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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24 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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885 views

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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21 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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MLEs multivariate normal distribution estimation

I’m a beginner in this field, I hope the problem will be clear… . Under some regularity assumption the MLE estimators of unknown parameters are unbiased and their distributions is a multivariate ...
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28 views

Use the delta method to find the approximate mean and variance

I dont know how to get E(W),Stuck for a long timeT^T I am trying to use E(Y3/(Y1+Y2)) but I dont know how to expand it and then use delta method to get variance.
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48 views

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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54 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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64 views

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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85 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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29 views

Sample size estimate for a ratio of sums

I need to do a sample size calculation for an A/B test. The metric is the sum of sales of items in a specific category divided by the total sum of sales of all items $$Metric = \frac{\sum_{...
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38 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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Test of independence for Bernoulli random variables

The qquestion is: Let X, Y be two Bernoulli random variables and denote by $p=\mathbb{P}[X=1]$, $q=\mathbb{P}[Y=1]$ and $r=\mathbb{P}[X=1, Y=1]$. Prove that X and Y are independent if and only if $...
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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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118 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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51 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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128 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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101 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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34 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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43 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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112 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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78 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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114 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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102 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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58 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 $Y_1, ..., Y_n$ ...
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125 views

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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37 views

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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117 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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624 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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1answer
275 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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517 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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391 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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525 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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62 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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58 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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86 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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157 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 the ...
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1answer
580 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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96 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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1answer
227 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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574 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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1answer
333 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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331 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
478 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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284 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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778 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)]...