Questions tagged [delta-method]

"The delta method, in its essence, expands a function of a random variable about its mean, usually with a one-step Taylor approximation, and then takes the variance." The term also refers to a method for showing that a function of an asymptotically normal statistical estimator is asymptotically normal.

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How to correctly estimate the ratio different of lower grain unit metric in Cluster randomized experiment?

I work on Education tech products that teachers/students would use in their learning journey. When we run experiment to test hypothesis of a feature, we need to do cluster randomization (cluster = ...
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Delta Method to calculate the standard error of ratio in an AB testing context

In AB testing context, if we have a control group and test group (2 groups), and I'd like to calculate the relative difference (Mean test/ Mean control -1) and the confidence interval of this ratio ...
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Use MLE to construct a 90% asymptotic confidence interval for $\alpha\beta$

Given $X_1,...,X_n$ are i.i.d random vectors from $p_{\alpha,\beta}$, $\alpha,\beta\in(0,\infty)$ the Fisher information matrix is $I(\alpha,\beta)=\begin{pmatrix} \beta/\alpha&0\\0&\alpha^2\\\...
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Covariance of ratio of dependent variables?

I am trying to use the Delta method (Please have a look at this link) to compute the covariances between the ratios of random dependent variables. I have 7 dependent variables $A_i$, $i\in\{1,2,3,4,5,...
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How to construct a confidence interval from a delta method approximation for the variance?

If I have a complicated function of multivariables $f(x_1,x_2,x_3,\ldots,x_n)$, and I were to find the variance approximation through the delta method, say $\sigma^2_{approx}$, would the 95% ...
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Using the Delta Method to get confidence intervals for a function of a parameter

I'm trying to use the Delta Method to get a confidence interval on some function of a population parameter $\theta$. Suppose we want a 95% confidence interval on $\frac{1}{\theta}$, where we're given ...
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When should I use the delta method rather than the parametric bootstrap?

Suppose I am willing to assume a particular likelihood function for an applied statistics problem. I am able to derive the MLE for the parameter $\theta$, which I will call $\hat{\theta}$. I can also ...
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Delta method for $\bar X$^2

I have a question about the delta method. The question is: $X_1,...,X_n \sim N(\mu,\sigma^2)$, where $\sigma^2=V(x)$ and $\mu=E(X)$ let T=$\bar X^2$ be an estimate for $\mu^2$. Find the asymptotic ...
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Assumption to apply the delta method

When proving the delta method of distributions in my textbook we make the following assumption: Let $X_{n}$ be a sequence of random variables. and: ${\sqrt{n}[X_n- c]\,\xrightarrow{D}\,\mathcal{N}(0,1)...
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Should I use EXP(confint(.)) or Delta method to calculate CI in Meta analysis?

I have a population in which I want to calculate the odds ratio and its CI for the treatment/outcome, calculations of the CI using the exp(confint) in R and the delta method provide slightly different ...
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Delta method for estimating a ratio involving variance and mean

Let $X$ be a binomial RV with parameters $(n,p)$. I am interested in the ratio given by $\hspace{5cm}\boxed{R=\frac{var[f(X)]}{\mu[f(X)](1-\mu[f(X)])}}$ where $\mu[f(X)]$ denotes the mean of $f(X)$. ...
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Evaluating the asymptotic distribution of a metric that's a function of both ML estimated parameters and data, generalizing the delta method

I have a particular problem with likelihood function $\mathcal{L}(\theta \mid X)$, in which I interested in the distribution of a metric $m(X) = f(X,\hat\theta)$ (even asymptotically, though knowing ...
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Delta method for Poisson ratio

Let $X_1,...,X_n$ be drawn from $Pois(\lambda)$ and $Y_1,...,Y_n$ from $Pois(\theta)$. I would like to find the asymptotic distribution of $$\frac{\overline X}{\overline X + \overline Y }$$ using ...
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Can I use the delta method with a function that depends on n to approximate the distribution of a function of the sum of iid random variables?

Let $X_1, X_2,...$ be i.i.d. random variables with finite mean $\mu$ and finite variance $\sigma^2$. From the Central Limit Theorem, we know that $\sqrt{n}(\bar{X_n}-\mu)$ tends in distribution to $N(...
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Derivation covariance between ratio of random variables

Suppose I am interested in computing the covariance between $\frac{A}{B}$ and $\frac{X}{Y}$. From Ratio of correlated vectors is uncorrelated? I understood that using the delta method, this amounts to ...
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Calculating the variance of a function which depends on the standard deviation and mean of a random variable

I am a bit new to statistics and had some conceptual questions regarding the calculation of variance. I want to calculate the variance of a function $y=\frac{\sigma_{X}}{g(\overline{X})}$. As seen in ...
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Stata vs. R: Delta Method provides different results for relative risk SE's from logit model

I've been trying to estimate the conditional mean treatment effect of covariates in a logit regression (using relative-risk) along with their standard errors for inference purposes. The delta method ...
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Delta Method around zero is a N(0, 0)

I have this problem: $\sqrt N \hat{\theta} \sim N(0, V)$ where $E(\hat{\theta}) = \theta_{0} = 0$. I must find the asymthotic distribution of $\frac{N}{V}\hat{\theta}^{2}$ but if I use the Delta ...
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Do I need delta method for calculating SE of absolute difference between two proportions?

