Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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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Variance of a function of one random variable

Lets say we have random variable $X$ with known variance and mean. The question is: what is the variance of $f(X)$ for some given function f. The only general method that I'm aware of is the delta method, but it gives only aproximation. Now I'm…
Tomek Tarczynski
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How to use delta method for standard errors of marginal effects?

I am interested in better understanding the delta method for approximating the standard errors of the average marginal effects of a regression model that includes an interaction term. I've looked at related questions under delta-method but none have…
Thomas
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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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Different ways to produce a confidence interval for odds ratio from logistic regression

I am studying how to construct a 95% confidence interval for odds ratio from the coefficients obtained in the logistic regression. So, considering the logistic regression model, $$ \log\left(\frac{p}{1 - p}\right) = \alpha + \beta x …
Márcio Augusto Diniz
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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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Use $\bar{X}^2$ for hypothesis test that $\mu=0$ because faster convergence rate?

Suppose that I have $X_1,\ldots,X_n$ are i.i.d. and I want to do a hypothesis test that $\mu$ is 0. Suppose I have large n and can use Central Limit Theorem. I could also do a test that $\mu^2$ is 0, which should be equivalent to testing that $\mu$…
Xu Wang
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Standard errors of hyperbolic distribution estimates using delta-method?

I want to calculate the standard errors of a fitted hyperbolic distribution. In my notation the density is given by \begin{align*} H(l;\alpha,\beta,\mu,\delta)&=\frac{\sqrt{\alpha^2-\beta^2}}{2\alpha \delta K_1 (\delta\sqrt{\alpha^2-\beta^2})}…
Jen Bohold
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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} Using a Taylor expansion approach, and noting…
Bertus101
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Taylor Series and Multivariate Delta Method

I asked this question on https://math.stackexchange.com/ but did not get any answer. Sorry for cross posting. I'm trying to understand delta method for matrices and vectors to find the variance-covariance matrices for the functions of matrices and…
MYaseen208
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95% confidence intervals on prediction of censored binomial model estimated using mle2 / maximum-likelihood

I am working on a problem in which I have multiple pairs of currently living males i that each have a presumed paternal ancestor ni generations ago (based on genealogical evidence) and where I have info on whether there is a mismatch in their Y…
Tom Wenseleers
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How do you calculate standard errors for a transformation of the MLE?

I need to make inference about a positive parameter $p$. To acomodate the positiveness I reparametrized $p=\exp(q)$. Using MLE routine I computed point estimate and s.e for $q$. The invariance property of the MLE directly gives me a point estimate…
Marcel
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How to compute asymptotic confidence intervals for differences in quantiles?

Can anyone give me advice on computing the asymptotic confidence intervals for a difference in quantiles of a distribution? For example, I have fit a log-normal distribution to doubly interval censored data with a maximum likelihood estimator and…
scottyaz
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Limiting distribution of a squared sum of random variables

Let $S_n = \frac{1}{n}\sum_{i=1}^n X_i$, and $T_n = \frac{1}{n}\sum_{j=1}^nY_i$, where The $X_i$ are iid, the $Y_i$ are iid (with a different law) $X_i$, and $Y_i$ are dependent For $i\neq j$, $X_i$ and $Y_j$ are independent. Is there a central…
Ian Langmore
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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 we estimate Sigma (variance) using these 4 largest…
user2719731
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Elementary approach to higher order asymptotics

I am trying to understand “higher order asymptotics”. I find several texts on Likelihood asymptotics, nothing’s easy to read... if you have any nice pointers on this direction, I’ll be interested; however my main question follows. The following…
Elvis
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