Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [delta-method]

Use this tag for questions about approximate probability distributions for functions of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator.

In statistics, the delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator.

Although the delta method generalizes easily to a multivariate setting, careful motivation of the technique is more easily demonstrated in univariate terms. Roughly, if there is a sequence of random variables $X_n$ satisfying $$\sqrt n \left(X_n - \theta \right) \xrightarrow D \mathcal N \left(0, \sigma^2 \right),$$ where $\theta$ and $\sigma^2$ are finite valued constants, and $\xrightarrow D$ denotes convergence in distribution, then $$\sqrt n (g(X_n) - g(\theta)) \xrightarrow D \mathcal N \left(0, \sigma^2 \cdot g'(\theta)^2 \right)$$ for any function $g$ satisfying the property that $g'(\theta)$ exists and is non-zero valued.

47 questions
12
votes
1 answer

What exactly do delta method estimates of moments for $1/\bar X_n$, $\bar X_n\sim\mathcal N(\mu,\sigma^2/n)$ approximate? (not as simple as you think)

Let me start with the excerpt out of Casella & Berger's Statistical Inference (2nd edition, pg. 470) that inspired this question. Definition 10.1.7 For an estimator $T_n$, if $\lim_{n\to\infty}k_n\mathrm{Var}T_n=\tau^2<\infty$, where $\{k_n\}$ is a…
Aaron Hendrickson
  • 6,388
7
votes
2 answers

Using CLT, Slutsky's theorem and delta method

Let $Y_n$ be a sequence of random variables with $\chi^2_n$ distribution. Using Slutsky' theorem or delta method prove that $$\sqrt{2Y_n}-\sqrt{2n-1}\stackrel{D}\to N(0,1)$$ In the first place I proved from CLT that…
zekolor
  • 73
4
votes
0 answers

Did I find a generating function for assigning values to $\mathsf EX^{-n}$, $X\sim\mathcal N(\mu,\sigma^2)$, $n=1,2,\dots$?

This topic has piqued my interest on and off for some time now and I'm curious if the methods used here have been discussed somewhere in the literature. Suppose we have $X\sim\mathcal N(\mu,\sigma)$ and we wish to come up with some meaningful…
Aaron Hendrickson
  • 6,388
3
votes
1 answer

Asymptotic distribution of $T= \frac{\hat{p_2}-\hat{p_1}}{\sqrt{\frac{2 \hat{p} \hat{q}}{n}}}$

Suppose there are two independent sequences of Bernoulli Random Variables $\{X_i\}_{1}^{n}$ and $\{Y_i\}_{1}^{n}$ with $P(X_i=1)=p_1$ and $P(Y_i=1)=p_2$. Let $\hat{p_1} = \frac{\sum_{i=1}^{n} X_i}{n}$ and $\hat{p_2} = \frac{\sum_{i=1}^{n} Y_i}{n}$.…
Iron Maiden 42
  • 243
3
votes
0 answers

Show that $\sqrt n(G_n − \frac1e) →^d N(0,\sigma^2)$

Let $X_1,X_2\ldots $ be a sequence of independent, identically distributed with $X_i\sim U(0,1)$ For the sequence of geometric means $G_n = \big(\prod_{i=1}^n X_i \big)^{1/n}$ show that $\sqrt n(G_n − \frac1e) →^d N(0,\sigma^2)$, for some $\sigma. …
chesslad
  • 2,623
2
votes
1 answer

Convergence of $( n^{3/2} (\bar{X}_n)^2)$ in distribution

Let $( X_1, X_2, \dots )$ be independent and identically distributed random variables. Denote $(\mu = \mathbb{E}[X_i])$ and $(\bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i)$. Assume that $( \mu = 0)$ and $( \text{var}(X_i) = 1)$. Determine to what and…
marek
  • 37
2
votes
1 answer

How to fully understand the definite integral of a Dirac Delta function?

EDIT In physics, most of us learned that \begin{align} \int_{a^-}^{a^+} f(x)\delta (x-a)dx&=f(a)\\ \frac{dH(x)}{dx}&=\delta(x). \end{align} So, would it be natural to have the…
MathArt
  • 1,568
2
votes
1 answer

4th moment of the sample mean estimator

I got a following setup: $(X_i)_{i \geq 1}$ are iid random variables with values in $\mathbb{R}$ and finite second moment. By the weak law of large numbers: $\sqrt{n}(\bar{X} - E(X))$ converges in distribution to $N(0, Var(X))$, so we could deploy…
wklm
  • 93
2
votes
1 answer

How do you obtain Variance of the "method of moment estimator" for the beta distribution using delta method?

I'm trying to solve a question where I have to find $V(\hat{\alpha})$ using the delta-method where $V$ is notation for variance and $\hat{\alpha}$ is the method of moment estimator for a beta distribution. To be more specific, I write down the…
kim
  • 317
2
votes
0 answers

Limit distribution of Bernoulli variance

Let $X_1 \dots X_n \sim B(1, p)$ be i.i.d. random variables. Then the variance of $X_i$ can be approximated through $Y_n = \bar{X}_n(1 - \bar{X}_n)$. What is the limit distribution of $Y_n$ as $n \to \infty$? I tried to solve this as follows. Let…
WafflesTasty
  • 739
1
vote
0 answers

Asymptotic distribution of estimator of the estimator of the standard deviation

I have the following task: Let $Y$ be a binomial random variable. Find the asymptotic distribution of the ML estimate and find the asymptotic distribution of the estimator $(p(1-p))^{\frac{1}{2}}$ of the standard deviation. What happens if $p=0.5$…
fabs
  • 31
1
vote
1 answer

Let $Y=g(X)$ with $X\sim F_X$ and $Y^\ast=g(\mu_X)+g^\prime(\mu_X)(X-\mu_X)$. Under what conditions does $Y\overset{d}{\to}Y^\ast$?

Let $Y=g(X)$ be a nonlinear transformation of some continuous random variable $X$. Assume $Y$ does not have any well-defined moments, e.g. $Y=1/X$ with $X\sim\mathcal N(\mu,1)$ and $\mu\neq 0$. If we expand $Y$ as a Taylor polynomial of order one…
Aaron Hendrickson
  • 6,388
1
vote
1 answer

What is the distribution of functions of random variables?

Is there a theorem that states what the distribution of a function of a random variable should be given the distribution of a random variable? For example, say $X_1$,$X_2$,...$X_n$ is a sequence of iid random variables drawn from a Bernoulli…
Ryan J
  • 11
1
vote
0 answers

Find the limit of $\Pi_n$ in distribution

Suppose that $X_k(1 \leqslant k \leqslant n )$ are positive i.i.d random variable with finite mean $\mu$ and variance $\sigma^2$. Define $P_n= (\prod \limits _{k=1}^n X_k)^{\frac{1}{n}}$ $(n\geqslant1)$and $\Pi_n=\frac{P_n-\mu}{\sigma / {\sqrt{n}}}$…
tcxrp
  • 37
1
vote
1 answer

Finding limiting distribution with delta method

Let $X_1, X_2, ..., X_n$ be iid $N \sim (\theta, \sigma^2)$, Let $\delta_{n} = \bar{X}^2 - \frac{1}{n(n-1)}S^2$, where $S^2 = \sum (X_i - \bar{X})^2$ . Find the limiting distribution of (i) $\sqrt{n} (\delta_{n} - \theta^2)$ for $\theta \neq 0$ (ii)…
DataBall
  • 65
1
2 3 4