I want to know if I need delta method for the below 3 scenarios for online experiment: % change of clicks per user between control and test group, (test clicks per user - control clicks per user)/...
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Delta method for third moment

Suppose $X_1,...,X_n$ is a sample from a population with mean $\mu$ and variance $\sigma^2$ and third central moment of $\mu_3$. I want to justify that: $$E[\left( h(\bar{X})-E(h(\bar{X}))\right)^3]=\...
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Why approximate delta-method Variance isn't multiplied by $\frac{1}{n}$?

I'm reading Casella-Berger chapter 10, where they introduce asymptotic evaluations. I don't seem quite to understand how the factor $\sqrt{n}$ works when we are using asymptotic evaluations in order ...
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Using delta method to get confidence intervals for multinomial logit?

I am working with some choice modeling data and am interested in trying to potentially use the delta method with the multinomial logit model that I'm analyzing the data with. Here's an example: First, ...
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variance estimation using order statistics

I have four largest samples drawn from a distribution of N i.i.d Gaussian R.V. with standard deviation (Sigma) where sigma is unknown. N is known to be between 50-200. Mean is given to be 0. How do ...
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confidence interval of $\beta$, where $X$'s are from exponential distribution

Suppose $X_i\overset{ind}{\sim}\mathcal{E}(\lambda_i)$, where $\lambda_i=(t_i\beta)^{-1}$, where $t_i$'s are positive known values and $\beta$ is positive unknown parameter. Here $i=1,\dots,n$. It can ...
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Second order with Delta method on a ratio to improve variance estimation accuracy

Following a previous post on math exchange without success, I have applied the "Delta method" that says : Delta method : There are alternative formulations of this expression which may be ...
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How to estimate the sample variance of the estimator of the parameter $P(x≤0)$ where $x \sim N(\mu,\sigma)$?

This question relates to How to estimate $P(x\le0)$ from $n$ samples of $x$? One way to make this estimate is to use estimates $\hat\mu$ and $\hat\sigma$ and compute from those $$ \hat p = \Phi \left(...
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Delta Method and Asymptotic Variance [duplicate]

I am working through a statistics course right now and struggling a lot with this question. I'm not really sure where to begin. Any reading or idea where I should begin? I really need to understand ...
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Confidence Interval for Estimator using Delta method

The statement I am given the following discrete distribution with $\theta>0$ $$p(x) = \left(\frac{\theta}{1+\theta}\right) ^{2-x}\left(\frac{1}{1+\theta}\right)^{x-1} \hspace{1cm} x=1,2$$ I need to ...
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What's the asymptotic distribution of $\exp(X_n)$, if $X_n$ is a sequence of asymptotically normally distributed random variables?

Let $(X_n)_{n\in\mathbb{N}} $ be a sequence of asymptotically normally distributed random variables, such that $\lim\limits_{n\to\infty}\sqrt{n}X_n\sim N(0,1)$. What's the asymptotic distribution of $...
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Confusion about the delta method

I'm reading Statistical Models by A. C. Davison and I'm really confused by this section on the Delta method. It's not mentioned explicitly, but is $h(T_n)$ a consistent estimator of $h(\mu)$? In the ...
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Method for obtaining an asymmetric confidence interval

I have an estimator $\hat{\theta}$=$\frac{XY}{Z}$ where $X$ and $Y$ are constants and $Z$ is a random variable. $Z$ ranges from [1, $Y$]. Further $Z$ $\rightarrow$ $Y$ in the limit (asymptotically), ...
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Why is the delta method defined the way it is?

The delta method begins with the assumption of $\sqrt{n} \left[X_n - \theta\right] \stackrel{D}{\to} \mathcal{N}(0, \sigma^2)$. Why is this? Wouldn't it make more sense to start in the more familiar ...
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9 votes
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Trying to approximate $E[f(X)]$ - Woflram Alpha gives $E[f(X)] \approx \frac{1}{\sqrt{3}}$ but I get $E[f(X)] \approx 0$?

