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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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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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Asymptotic distribution using the Delta method

Let $X_1, \ldots, X_n$ be i.i.d. normal random variables, where $X_i \sim N(\theta, \theta^2)$ with an unknown $\theta > 0$. We could for example estimate $\theta$ using the average $$\hat\theta_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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60 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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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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1answer
69 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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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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165 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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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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What is the stop condition of delta learning method in training a neural network? [duplicate]

I know that in training a neuron using delta rule, for each epoch, I must update the weights using the following rule for all training examples: ...
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55 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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Loss Function Asymptotic Distribution

I am trying to find the asymptotic distribution of a maximum estimator loss function of the type: $$ \hat{\theta} = \arg \max_\tilde{\theta} M_n(\tilde{\theta},x) $$ $$ \theta = \arg \max_\tilde{\...
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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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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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$\sqrt{n}$-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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1answer
213 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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60 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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137 views

Proving delta method

when proving the delta method where we show that $$\sqrt{n}(g(Y_n) - g(\theta)) \to n(0, \sigma^2 (g'(\theta))^2)$$ in distribution, we use taylor expansion where $$g(Y_n) = g(\theta) + g'(\theta)(...
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Pooled standard error of an estimate that is the product of combined estimates

I have two independent mean estimates (with SE and sample size) that I want to multiply to create a combined estimate. I'm having trouble interpreting what to use for sample size in the combined ...
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How to calculate the variance of a ratio when numerator and denominator were estimated independently?

I would like to calculate the variance for a ratio. The ratio is Ne/Nc (i.e., genetic effective population size/census population size). I estimated Ne using a genetics software (NeEstimator) and I ...
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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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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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106 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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184 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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1answer
198 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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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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118 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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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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245 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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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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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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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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295 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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209 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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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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124 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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1answer
62 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 ...
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179 views

Where does the delta method's name come from?

Where does the delta method's name come from? I don't see anything related to "$\epsilon$-$\delta$", nor Dirac delta for example.
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287 views

Ratio distribution derived from multinomial distribution

Suppose we have a multinomial distribution with support $(X_1,...,X_n)$ and $\sum_{i=1}^nX_i=N$. Consider the probability distribution of $X_1/X_2$, say. Does this distribution have an expected value ...
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Proving convergence to normal distribution using the delta method

The following is a question and instructor provided solution from a recent exam I took. Question: Prove that $\sqrt{n}T_n$ converges in distribution to N(0, 4). $T_n$=$(\bar{X}^2 - \hat{\sigma}^2)/\...
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105 views

Asymptotics - Delta Method used in proof

How can I use the delta method to show that $\sqrt n$ (1/$\bar Y_n$ - 1/$\mu)$ $\rightarrow$ N (0, $\sigma^2$/$\mu^4$) ? We know that: $\bar Y_n$ = $1/n \sum_{n=1}^n Y_i$ $\ Y_i$ is i.i.d; $E(\ ...
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How to interpret the Delta Method?

I'm reading through https://www.statlect.com/asymptotic-theory/delta-method it defined the Delta Method as: The delta method is a method that allows us to derive, under appropriate conditions, ...
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Use $\bar{X}^2$ for hypothesis test that $\mu=0$ because faster convergence rate?

Suppose that I have $X_1,\ldots,X_n$ are i.i.d. and I want to do a hypothesis test that $\mu$ is 0. Suppose I have large n and can use Central Limit Theorem. I could also do a test that $\mu^2$ is 0, ...
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193 views

Standard error of Pietra index with Pareto assumption

I am working on this problem of income distribution. I am assuming that my income data $X_i$ is Pareto : $f(x_i;\alpha) = {\alpha \over x_0}({x_i \over x_0})^{-(\alpha + 1)}$ I found my MLE ...
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Delta method with mix of continuous and discrete variables

This is my first question on Cross Validated so please bear with me if my question is lagging in any dimension. My question regards how to evaluate a Jacobian matrix when one variable is binary. I ...