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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19 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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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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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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59 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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39 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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29 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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31 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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54 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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73 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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49 views

Standard error for the difference between two predicted probabilities

Imagine a logit model with a continuous independent variable, a binary independent variable (the treatment) and an interaction between the two. In a first step, I want to predict probabilities at ...
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29 views

Standard error of conditional survival probability using delta method

I need to estimate the standard error of the conditional survival probability using the delta method. I fitted a kaplan meier curve for these probabilities.I know how the delta method works, I just ...
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29 views

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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Delta Method Asymptotic Variance

Hello I am struggling with this question I would like to have some help from you, here it is : $X_1,X_2,...X_n ~ N ( \mu,\sigma^2)$ we want to to find with asymptotic level of 5% for the hypothesis : ...
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53 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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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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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 $Y_1, ..., Y_n$ ...
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Terminology: “Central Limit Theorem” for Delta Method

This is a question about when is it appropriate to call an asymptotic normality statement, the "Central Limit Theorem" (CLT). More specifically, suppose I have $X_1, X_2, \dots X_n$ iid from a ...
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why is method of moments estimates asymptotically normal

I have noticed that a lot of statistics textbooks contain lengthy discussions and detailed proofs on showing that MLE estimates are asymptotically normal (under regularity conditions). On the other ...
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Multinomial logit with ridge penalization and value of time

I am fitting a multinomial logit model with ridge penalty and in turn estimating the value of time (VOT) or availability to pay (WTP). I want to work with real and simulated data. For the real data I ...
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80 views

Standard Error of a function of ML estimators

The background of the problem is as follows: Suppose $X_1,...,X_n \sim U(a,b)$ independently where $a$ and $b$ are unknown parameters and $a < b$. Let $\hat\tau$ be the MLE of $\tau$, where $\tau =...
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346 views

Asymptotic distribution of $\sqrt{n}\left(\hat{\sigma_{1}^{2}}-\sigma^2\right)$

I'm trying to find a confidence interval for variance $\sigma^{2}$ when some sample $X_{1},...,X_{n}$, with mean $\mu$ known, may have violated normality assumption. To do this I'm investigating the ...
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193 views

Resampling-based Confidence Intervals for RERI

Relative Excess Risk Due to Interaction (RERI) has been used to quantify the joint effects of 2 exposures in epidemiology. RERI is the proportion of disease among those with both exposures that is ...
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328 views

Delta Method Average Marginal Effects Multinomial Logit

Following the incredible demonstration in Statalist by Jeff Pitblado on how to calculate - using the Delta Method - the Standard Errors for Average Marginal Effects of a Logit Model. Q: What would ...
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283 views

Asymptotic distribution of sample variance via multivariate delta method

I was trying to get the asymptotic distribution of sample variance using multivariate delta method and without normality assumption. So I defined the random vector $ z = \left( \begin{matrix} X \\...
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346 views

Is delta method better than bootstrap to generate standard error for marginal effects?

I read here, here, here, here, and elsewhere that " Parametric bootstrap closely related to objective Bayes. (That’s why it’s a good importance sampling choice.) When it applies, parboot approach ...
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60 views

Dependence of estimator covariance on sample count

Say that $X$ is a set $\{X_1, X_2, ..., X_N\}$ of (non-independent) random variables, and that $\hat{\mu}$ is a set $\{\hat{\mu}_1, \hat{\mu}_2, ..., \hat{\mu}_N\}$ of estimators. Each $\hat{\mu}_i$ ...
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35 views

Can delta method be applied for determining the between subject variability (random variance) of a function of X?

Say, for example, I square root transformed X such that it follows normal distribution, fitted a linear mixed effects model and obtained between subject variability (BSV) of sqrt(X). How do I now ...
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52 views

Confidence Interval for 10 unit change in hazard ratio?

I'm fitting a Cox model with one predictor, X. That is, $$h(t) = h_0(t) exp(X_i \beta).$$ I'm an interested in getting a confidence interval for the hazard ratio of a 10 unit change in X instead of a ...
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143 views

$\sqrt{n}$-equivalence of M-estimator based on plug-in estimator

Suppose our model has a nuisance parameter $\eta_0$ of which we possess a consistent estimator $\hat{\eta}_0$. We obtain an estimator $\hat{\theta}$ of a parameter of interests $\theta$ by finding ...
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414 views

Standard errors in R, package emmeans

I am fitting a multinomial logit model in R by using the multinom() function in the nnet package. I would like to retreive the proportions in each class for the two ...
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84 views

Delta method when function depends on n (and related question)

I had a delta method question that I may be misunderstanding. Suppose we have some estimator $\hat{Z}$ that is consistent and asymptotically normal such that $\sqrt{n}(\hat{Z} - Z)\stackrel{d}{\to} N(...
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212 views