Let $X \sim \mathcal{N}(\mu_X,\sigma_X^2) = \mathcal{N}(0,1)$. Let $f(x) = e^{-x^2}$. I want to approximate $E[f(X)]$. Wolfram Alpha gives \begin{align} E[f(X)] \approx \frac{1}{\sqrt{3}}. \end{align} ...
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The two estimators of mean of Gamma distribution and the estimators' variances

In Casella Example 10.1.18, the author says it is not easy to calculate the mean of gamma distribution. It seems that we CAN use the easy way $\bar X=\frac{\sum X_i}n$, but the variance of the mean we ...
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Delta method and Fisher information

Delta method (Casella Theorem 5.5.24) says if the distribution of $\sqrt{n}|Y_n-\theta|\to \mathrm{n}(0, \sigma^2)$ as $n\to\infty$, (where we use sequence of $Y_n$ to estimate $\theta$), then we can ...
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How is delta method used here in approximating the square root of a normal random variable?

I am reading this example where the distribution is given by $Y=\frac{\sigma^2\chi^2_{n-1}}{n-1}.$ By CLT, $Y\sim\mathcal{N}(\sigma^2,\frac{2\sigma^4}{n-1}).$ Up to here it was all clear to me. Then ...
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Limiting Distribution Given Samples from Joint Distribution

Let $(X_i, Y_i)$, $1 \leq i \leq n$ be independent and identically distributed samples from a joint distribution $F (x, y)$. Suppose that $E[X^4], E[Y^4] < \infty$. Now define $\sigma_{XY} = E[(X - ...
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Confidence interval for expected difference in Y for poisson multiple regression, changing one X variable

Suppose I have $y_i \sim \text{Poisson}(\exp(\beta_0 + \beta x_{1,i} + \beta_2 x_{2,i}))$. I run my regression, and I get estimates of $\beta$. I want to conduct inference on the following: $E_{x_1}[y|...
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Does a version of the Delta Method exist for non-i.i.d. sequences?

I have a sequence of random variables that are non-independent, but usually identically distributed. I am wondering if a version of the Delta Method exists under the case when I only have that the ...
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Asymptotic distribution after replacing quantities by consisent estimators

Suppose that we wish to estimate $T(\theta_1,\theta_2)$, a continuous function of several parameters. Suppose that we know the asymptotic distribution when $\theta_1$ is replaced by an estimator $\hat{...
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Question about delta method and variance-stabilization

The delta method or variance-stabilizing transformation can be applied to make the variance be "nearly constant" (https://en.wikipedia.org/wiki/Variance-stabilizing_transformation). They use ...
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How can the square of an asymptotically normal variable also be asympotically normal?

The Delta method states that, given $$ \sqrt{n} (X_n - \mu) \xrightarrow{d} N(0, 1) $$ then $$ \sqrt{n} (g(X_n) - g(\mu)) \xrightarrow{d} N(0, g'(\mu)) $$ I'm surprised that this can be true. As a ...
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Confidence Interval around a predictor

I have a logistic regression as follows: $\log \frac{p}{1-p} = \beta_0 + \beta_1x$. I'm looking for a confidence interval around a value of $x$, which would correspond to a specific value of $p$. ...
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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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Delta method confusion

I am supposed to use the delta method to find the limiting distribution for $$\sqrt{n}\left(\frac{\overline{X}_n}{1-\overline{X}_n} - \frac{E(X)}{1-E(X)}\right)$$ where $f(x, \theta)=\theta x^{\theta-...
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7 votes
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The "correct" way to approximate $\text{var}(f(X))$ via Taylor expansion

tl;dr: There are two commonly reported formulas for approximating $\text{var}(f(X))$, but one is notably better than the other. Since it isn't the "standard" Taylor expansion, where does it come from, ...
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Standard errors with delta method

Trying to recreate other author's results. E.g. this paper. Introduction to the model is on page 10, while table with results is presented on page 13. Under the table there's a small note that SE were ...
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How to find the asymptotic distribution of an estimator given the mean and variance of an estimator

I understand that the Delta Method can be used to find asymptotic distribution of estimators. I have a MLE Estimator with $ E[\hat\Theta] = \frac{n\Theta_0}{n+1} $ $ Var[\hat\Theta] = \frac{\Theta^...
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Confidence interval of transformed random variable

Let $X \sim \mathcal{N}(0, \sigma^2)$. I can construct a level-$\alpha$ confidence interval for $X$ as $(X-q\sigma, X+q\sigma)$, where $q=\Phi^{-1}(\alpha/2)$ and $\Phi$ is the standard normal CDF. I ...
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Using delta method (deviation of transformed variable)

How can in prove the following statement with delta method: "If I divide a variable by its deviation, the deviation of the transformed variable is 1."
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