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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Conceptually Interesting Applications of the Delta Method [closed]
What are some nice, insightful applications of the delta method?
From Casella & Berger's Statistical Inference book (2nd ed.), the following example appears under The Delta Method:
Example 5.5.19 ...
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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 ...
