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 measurability question
Suppose we have a random variable $\hat \theta_n$ such that $\hat \theta_n \to \theta_0$ in probability. Let $f \colon \mathbb{R} \to \mathbb{R}$ be infinitely differentiable function. Then, the delta ...
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Frequentist SVAR sign restrictions IRF error bands construction
I am currently writing a small library for sign restricted SVAR, and I ran into a problem of constructing error bands for impulse responses. At this moment, I use Lutkepohl delta-method to construct ...
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Delta method application
consider
$$C = (\Sigma B) \circ B$$
where $B$ is a $K\times 1$ vector of parameters
$\Sigma$ is a $K\times K$ covariance matrix and
$\circ$ denoted the element-wise multiplication.
$\Sigma$ is to be ...
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Standard Error of fitted value at breakpoint (segmented regression)
I am currently using the "segmented.lm" function to detect a change point in my data. At this stage I am trying to figure out how to derive the SE of the y value of the corresponding change ...
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What to do when the denominator in a proportion also has uncertainty?
This is related to my previous questions Help using the delta method and Help using the delta method. However this time, the denominator used in the rates has uncertainty.
I have two years of data (...
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Help using the delta method
I have the following information:
$r_1$ = response count in year 1
$r_5$ = response count in year 5
$p_1$ = population in year 1
$p_5$ = population in year 5
$\text{Rate}_1 = \frac{r_1}{p_1}$ = rate ...
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How do I compute a 95% CI for the percentage change in the average cost per observation using aggregated data and the delta method?
I have aggregated data from two independent groups (or experiments). For each group, I have the following summary statistics:
Total cost,
Total number of observations, and
Sum of squared costs.
For ...
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Condition for Delta Method Approximation for $|μ| \gg σ$
I'm trying to understand the mathematical justification for the Delta method approximation when $|μ|$ is substantially larger than $σ.$ Specifically, I'm looking for a proof of the following formulae:
...
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Delta Method and the Variance Convergence
In the references I've seen, the $\Delta$-method is typically formulated in terms of convergence in distribution: for $X_i$ i.i.d., $\mathbb{E}[X_i]=\mu$ and $\mathrm{Var}_\mu(X_i)=\sigma^2<\infty$ ...
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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 ...
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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_{...
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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 $\...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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 ...
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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
$$\sqrt{n}[X_n - \theta]\xrightarrow{\text{D}} \...
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Variance of Kaplan-Meier estimator
Here and here and on the Wikipedia page it is stated that for estimating the variance of Kaplan Meier estimator $S(t)$ using delta method one can use the fact that: $$Var(log\hat{S}(t)) \approx \frac{...
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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 ...
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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 ...
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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 ...
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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,...
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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% ...
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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 ...
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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 ...
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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)...
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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 $f(X)$.
...
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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 ...
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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(...
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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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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 ...
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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 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]=\...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
user226073
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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(...
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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 ...
