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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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Delta method vs actual expectation

If $x \sim N(\mu,\sigma^2)$, then by first principles, $$\mathbb{E}(e^x) = e^{\mu + \sigma^2 / 2}.$$ I am trying to figure out where the "Delta method" is wrong here: If $(x-\mu) \sim N(0,\...
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Derive the expectation and variance of squared sample correlation: delta-method or else?

I would like to obtain the expectation and variance of the squared Pearson sample correlation ($\operatorname{E}(R_{lk}^2)$ and $V(R_{lk}^2)$) between two random variables $l$ and $k$ following a ...
CafféSospeso's user avatar
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Understanding Asymptotic Relative Efficiency and how to compute it

I am learning about asymptotic relative efficiency (ARE) in class, and I am trying to understand exactly how to compute the ARE. From my understanding, asymptotic relative efficiency refers to ...
Harry Lofi's user avatar
2 votes
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95 views

A test of the difference between two r-squared?

According to Olkin and Finn (1995) and Alf and Graf (1999), the variance of the difference in r-squared is $$ var(r_1^2 - r_2^2) = a \phi a^\mathsf{T}, $$ where $a = \begin{bmatrix}2 r_{1} & -2 r_{...
Kniven's user avatar
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Coefficient covariance matrix of inverse probability weighted regression

I am interested in computing an estimate $\hat\Sigma_\hat\beta$ of the asymptotic covariance matrix of the parameter estimates $\hat\beta$ in a regression of $Y$ on $\{X, Z\}$, weighted by weighs $\...
Noah's user avatar
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Creating a confidence interval for the natural log of the proportion of successes [duplicate]

In a random sample of n subjects with n being very large, let X be the number of successes. Now I want to create the confidence interval for the natural log of the proportion of successes. Can I ...
Rishav Dhariwal's user avatar
1 vote
1 answer
94 views

Reparameterization of the variance-covariance matrix (`apVar`) of the random-effect parameters estimated by `lme`

The question is about computing the variance of the random-effect parameters estimated when fitting a linear mixed-effect model when the parameterization of the random-effect parameters changes. This ...
gavril's user avatar
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1 answer
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Reference needed for Delta method

I came to know from somewhere that there is a technique called delta method which can be used to approximate the distribution of a function of a random variable using the distribution of the ...
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32 views

Correctness of derivation for binary F1 variance for F1 confidence intervals

I'm developing a python library for confidence intervals for common accuracy metrics, with both analytic and bootstrap computations. Following this paper, I implemented the Macro and Micro F1 scores ...
Jacob G's user avatar
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1 answer
281 views

Use the delta method to find confidence intervals

Given $X_1 ... X_n \sim \textrm{Exp}(\lambda)$, I found the MLE : $$\hat{\lambda} = \frac{1}{\bar{X}}$$ Now I need to find confidence intervals for: $$\eta = \lambda \cdot \log(\lambda)$$ To do so, I ...
CORy's user avatar
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Estimating covariance based on Delta approach

I have a bivariate normal distribution $\left(X_1, X_2\right)$ with mean vector $\left(\mu_1, \mu_2\right)$ and some VCV matrix ...
Brian Smith's user avatar
2 votes
1 answer
713 views

Delta method for ratio metrics

I have the following issue: I would like to do a power analysis (find the right sample size) for a ratio metric ($Z = \frac{X}{Y}$). The in-house statistical software I inherited uses a delta ...
Ben Labosch's user avatar
2 votes
3 answers
330 views

Deriving the asymptotic distribution using delta method

I have the density function: $P_Y(y) = \sqrt{\frac{1}{2\pi y^3}} \exp\left(-\frac{(y-\mu)^2}{2\mu^2y}\right)$ If we define $r := \mu^2$ what is its asymptotic distribution? The right answer is $\sqrt{...
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1 answer
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how to estimate the number of known words in a text

I would like to estimate the proportion of known words in a text from a sample of tested words, where a subject answers if they know the meaning or not, and the frequency of how often they appear in a ...
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1 answer
251 views

Negative variance with Delta Method in A/B test analysis?

I'm analyzing a ratio metric in the context of an A/B test (e.g. "Clicks / Impressions"). Since the randomization unit and analysis unit are different (users vs impressions), I'm applying ...
jdorn's user avatar
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3 votes
1 answer
178 views

Variance stabilizing transformation for logistic regression

Question: Are there (known) variance stabilizing transformations for logistic regression? Backgound: As an M-estimator, logistic regression is asymptotically normal, under suitable regularity ...
Idontgetit's user avatar
2 votes
0 answers
104 views

How Best to Use the Delta Method to Approximate True Variance in a T-test Given Correlated Data?

