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.
48 questions
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Delta method measurbility question.
Suppose we have a random variable $\hat \theta_n$ such that $\hat \theta_n \to \theta_0$ in probability. Let $f \colon \mathbb{R} \to \mathbb{R}$ be infinitely differentiable function. Then, the delta ...
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1
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59
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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 ...
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439
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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 ...
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102
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Expectation of the inverse of an estimator using delta method linearization
I'm working on a research internship about the precision of consumption price indexes and face some probems that leads me to ask a lot of questions about the methods I used and need to be answered.
...
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96
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Expected and Variance of the Squared Sample Correlation
I would like to obtain the expectation and variance of the squared sample correlation ($\operatorname{E}(R_{lk}^2)$ and $V(R_{lk}^2)$) between two random variables $l$ and $k$ following a bivariate ...
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55
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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 ...
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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 below?
\begin{...
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83
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Why is $\frac{1}{2\pi} \int_{-\infty}^\infty \exp \left (i k x \right) \text{d}k = \delta\left(x\right)$? [duplicate]
I want to understand this and the step that I'm struggling with is that apparently
$$
\frac{1}{2\pi} \int_{-\infty}^\infty \exp \left (i k x \right) \text{d}k = \delta\left(x\right)
$$
with $\delta$ ...
- 143
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How to put the bivariate/multivariate delta method into linear algebra notation?
So I understand that the delta method allows us to approximate the expectation and variance of a random variable. Mathematically,
Let $Z = g(X,Y)$ where $X$ and $Y$ are random variables with $E[X] = \...
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136
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Finding the asymptotic variance of an estimator
Suppose X1, . . . , Xn are independent and exponential with parameter θ.
Let p me an estimator such that
p = #{i : Xi > 1}/n
Where #A is the number of of elements in A.
The original question is to ...
2
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1
answer
464
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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 ...
- 93
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474
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Asymptotic variance of a consistent estimator of gamma random variables
Given $X_1,...,X_n \sim Gamma(\alpha,1/\alpha)$ random variables for some $\alpha>0$, let $\hat{\alpha}$ be a consistent estimator of the sample average $\bar{X}_{n}$ of the sample in terms of $\...
- 207
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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 $\hat{p_1} = \frac{\sum_{i=1}^{n} X_i}{n}$ ...
- 243
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Covariance of normalized variables
I have 7 variables $A_i$, $i\in\{-3,-2,-1,0,1,2,3\}$. ($A_{-1}$ and $A_1$) are identically distributed. ($A_{-2}$ and $A_2$) are identically distributed. ($A_{-3}$ and $A_{3}$) are identically ...
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654
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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}...
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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 ...
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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 ...
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2
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83
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Probabilities , Central limit theorem
So I am working on a problem where I have to find the approximate law of $\frac{1}{\bar{X}}$ where $X_{j}$ follow the Normal distribution law. Everything is fine up until the point I have to check the ...
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59
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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)...
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667
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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} (...
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548
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Proof of general delta method
I have found proof of the "delta method", (From Mathematical Statistics by Shao Jun P61) but I cannot understand some steps in this proof.
Theorem :
Let $X_1, X_2,...$ and $Y$ be random k-...
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518
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Does sequence convergence implies convergence in probability?
I am currently learning for my statistic final exam and I want to understand one part of the proof for the Delta method. I am going to skip some details of the proof and only ask one important ...
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130
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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 $X\sim\mathcal N(\mu,\sigma)$ ...
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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 ...
- 317
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386
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Sample of Random Bernoulli Trials Delta Method
I am trying to solve the following problem
I am trying to find the estimator of the Bernoulli trials, which since we took a sample of them, they would be computed using the Binomial Variance. I am ...
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1
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994
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Method of moments estimator and Delta Method
I’ve been a couple of days trying to figure out a satisfactory solution to this exercise but I’m stuck. By how the exercise is phrased I think it wants me to use the moments method and I also think it ...
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532
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Delta method Taylor Expansion question
In the Taylor series expansion of the delta method, could someone help me why it leads to this equation?
I understand the first and second components, but the little-o function on the very right is ...
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125
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The formula of matrix calculus
Suppose $A \in \mathbb{R}^{p \times p}$ is a semi-positive defined symmetric matrix. Then $A^{1/2}$ is well-defined. Now I want to know does there exists a formula for
$$\frac{\partial A^{1/2} }{\...
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548
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How to prove the Delta Method?
As exercise 9.14 of "Measure theory and probability theory" by Krishna and Soumendra, I'm trying to demonstrate the delta method.
We have a random variable sequence $X_n$, a divergent ...
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101
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Mean square error of $\phi(x-\bar X_n)$
I'm asked to calculate the mean square error of $\phi(x-\bar X_n)$ given a sample $X_1,\ldots,X_n$ from $\mathcal N(\mu,1)$. Here $\phi$ denotes the standard normal density. Now by the Delta method, ...
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2k
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Approximating the log of a chi-square distribution
I'm trying to solve this problem:
Let $X_1,...,X_n$ be a random sample from a $N(0,\sigma^2)$ distribution. Let $\bar{X}$ be the sample mean and let $S$ be the sample second moment $\sum X_i^2/n$. ...
