Questions tagged [asymptotics]
Asymptotic theory studies the properties of estimators and test statistics when the sample size approaches infinity.
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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 ...
5
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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$$\beta$ + $\eta_i$
And we are not assuming normality of $\eta_i$.
My understanding is that as long as your sample size is relatively large (and i know ...
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30 views
Asking for feedback on the application of a Central Limit Theorem
Let $\{X_{n,i}:1\leq i \leq d_n\}$ be a triangular array of mean zero random variables where $d_n$ is a positive increasing sequence ($d_n\leq n$). Under some conditions, a Central Limit Theorem ...
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1answer
35 views
What's the relationship between $\frac{1}{n}\sum_{i=1}f(X_{1i},X_{2i})$ and $\frac{1}{n^2}\sum_{i}\sum_{j}f(X_{1i},X_{2j})$?
Suppose $(X_1,X_2)$ is a bivariate random vector following distribution $G$. $f(x_1,x_2)$ is a known bivariate smooth function. Suppose we are interested in estimating $E[f(X_1,X_2)]$ using a random ...
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2answers
31 views
For arbitrary random variable $Z$, prove $P(\lvert Z1_{B^{c}}\lvert > \epsilon) \leq P(B^{c})$?
This question is asked to understand proof of Lemma 9.15 from Keener.
For arbitrary random variable $Z$, show that
$$P(\lvert Z1_{B^{c}} \lvert > \epsilon) \leq P(B^{c})$$
for event $B$ and ...
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13 views
How to simulate the supremum of a Gaussian Process
I have a problem where I need to estimate the quantiles of the supremum of a Gaussian Process certain point $t_0$ in time. This should be achieved by simulations.
I have a centered Gaussian Process $...
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19 views
Is maximum-a-posteriori estimation consistent?
I am wondering if Maximum-a-Posteriori (MAP) estimates are consistent in the frequentist sense.
When I am searching for this, usually what pops up is posterior consistency, for example in the sense ...
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59 views
Proving convergence in probability to zero, is this derivation correct? [closed]
I want to show that $A=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(\widehat{B}_{i}-B_{i})X_i$ converges in probability to 0, where $B_i=E(C_i|Z_i)$ and $C_i$ is i.i.d. binary and $Z_i$ is a discrete random ...
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33 views
Is kernel density estimator a linear transformation?
I am reading the book Nonparametric econometrics, I am thinking since the kernel density estimator is given as
$$\hat{f}(x)=\frac
{1}{nh}\sum_{i=1}^nK\left(\frac{X_i-x}{h}\right),$$
where $K(\cdot)$ ...
3
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1answer
32 views
What is the asymptotic distribution of $\cos(\bar{X})$ from IID standard normal data?
Suppose that $\bar{X}$ is the mean of $n$ independent standard normal random variables. I have seen it asserted that:
$$n (\cos(\bar{X}) - 1) \overset{\text{dist}}{\longrightarrow}
-\frac{1}{2} \...
1
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1answer
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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1answer
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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36 views
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 ...
3
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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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1answer
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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0answers
10 views
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 (...
3
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1answer
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 $k=0, 1,\dots$:
$$\lim_{n\to\infty}\binom nkp^k (1-p)^{n-k}=\...
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0answers
22 views
Asymptotics of Marginal Likelihood
I'm working with Bayes factors, and I want to develop some intuition for the result
$$
\frac{m_1(\mathbf{X})}{p_n(\mathbf{X}|\hat\theta_n)}\xrightarrow{p}\frac{\pi_1(\theta_0)\sqrt{2\pi}}{\sqrt{\...
5
votes
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 ...
3
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0answers
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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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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0answers
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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votes
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 $$\hat f_h(x) = \frac{1}{nh}\sum_{i=1}^nK\left(\frac{x - X_i}{h}\right)....
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0answers
9 views
rate of convergence for cross derivative estimation in local linear regression
Suppose $Y_{i}=m(X_{1i},X_{2i})+\epsilon_{i}$, with $E(Y_{i}|X_{1i},X_{2i})=m(X_{1i},X_{2i})$ where $m(\cdot,\cdot)$ is an unknown smooth function. If the estimator $\widehat{m}(x_{1},x_{2})$ is ...
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0answers
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Intuition of the regression model under fixed design case (nonparametric regression)
Let $(x_1,Y_1), \dotsc, (x_n,Y_n)$ be a random sample from the regression model
$$Y_t=m(x_t)+\epsilon_t.$$
When authors want to develop the asymptotic properties of the local linear estimator of $m$ ...
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0answers
88 views
Is high dimensional PCA regression consistent?
Consider a set $(y_i, x_i), i=1\ldots,n$. The OLS estimator, which is a $\sqrt{n}$-consistent estimator of $\beta$ is obtained as $$\hat\beta=(X^tX)^{-1}X^ty$$.
Now perform a PCA on the matrix $X\in\...
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0answers
115 views
How are the variance of an estimate to $\int_Bf\:{\rm }d\mu$ and the deviation of $f$ from the mean $\frac1{\mu(B)}\int_Bf\:{\rm d}\mu$ related?
Let $(E,\mathcal E,\mu)$ be a probability space, $(X_n)_{n\in\mathbb N_0}$ be an $(E,\mathcal E)$-valued ergodic time-homogeneous Markov chain with stationary distribution $\mu$ and $$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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0answers
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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0answers
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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1answer
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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0answers
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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0answers
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 $$Z_m =\max_{k=1,\dots,m}X^{(m)}_k.$$...
1
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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 $5.7.$...