Let's say that the analytical infrastructure at my place of work heavily centers around t-tests. I'd like to use this platform to perform a t-test on correlated data, e.g. the extent to which an order ...
StatStudent19's user avatar
2 votes
1 answer
154 views

A question about the delta method in asymptotic distributions

I am reading up on the delta method from its Wikipedia page. Under the heading Univariate delta method the statement of the method is as follows: If $$\sqrt{n}[X_n - \theta]\xrightarrow{\text{D}} \...
figs_and_nuts's user avatar
1 vote
1 answer
203 views

Variance of Kaplan-Meier estimator

Here and here and on the Wikipedia page it is stated that for estimating the variance of Kaplan Meier estimator $S(t)$ using delta method one can use the fact that: $$Var(log\hat{S}(t)) \approx \frac{...
Hooman's user avatar
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1 answer
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Computing and simulating average marginal effect standard error using Delta Method with reproducible codes

I am trying to simulate calculating Average Marginal Effects on a basic linear regression with interaction on a binary variable and compare the empirical standard deviation I get from simulations and ...
StatsNoob's user avatar
5 votes
1 answer
331 views

is ab test valid for concluding causal relation, when analysis unit differs from randomization unit?

In a typical A/B test, the randomization unit is user level, sometimes the analysis unit may be page/visit level, like a cluster randomization experiment. In this situation, the iid assumption doesn't ...
wei's user avatar
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2 answers
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Delta Method to calculate the standard error of ratio in an AB testing context

In AB testing context, if we have a control group and test group (2 groups), and I'd like to calculate the relative difference (Mean test/ Mean control -1) and the confidence interval of this ratio ...
user1456579's user avatar
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132 views

Covariance of ratio of dependent variables?

I am trying to use the Delta method (Please have a look at this link) to compute the covariances between the ratios of random dependent variables. I have 7 dependent variables $A_i$, $i\in\{1,2,3,4,5,...
DarkBulle's user avatar
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How to construct a confidence interval from a delta method approximation for the variance?

If I have a complicated function of multivariables $f(x_1,x_2,x_3,\ldots,x_n)$, and I were to find the variance approximation through the delta method, say $\sigma^2_{approx}$, would the 95% ...
user321627's user avatar
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When should I use the delta method rather than the parametric bootstrap?

Suppose I am willing to assume a particular likelihood function for an applied statistics problem. I am able to derive the MLE for the parameter $\theta$, which I will call $\hat{\theta}$. I can also ...
frelk's user avatar
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226 views

Delta method for $\bar X$^2

I have a question about the delta method. The question is: $X_1,...,X_n \sim N(\mu,\sigma^2)$, where $\sigma^2=V(x)$ and $\mu=E(X)$ let T=$\bar X^2$ be an estimate for $\mu^2$. Find the asymptotic ...
stat_student's user avatar
1 vote
1 answer
261 views

Assumption to apply the delta method

When proving the delta method of distributions in my textbook we make the following assumption: Let $X_{n}$ be a sequence of random variables. and: ${\sqrt{n}[X_n- c]\,\xrightarrow{D}\,\mathcal{N}(0,1)...
Daniel De Wet's user avatar
3 votes
1 answer
751 views

Delta method for estimating a ratio involving variance and mean

Let $X$ be a binomial RV with parameters $(n,p)$. I am interested in the ratio given by $\hspace{5cm}\boxed{R=\frac{var[f(X)]}{\mu[f(X)](1-\mu[f(X)])}}$ where $\mu[f(X)]$ denotes the mean of $f(X)$. ...
wanderer's user avatar
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2 answers
878 views

Delta method for Poisson ratio

Let $X_1,...,X_n$ be drawn from $Pois(\lambda)$ and $Y_1,...,Y_n$ from $Pois(\theta)$. I would like to find the asymptotic distribution of $$\frac{\overline X}{\overline X + \overline Y }$$ using ...
Bihu Duo's user avatar
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1 vote
1 answer
523 views

Can I use the delta method with a function that depends on n to approximate the distribution of a function of the sum of iid random variables?

Let $X_1, X_2,...$ be i.i.d. random variables with finite mean $\mu$ and finite variance $\sigma^2$. From the Central Limit Theorem, we know that $\sqrt{n}(\bar{X_n}-\mu)$ tends in distribution to $N(...
DM-97's user avatar
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2 votes
1 answer
445 views

Derivation covariance between ratio of random variables

Suppose I am interested in computing the covariance between $\frac{A}{B}$ and $\frac{X}{Y}$. From Ratio of correlated vectors is uncorrelated? I understood that using the delta method, this amounts to ...
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140 views

Calculating the variance of a function which depends on the standard deviation and mean of a random variable

I am a bit new to statistics and had some conceptual questions regarding the calculation of variance. I want to calculate the variance of a function $y=\frac{\sigma_{X}}{g(\overline{X})}$. As seen in ...
Ab21's user avatar
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0 answers
564 views

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 ...
cwh's user avatar
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3 votes
1 answer
225 views

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 ...
HolParadise's user avatar
1 vote
0 answers
25 views

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)/...
ReichieLee's user avatar
1 vote
0 answers
121 views

Delta method for third moment

Suppose $X_1,...,X_n$ is a sample from a population with mean $\mu$ and variance $\sigma^2$ and third central moment of $\mu_3$. I want to justify that: $$E[\left( h(\bar{X})-E(h(\bar{X}))\right)^3]=\...
statwoman's user avatar
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1 vote
1 answer
164 views

Why approximate delta-method Variance isn't multiplied by $\frac{1}{n}$?