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Show that $\sqrt n(G_n − \frac1e) →^d N(0,\sigma^2)$
Let $X_1,X_2\ldots $ be a sequence of independent, identically
distributed with $X_i\sim U(0,1)$
For the sequence of geometric means $G_n = \big(\prod_{i=1}^n X_i
\big)^{1/n}$ show that $\sqrt n(G_n − ...
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347
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Delta method for asymptotic variance
Let $X_i$ be i.i.d. r.v. with $\sim Exp(\lambda)$ and for n to $\infty$
$$\sqrt{n} \cdot (Z_n - \theta) \overset{d}{\to} N(0,\sigma^2) $$
$\tilde{\lambda} = - ln(\bar{Y_n})$, where $Y_i = 1 \{X_i = 0\...
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152
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Show that $\sqrt{n}(g(Y_n)-g(m))$ converges in distribution to $N(0,(g'(m))^2σ^2)$
I want to solve the following exercise, but I'm having problems handling all the information given.
Let $Y_n $ be a sequence of real random variables and $m, σ > 0$ be real numbers such that as $n ...
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$\delta$-method on $\mathbb{R}^d$
Let $(X_n)_n$ be a sequence of i.i.d random variables taking values in $\mathbb{R}^d,d \geq 1,$ such that $E[||X||^2]<+\infty$ where $||.||$ is a norm on $\mathbb{R}^d,$ and let $\overline{X}_n=\...
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Question about variance stabilizing transformation
Let $X_1$, $X_2$, $\ldots$ iid, $\sim$ $N(\theta,\sigma^2)$; $\theta \in \mathbb{R}$ unknown, $\sigma > 0$ and $a \in \mathbb{R}$ given. $g(\theta) := P(X_i > a)$.
We want to estimate $g(\theta)...
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Finding constants $\beta_{1}$ and $\beta_{2}$ using the delta method
Suppose that $\sqrt{N} \left ( \widehat{\theta} - e \right ) \overset{d} N\left ( 0, \sqrt{2} \right )$,
If $Y_{n} \equiv \sqrt{N} \left ( log \widehat{\theta} - \beta_{1}\right )$, provide constants ...
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170
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Difference of limit - Delta method vs. Slutsky
Let $(X,Y)$ be bivariate normal distributed and
$$
S_X^2 = \sum_i (X_i - \bar{X})^2 \\
S_Y^2 = \sum_i (Y_i - \bar{Y})^2.
$$
Now I want to derive the limit of
$$
T_n:= \left ( \sqrt{n} \left ( \frac{\...
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152
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Epsilon-delta proof method [duplicate]
Use the definition of continuity and the formal definition of the limit value
("$\epsilon$-$\delta$ definition") to show the following:
Let $f: \mathbb{R} \to \mathbb{R}$ be a function that is ...
- 547
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590
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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 \...
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Asymptotic distribution of MLE of $\theta$ for the pdf $f(x)=\frac{\theta}{(1+x)^{\theta+1}}$
There's some questions:
Suppose $X_1, \dots, X_n$ be iid random variables with common density function
$$f(x) = \frac{\theta}{(1+x)^{\theta+1}} , x > 0 , \theta > 0 $$
(a) Find the ...
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0
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303
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Asymptotic distribution of MLE of joint exponential distributions
Given $X_1,...,X_n \sim \text{Exp}(\theta)$ and $Y_1,...,Y_n \sim \text{Exp}(\frac{1}{\theta})$, where $\theta>0$ and have the same $\theta$ in both distributions. $X$ and $Y$ are independent.
I ...
- 338
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vote
0
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666
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Delta method around 0 derivative point
First let me write the statement of delta method from wikipedia:
If there is a sequence of random variables $(X_n)$ satisfying
$\sqrt{n}[X_n-\theta]\,\xrightarrow{D}\,\mathcal{N}(0,\sigma^2)$
...
- 151
1
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1
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260
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Limiting Distribution on order statistics
Here is the question:
Find the limiting distribution of the quartile ratio defined as $\frac{X_{(3n/4):n}}{X_{(n/4):n}}$ (the third quartile divided by the first quartile)for the exponential ...
- 66
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2
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798
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Delta method to find E(Y) and V(Y)
I know a Poisson random variable X has E(X) = μ and also Var(X)
= μ. I'm given symmetrizing chance of variables, $$ Y= X^{2/3} $$ and I need to use the delta method to
approximate E(Y) and Var(Y) for ...
- 83
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0
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52
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How do you calculate the average?
I've working on a project which involves datasets with points at, what should be, regular intervals.
However, due to things such as down times / reboots / outages, the data collected (from sensors) ...
- 862
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2
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78
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Δ meaning in mathematics
I am reading an article for percentage calculations...
"This means that if you want to calculate the price change in a given stock, you subtract the current price from yesterday's closing price and ...
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0
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1k
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Using the delta method to find the asymptotic distribution
The Question:
Given independent random variables $X_1, \dots,X_n \sim N(\mu,\mu^2)$, let $Q=\sum_{i=1}^n(X_i-\overline X)^2$ where $\overline X = \frac 1n \sum_{i=1}^nX_i$.
Use the delta method to ...
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