Proving the delta method

I am interested in proving the delta method, where we show that $$\sqrt{n}(g(Y_n) - g(\theta)) \overset{\text{Dist}}{\to} \text{N}(0, \sigma^2 g'(\theta)^2).$$ We use Taylor expansion where $$g(Y_n)...
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Are the Inverses of two asymptotically equivalent matrices themselves asymptotically equivalent

Suppose $M_n = P_n + op(1)$. Is it the case that $M_n^{-1} = P_n^{-1} + op(1)$, if both $M_n^{-1}$ and $P_n^{-1}$ exist with probability going to 1 as $n$ increases? Can the Continuous Mapping ...
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392 views

Implicit hypothesis testing: mean greater than variance and Delta Method

I am struggling with a hypothesis test between the mean and variance of a sample of i.i.d Gaussian random variables. This (self-study) question arises in the context of the Delta Method (first or ...
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197 views

Is the t-value (or z-value) of the margins of a logit model equal to the t-value of the coefficient?

The accepted answer of this question seems to indicate that the z-value of the margin is the same as the z-value of the coefficient in the logit model. This, it seems, is true with how Stata ...
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249 views

Estimator for Pareto Distribution & Delta Method

Assume that $Y$ has a Pareto distribution with parameters ($\theta, t$ = 1). An estimator of $\theta$ is $\tilde{\theta}$ where $\bar{Y} = \frac{\tilde{\theta}t}{\tilde{\theta} - 1}$. Solve for $\...
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349 views

Degrees of freedom for t-test after delta method

Say I have a dataset with $n$ observations of two independent variables, X and Y. From the observations of the X variable I estimate a parameter $\hat{\alpha}$ and corresponding variance $\hat{\sigma}...
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Showing p-th sample quantile is asymptotically normal [duplicate]

I'm working through van der Vaart to improve my knowledge on asymptotic statistics, and I'm attempting the following problem. Let $F_n^{-1}(p)$ be the p-th sample quantile of a sample from a ...
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255 views

Why is delta method only for asymptotic distributions?

I'm wondering why the delta method, , "is only used for asymptotic distributions", as @mpiktas write in this post: Variance of a function of one random variable, or as is written here: "The delta ...
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329 views

Standard error of the estimate in logistic regression

We usually get an estimate of $\beta$ in the logistic regression by finding the $MLE$ of the observed random samples of $X_1,X_2....,X_N$. Then we use Wald's test i.e. ${[\hat \beta / S.E.(\hat \beta)]...
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109 views

Higher order delta / taylor series approximation relationship to normal distribution?

For a normally distributed variable X, one can call on the delta method to provide an asymptotically normally distributed variable for a non-linear function of it, g(X). This is based on a linear ...
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413 views

Delta Method Confidence Intervals

In a standard linear regression setup: $$ y_t = \beta X + \varepsilon _t$$ where $$ e_t \sim N(0, \sigma ^{2}) $$ I have found the Maximum Likelihood Estimators for β and σ (from OLS), but now ...
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246 views

Delta Method with Ratio that Cancels

Say I have Gas Consumption = Energy Used Per Hour * Heating Hours and I want to construct a CI for Gas consumption. Do I have to use the delta method and write: $$\mathrm{Var}(GC) = \mathbb{E}(\...
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63 views

Prelude to the delta method

Given a probability density say $f(x)$ with parameters $\theta$ and a sample of size n say $x_1,\ldots,x_n$ we can compute the MLE estimate say $\theta_n$ by passing $f$ and $x_1,\dots,x_n$ to ...
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369 views

Understanding the delta method

I'm trying to work on #1 from Chapter 15 of Greene's Econometric Analysis and I'm confused about how to use the Delta method. The problem statement is: For the normal distribution $μ_{2k} = \frac{\...
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468 views

Delta Method for Marginal Effects of Generalized Linear model

Consider the generalized regression model (in my case a probit, but I'll leave it more generally): $$ E[y|X] = F(x'\beta) $$ where both $x$ and $\beta$ are ($K \times 1$) column vectors and $F$ is ...
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272 views

Delta method for non-normal variables

Is the delta method valid also for non-normal variables? Claim: Let $\sqrt{n}(\hat{X}_n-\theta) \xrightarrow{d} \hat{f} $. With $\hat f$ having a finite distribution. Then for every $g$ such ...
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225 views

Misunderstanding the delta method

For my understanding I am using the delta method in 2 functions.I am computing the MLE of 2 functions. $function 1 = f_1 = f $ vs $function 2 = f_2 = 2 * f$ Main Query: Can I assume that the ...
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183 views

delta method variance

This is probably a basic question but I will ask it anyway b/c Im stuck. I do bird research and I have bird density estimates for thousands of sampled patches along with the associated variance. To ...
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90 views

Sampling instead of delta method

Let's say I estimating some parameter vector $\theta$ with some estimator $\hat \theta$. I have that asymptotically, $\hat \theta \text{ }\dot \sim \text{ } N(\theta, \Sigma)$ Now let's say I want ...