# 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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### Confusion on delta method [closed]

I am learning Delta Method. One confusion is that, Delta method is for asymptotic distribution of $g(\bar X_{n})$, but from CLT, we know that \begin{align*} \bar X_{n} \rightarrow_{d} N(E(X), \frac{...
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### 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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### 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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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_{... 0answers 44 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 ... 0answers 77 views ### 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 ... 0answers 26 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." 0answers 211 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 ... 0answers 76 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 ... 1answer 165 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))? ... 1answer 130 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}) - ... 0answers 36 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 ... 1answer 47 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 ... 0answers 136 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\... 1answer 79 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 ... 0answers 33 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 ... 0answers 138 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. ... 0answers 113 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)... 0answers 61 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 ... 0answers 137 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 ... 0answers 39 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 ... 1answer 128 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 =... 1answer 764 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 ... 1answer 296 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 ... 0answers 592 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 ... 1answer 439 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 \\... 0answers 582 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 ... 1answer 63 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 ... 1answer 72 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 ... 0answers 103 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).$$I'm an interested in getting a confidence interval for the hazard ratio of a 10 unit change in X instead of a ... 2answers 162 views ### \sqrt{n}-equivalence of M-estimator based on plug-in estimator Suppose our model has a nuisance parameter \eta_0 of which we possess a consistent estimator \hat{\eta}_0. We obtain an estimator \hat{\theta} of a parameter of interests \theta by finding the ... 1answer 626 views ### Standard errors in R, package emmeans I am fitting a multinomial logit model in R by using the multinom() function in the nnet package. I would like to retreive the proportions in each class for the two ... 0answers 104 views ### Delta method when function depends on n (and related question) I had a delta method question that I may be misunderstanding. Suppose we have some estimator \hat{Z} that is consistent and asymptotically normal such that \sqrt{n}(\hat{Z} - Z)\stackrel{d}{\to} N(... 1answer 235 views ### Proving the delta method I am interested in proving the delta method, where we show that$$\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)...
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 ...