Questions tagged [robust-statistics]
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normally distributed. Robust statistical methods have been developed for many common problems, such as estimating location, scale and regression parameters.
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Mixture Model Expectation
We know that if an i.i.d. sample is drawn from $p_{\theta}=\text{Ber}(\theta)$, $\theta\in (0,1)$ then
$$\mathbb{E}_{p_{\theta}}[\bar{X}] = \theta,$$
where $\bar{X}$ denotes the sample mean.
Now, ...
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Expectation under true distribution with mixture samples
Let $X_1, X_2, \dots , X_n$ be an i.i.d. sample from the mixture distribution
\begin{equation} \label{eqn:mixture distribution}
p_{\epsilon,\theta} = (1 - \epsilon)p_{\theta} + \epsilon \delta,
\end{...
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How to rigorously robustness to outliers and prove that Least Absolute Deviation is more robust than Ordinary Least Square?
I am an engineer who really love math, and recently watched an educational video "Fitting a line WITHOUT using least squares?" where at timestamp 7:10, the presenter demonstrates that Least ...
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Is it possible to derive an algebraic solution of optimal parameters given an l1-norm misfit statistic, rather than l2-norm/chi2?
I have been reading about maximum likelihood parameter estimation using $\chi^2$ as the misfit statistic. In particular, I've followed a derivation where a linear model $y = mx + c$ is used, leading ...
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Why does Least Absolute Deviation regression exactly fit n measureemnts for a linear system with n independent variables? [closed]
I am applying LAD regression to conduct some research. I have the following questions regarding LAD:
I know LAD exactly fits n measurements for a linear system with n variables. But I cannot easily ...
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Does the Tyler's M-estimator lose the estimator of scale?
I was learning some robust estimation methods dealing with outliers and heavy-tail. I noticed that Tyler's M-estimator, whose key idea is to standardize the sample data by the distance to the mean, ...
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Is there a robust version of the moving least squares or of the Savitzky–Golay filter?
Is there a name for the following type of filter?
I want to filter a noisy signal $f(x) = f_0(x) + noise(x)$ (where $f_0$ is a noiseless signal), to get a filtered signal $f_\text{F}(x)$ while ...
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how does huber compute the $var(s_n)/E[s_n]$ and $var(d_n)/E[d_n]$?
How does Huber in book 'Robust statistical procedures' in chapter 1 compute the variance of certain statistical functions?
He defines the mean square deviation to be $$s_n = \sqrt{\frac{1}{n} \sum \...
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About the closest linear function to an arbitrary function in L1 norm
Let $\mu$ be a probability distribution over $\mathbb{R}^n$. All functions discussed henceforth are from $\mathbb{R}^n$ to $\mathbb{R}$. Let $l^\ast$ be a linear function and $f$ be a function such $f=...
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Proving upper bound for Bias of truncated sample mean
So we have the truncated sample mean:
$\begin{align}
\hat{\mu}^{\tau} := \frac{1}{n} \sum_{i =1}^n \psi_{\tau}(X_i)
\end{align}$
Where the truncation operator is defined as:
$\begin{align}
\...
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M-estimator as a quantile estimator
According to the answer https://stats.stackexchange.com/a/497785/310702,
$\alpha$-quantile sample estimator can be considered as M-estimator with
$\rho(y_i,\theta)=\alpha(y_i-\theta)_+ + (1-\alpha)(\...
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confidence interval methods for the mean are "robust against departures of normality"---does that refer to the population or sampling distribution?
It is sometimes said that confidence interval methods for the mean are robust against departures of normality. But does this refer to the population distribution, or the sampling distribution (of the ...
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How can the tangent set be not closed or not linear or both?
This question is related to my previous question: Question about a statement: why taking linear span?
There the answer was satisfactory but I am wondering now about some examples of tangent sets that ...
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Question about a statement: why taking linear span?
I am reading some lecture notes about semiparametric statistics. We are in the context of determining some basic properties about the efficient influence function, here denoted by $\tilde{\psi}_P$ ...
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What is the formal definition of the breakdown value of a statistic
On page 482 of Statistical Inference (Second Edition) by Casella & Berger, the authors define the breakdown value as follows:
Defintion 10.2.2 Let $X_{(1)} < \dots < X_{(n)} $ be an ordered ...
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