I'm reading Casella-Berger chapter 10, where they introduce asymptotic evaluations. I don't seem quite to understand how the factor $\sqrt{n}$ works when we are using asymptotic evaluations in order ...
Niccolò Cavagnola's user avatar
2 votes
1 answer
433 views

Using delta method to get confidence intervals for multinomial logit?

I am working with some choice modeling data and am interested in trying to potentially use the delta method with the multinomial logit model that I'm analyzing the data with. Here's an example: First, ...
andy_d's user avatar
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7 votes
2 answers
282 views

variance estimation using order statistics

I have four largest samples drawn from a distribution of N i.i.d Gaussian R.V. with standard deviation (Sigma) where sigma is unknown. N is known to be between 50-200. Mean is given to be 0. How do ...
user2719731's user avatar
4 votes
1 answer
170 views

confidence interval of $\beta$, where $X$'s are from exponential distribution

Suppose $X_i\overset{ind}{\sim}\mathcal{E}(\lambda_i)$, where $\lambda_i=(t_i\beta)^{-1}$, where $t_i$'s are positive known values and $\beta$ is positive unknown parameter. Here $i=1,\dots,n$. It can ...
Tan's user avatar
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2 votes
0 answers
275 views

Second order with Delta method on a ratio to improve variance estimation accuracy

Following a previous post on math exchange without success, I have applied the "Delta method" that says : Delta method : There are alternative formulations of this expression which may be ...
user avatar
6 votes
1 answer
93 views

How to estimate the sample variance of the estimator of the parameter $P(x≤0)$ where $x \sim N(\mu,\sigma)$?

This question relates to How to estimate $P(x\le0)$ from $n$ samples of $x$? One way to make this estimate is to use estimates $\hat\mu$ and $\hat\sigma$ and compute from those $$ \hat p = \Phi \left(...
Sextus Empiricus's user avatar
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1 answer
442 views

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 ...
Thomas DeWaters's user avatar
2 votes
0 answers
202 views

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 ...
Suriya's user avatar
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197 views

What's the asymptotic distribution of $\exp(X_n)$, if $X_n$ is a sequence of asymptotically normally distributed random variables?

Let $(X_n)_{n\in\mathbb{N}} $ be a sequence of asymptotically normally distributed random variables, such that $\lim\limits_{n\to\infty}\sqrt{n}X_n\sim N(0,1)$. What's the asymptotic distribution of $...
stats19's user avatar
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3 votes
0 answers
210 views

Confusion about the delta method

I'm reading Statistical Models by A. C. Davison and I'm really confused by this section on the Delta method. It's not mentioned explicitly, but is $h(T_n)$ a consistent estimator of $h(\mu)$? In the ...
Yandle's user avatar
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2 votes
1 answer
584 views

Method for obtaining an asymmetric confidence interval

I have an estimator $\hat{\theta}$=$\frac{XY}{Z}$ where $X$ and $Y$ are constants and $Z$ is a random variable. $Z$ ranges from [1, $Y$]. Further $Z$ $\rightarrow$ $Y$ in the limit (asymptotically), ...
compbiostats's user avatar
  • 1,557
3 votes
1 answer
142 views

Why is the delta method defined the way it is?

The delta method begins with the assumption of $\sqrt{n} \left[X_n - \theta\right] \stackrel{D}{\to} \mathcal{N}(0, \sigma^2)$. Why is this? Wouldn't it make more sense to start in the more familiar ...
Jarrett Meyer's user avatar
9 votes
2 answers
300 views

Trying to approximate $E[f(X)]$ - Woflram Alpha gives $E[f(X)] \approx \frac{1}{\sqrt{3}}$ but I get $E[f(X)] \approx 0$?

Let $X \sim \mathcal{N}(\mu_X,\sigma_X^2) = \mathcal{N}(0,1)$. Let $f(x) = e^{-x^2}$. I want to approximate $E[f(X)]$. Wolfram Alpha gives \begin{align} E[f(X)] \approx \frac{1}{\sqrt{3}}. \end{align} ...
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The two estimators of mean of Gamma distribution and the estimators' variances

In Casella Example 10.1.18, the author says it is not easy to calculate the mean of gamma distribution. It seems that we CAN use the easy way $\bar X=\frac{\sum X_i}n$, but the variance of the mean we ...
Charlie Chang's user avatar