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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.

For the quotation, see http://www.stata.com/support/faqs/statistics/delta-method/. For the second sense of the definition, refer to http://en.wikipedia.org/wiki/Delta_method.

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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 counter-example, consider a sequence of random…
Heisenberg
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How to use Delta Method while the first-order derivative is zero?

http://en.wikipedia.org/wiki/Delta_method In the Wikipedia article, it was assumed that $g'(\theta)$ must exist and that $g'(\theta)$ is non-zero valued. Is it possible to find the asymptotic distribution for $\sqrt{n}(g(X_n)-g(\theta))$ given…
Eddy Chen
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Calculating the Variance using Delta Method

I'm trying to find the variance of $L$, $Var(L)$, using the delta method (I want to find a closed form). $L$ is defined as: $$L = \frac{A}{B} + \frac{C}{D}$$ All $A$, $B$, $C$, and $D$ are dependent. But I'm not sure how to proceed.
user9292
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Delta method for non-normal variables

Is the delta method valid also for non-normal variables? Claim: Let $\sqrt{n}(\hat{X}_n-\theta) \xrightarrow{d} \hat{f} $. With $\hat f$ having a finite distribution. Then for every $g$ such that $g'(\theta)$ is nonzero we…
Thomas
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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 arrangement of $X_n \stackrel{D}{\to}…
Jarrett Meyer
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Delta Method to find Asymptotic distribution

Hi, so I understand that the delta method means finding the distribution of some function of B, F(b) through taylor expansion. And through this process, we get that: $$ F(b) = N( f(b), \Gamma * \Sigma_\beta * \Gamma') $$ Where gamma = df/db So in…
BNA
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Authorship of a delta method - Does R. Dorfman invented it?

I would like to define an authorship of a delta method, which I need to make a proper reference in my work. As far as I know, the earliest record is dated back to 1936 when Robert Dorfman has published his A Note on the δ-Method for Finding Variance…
Adam Przedniczek
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The mean and Variance of $log(Combination Index)$

In biological sciences, ($CI$) stands for the Combination Index which is a conventional method for dose-response assessment and drug interaction analysis. It can be defined as: $$CI = \frac{\bar{D}_{1} \hat{\beta}_{1}}{ln(\bar{SF}_{1}) -…
user9292
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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)/control clicks per user % change of click through…
ReichieLee
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Why is delta method only for asymptotic distributions?

I'm wondering why the delta method, , "is only used for asymptotic distributions", as @mpiktas write in this post: Variance of a function of one random variable, or as is written here: "The delta method is a general method for deriving the variance…
HeyJane
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delta method variance

This is probably a basic question but I will ask it anyway b/c Im stuck. I do bird research and I have bird density estimates for thousands of sampled patches along with the associated variance. To estimate the total population size of all my sample…
Nahanian
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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."
Norbert
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Prelude to the delta method

Given a probability density say $f(x)$ with parameters $\theta$ and a sample of size n say $x_1,\ldots,x_n$ we can compute the MLE estimate say $\theta_n$ by passing $f$ and $x_1,\dots,x_n$ to optim in R. We will have by the asymptotic normality of…
user2338823
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