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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72 views

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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35 views

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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54 views

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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35 views

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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44 views

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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111 views

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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77 views

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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16 views

Delta method and bias correction

I'm learning Asymptotic statistics and key methods like Delta Method. I came across this paper on application of delta method: Applying the Delta method in metric analytics: A practical guide with ...
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23 views

Getting standard errors for a counterfactual

I fit a linear model and want to know what the percentage increase in aggregate f(y) of setting regressor x to 0. The point ...
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50 views

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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66 views

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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How to estimate Standard error with delta method

I am estimating quantile ARDL model introduced Cho et al. (2015). I have Long-run parameter and variance-covariance matrix. How can I calculate standard errors using the delta method? Thank you.
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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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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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196 views

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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77 views

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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43 views

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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28 views

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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114 views

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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43 views

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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129 views

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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Another 3-part question, this time on limiting distributions. Care to critique my work?

Let $X \stackrel{d}{\sim} Geometric(p)$ for $0 < p < 1$. E.g., $X$ has the pmf $f(x|p) = p(1-p)^{x-1}, x = 1, 2, ...$ with $E(X) = \frac{1}{p}$ and $Var(X) = \frac{1-p}{p^2}.$ a.) Find the limit ...
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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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30 views

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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98 views

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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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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111 views

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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325 views

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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57 views

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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28 views

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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484 views

How to calculate confidence interval and p-value for percent change of treatment relative to control?

I'm analyzing the result of an experiment where the dependent variable is a count variable (# of purchases), and the unit of observation is an individual. The way I'm calculating the treatment effect $...
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119 views

Pairwise Contrast on relative change emmeans package

So I have it a Generalized Linear Mixed Model and am looking to do contrasts. However, in this case, the biochemically relevant contrast is not a simple difference of differences. It is the difference ...
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304 views

How to use inverse information matrix and Delta method to find sample variation?

Explain how the inverse of the information matrix and the Delta Rule be used to generate an approximate sampling variance for the estimated proportion exp(beta 0 + beta 1)/(1+exp( beta 0 + beta 1))? ...
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217 views

Delta Method Confidence Interval: Dividing by $\sqrt{n}$

To compute the (approximate) limiting (asymptotic) distribution of a function of a statistic with known (asymptotically normal) variance, the delta method can be invoked: $\sqrt{n}[g(\hat{\theta}) - ...
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48 views

How can I construct a confidence interval for the age dependency ratio?

Say I have a simple random sample of people from a given country, and I estimate the elderly dependency ratio in the population by taking the ratio of the number of people aged 65+ over the number ...
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60 views

standard errors when bootstrap is not possible. delta method?

I run standard fixed effects regressions on a panel of aggregate firm level data (variables like average value added, average labor). I use the fixed effects that I estimate to simulate firm level ...
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224 views

Delta method for vector valued functions

Suppose I have an estimator $B\in\mathbb{R}^m$ converging to $\beta$, such that $$ \sqrt{n}(B-\beta)\rightarrow\mathcal{N}(0,\Sigma). $$ I am interested in a quantity $\mathbf{h}(B):\ \mathbb{R}^m\...
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86 views

Estimate for the standard error of the probability of a residual lifetime

Suppose that we estimate the survival function using the Kaplan-Meier estimator. Based on that KM-curve $\hat{S}(\cdot)$, one can then estimate the probability that the residual lifetime is larger ...
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hypothesis testing doubt

I have the OLS regression model: $Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_3 + \beta_4X_4 + \epsilon$ I want to check the hypothesis: Ho : $\beta_2*\beta_3$ = 1 Will I use the Delta ...
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212 views

Fisher Information for a Gaussian Process

Suppose I fit a Gaussian process to data such that the posterior distribution over any output is also a Gaussian process, $\mathcal{G}\mathcal{P}(\mu(x),\sigma^2(x))$ where $x$ is some valid input. ...
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163 views

confidence intervals of function of predictions

I would like to know how to get confidence intervals of function of predictions of a gam (via R package mgcv) model. In detail, I got $h\left( y_i \right) = E\left(y_i\right)$ and $std\left(y_i\right)$...
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67 views

What is the probability a confidence interval will contain the sample mean from future samples?

Suppose I observe $X_1, ..., X_n$ independent and identically distributed random variables and I calculate a confidence interval $(L, U)$ from this data. Now I take a second sample $Y_1, ..., Y_n$ ...