$, where $X_i>0$ for all $i$, and $\epsilon_1, \epsilon_2,...,\epsilon_n$ are iid exponential random variables with parameter $\lambda$. $\widehat{\beta}_n$...
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1answer
60 views

Asymptotic Distributions of form: $\sqrt{n}(\hat{\mu} - \mu, \hat{\sigma}^2 - \sigma^2)$

Suppose $X_1, \dots, X_n$ iid normals $N(\mu, \sigma^2)$, and $\hat{\mu}$ and $\hat{\sigma}^2$ are the MLE. How would one go about finding $$\sqrt{n}(\hat{\mu} - \mu, \hat{\sigma}^2 - \sigma^2).$$...
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1answer
38 views

KDE for $h \rightarrow 0$

Let $K$ be a kernel and $X_1,\dots, X_n$ a sample drawn from some distribution with density $f$. The KDE of $f(x)$ is defined by $$\hat f_h(x) = \frac{1}{nh}\sum_{i=1}^nK\left(\frac{x - X_i}{h}\right)....
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9 views

rate of convergence for cross derivative estimation in local linear regression

Suppose $Y_{i}=m(X_{1i},X_{2i})+\epsilon_{i}$, with $E(Y_{i}|X_{1i},X_{2i})=m(X_{1i},X_{2i})$ where $m(\cdot,\cdot)$ is an unknown smooth function. If the estimator $\widehat{m}(x_{1},x_{2})$ is ...
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Intuition of the regression model under fixed design case (nonparametric regression)

Let $(x_1,Y_1), \dotsc, (x_n,Y_n)$ be a random sample from the regression model $$Y_t=m(x_t)+\epsilon_t.$$ When authors want to develop the asymptotic properties of the local linear estimator of $m$ ...
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Is high dimensional PCA regression consistent?

Consider a set $(y_i, x_i), i=1\ldots,n$. The OLS estimator, which is a $\sqrt{n}$-consistent estimator of $\beta$ is obtained as $$\hat\beta=(X^tX)^{-1}X^ty$$. Now perform a PCA on the matrix $X\in\...
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115 views

How are the variance of an estimate to $\int_Bf\:{\rm }d\mu$ and the deviation of $f$ from the mean $\frac1{\mu(B)}\int_Bf\:{\rm d}\mu$ related?

Let $(E,\mathcal E,\mu)$ be a probability space, $(X_n)_{n\in\mathbb N_0}$ be an $(E,\mathcal E)$-valued ergodic time-homogeneous Markov chain with stationary distribution $\mu$ and $$A_nf:=\frac1n\...
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1answer
33 views

Scoring rules for time series data

I have found quite a lot of articles about scoring rules that seem to first work out theorems and proofs for scoring rules in an iid setting, after which they proceed to apply them to some time series ...
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1answer
22 views

Asymptotic distribution of variant of Fisher's $t$. What is wrong with my argument?

This is related to Asymptotic distribution of independent two-sample t-test. Consider two independent random samples of sizes $n_1$ and $n_2$ on independent random variables $x_1$ and $x_2$. Assume ...
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1answer
25 views

Asymptotic test of equality of coefficients from two different regressions

This question is a follow-up to Testing equality of coefficients from two different regressions. Consider the two data generating processes $$y_1=x_1'\beta_1+e_1$$ and $$y_2=x'_2\beta_2+e_2$$ where $...
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1answer
74 views

Asymptotic distribution of independent two-sample t-test

Consider two independent random samples of sizes $n_1$ and $n_2$ ($n_1\neq n_2$ may be the case) on independent random variables $x_1$ and $x_2$. That is, we have one iid sample of size $n_1$ from the ...
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46 views

Variance of $\hat{\sigma}^2=\frac{1}{n}\sum_{i=1}^n(Y_i - \hat{Y}_i)^2$ in regression

Consider a regression problem $Y_i = X_i'\beta+e_i$ with $\beta \in \mathbb{R}^p$. $e_i$'s are i.i.d. with $E(e)=0$ and $Var(e)=\sigma^2<\infty$. My question is about the (asymptotic) variance ...
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17 views

Time series - asymptotic distribution of autocovariances

What is the (joint) asymptotic distribution of a vector of autocovariances $\gamma_{j}$ of a Normal, stationary and ergodic time series $w_t$?
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28 views

Covariance of conditional poisson random variable sequence

Suppose $X_0,X_1,\cdots$ are iid $Poisson(\theta)$ r.v. Define $Y_k = X_k I_{\{ X_{k-1} = 0 \}}$ for $k=1,2,3,\cdots$ Find the limit of $Var(\sqrt{n}\overline{Y_n})$ and asymptotic distribution of $...
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1answer
53 views

the approximation of the variance of MLE (Cramer-Rai Lower Bound)

This is in In Casella's Statistical Inference,page 473, the approximation of the variance of MLE (Cramer-Rao Lower Bound). I really confused with the conclusion: $Var_{\hat{\theta}}h(\hat{\theta})$ ...
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30 views

limiting variance and asymptotic variance of the inverse sample mean

In Casella's [Statistical Inference],page 470. The definitions of limiting variance: For an estimator $T_n,$ if $\lim\limits_{n\rightarrow\infty} k_nVar T_n = \...
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16 views

Score asymptotically belongs to the span of Fisher's information matrix

I am reading a paper from Poskitt and Tremayne titled "Testing the specification of a fitted autoregressive-moving average model". The paper is concerned with a solution to the singularity of Fisher'...
2
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1answer
68 views

Limiting distribution of maximum of i.i.d. Gaussians with decreasing variance

Consider a random vector $X^{(m)} = (X^{(m)}_1,\dots,X^{(m)}_m)$ where, for fixed $m$, the elements of $X^{(m)}$ are i.i.d. $\mathcal{N}(0,\sigma^2 / m)$. Define $$Z_m =\max_{k=1,\dots,m}X^{(m)}_k.$$...
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1answer
46 views

Consistency of M-estimator?

The reported results can be found in van der Vaart's "Asymptotic Statistics". I am having some difficulties to understand the logic behind the following proof provided by the author: Theorem $5.7.$...

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