# 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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### 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 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $Y_1, ..., Y_n$ ...