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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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Measuring Robustness in Variational Bayesian Inference and Nonlinear Filtering

I am interested in how to properly pose/measure robustness, in a qualitative or potentially quantitative manner, when inferring a probability density function (pdf) either by Bayes' rule or a ...
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Distance of a hyperplane from Geometric Median

Let $d,n \in \mathbb{N}$. Assume we have $n$ points, $x_1,\dots,x_n$ where for every $i \in \{1,\dots,n\}$, we have $x_i \in \mathbb{R}^d$. Define the geometric median as $$ \theta^\star \in \arg\min_{...
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Query in "Certified adversarial robustness via randomized smoothing" Cohen et al paper

I am going through the proof sketch for Randomized Smoothing from paper: Cohen, Jeremy, Elan Rosenfeld, and Zico Kolter. "Certified adversarial robustness via randomized smoothing." In ...
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Incorporating the absolute difference and the relative difference in a single metric

I have a series of 12 connected 'zones' that have a specific numerical value at any given point in time, either negative or positive. This amounts to time series data. I have some forecasted values ...
Tom's user avatar
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Robustness of Geometric Median to Change of k Points

Let $N \in \mathbb{N}$ be a constant. Let $B_d(1)$ denote the ball of radius one in $\mathbb{R}^d$. Let $(a_1,\dots,a_N)\in (B_d(1))^N$ and $(b_1,\dots,b_N)\in (B_d(1))^N$ be two sets of N data ...
MMH's user avatar
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Principal Component Analysis Based on Scatter Matrices

There is a paper on robust PCA based on scatter matrices: https://wis.kuleuven.be/statdatascience/robust/papers/2005/hubertrousseeuwvandenbranden-robpca-technom-2005.pdf. PCA is generally performed ...
Lyricist's user avatar
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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, ...
Erica Gao's user avatar
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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 ...
HelloGoodbye's user avatar
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205 views

distance between coordinate-wise median and geometric median

Let $N \in \mathbb{N}$ be a constant. Let $B_d(1)$ denote the ball of radius one in $\mathbb{R}^d$. For $i \in \{1,\dots,N\}$, let $a^{(i)}$ be a vector in $B_d(1)$. Define the geometric median of $a^...
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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 \...
PLee's user avatar
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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=...
Mathews Boban's user avatar
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1 answer
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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} \...
Dylan Dijk's user avatar
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153 views

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)(\...
orematasaburou's user avatar
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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 $\tilde{\psi}_P$ ...
dual 's user avatar
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1 answer
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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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Interpretation of $l_p$ norm inequality

If $1\le p\le q\le \infty$, we know that the following inequality holds: $$\|a\|_q\le \|a\|_p.$$ What could be a possible interpretation of this inequality for a non-mathematician? For example, can we ...
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Minimiser of risk for linear-exponential error loss

Question: Solve the following optimisation problem: $$\arg\min_{f} \mathbb{E} \left( \exp (-(Y- f(X))) + (Y - f(X)) - 1 \right)$$ Context: The linear-exponential loss function (LINEX loss for short) ...
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Confused about the relationship between PCA and robust PCA

I recently learned about PCA and robust PCA. I understand that PCA is identifying the principal components by finding the eigenvectors of the covariance matrix (which of course contains information ...
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Determining if a bag of wheat cent pennies has been searched - using statistics.

I have collected a sample of 5000 wheat cent pennies, and recorded the number of each year and mint mark $\in \{P, D, S\}$. I have also found out how many wheat cent pennies were minted at each mint, ...
Sheldon Skaggs's user avatar
2 votes
2 answers
376 views

Robust Least Squares for general 2D lines

Question My goal is to robustly estimate a general 2D line from $n$ data points, where the line is parameterized by $\rho > 0$, the distance from the origin to the line and $\varphi$, the angle ...
Flo Ryan's user avatar
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Computing the pdf of two random variables with different supports.

I am fairly new to stack exchange, but I need some guidance on the following problem: Let $X$ and $Y$ be two independent, continuous random variables described by probability density functions $f_{X}$ ...
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Huber's derivation of asymptotic properties of M-estimator (Robust Statistics)

Below are snippets of Chapter 3 of Huber's "Robust Statistics. They are needed to derive the asymptotic normality of M-estimator. However, I am concerning the step going from (2.17) to (2.18) i....
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Argmin operation on loss function of a matrix

I'm trying to solve the following equation for $\beta$, the only unknown in the equation. I have two questions. I'm not quite sure how a matrix could fit into the Huber loss function and subsequent ...
Aaron Ahn's user avatar
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39 views

For this probability is the Binomial Distribution more appropriate or just exponentiating the odds?

I have a somewhat of a unique problem, I am trying to determine the odds of the following, A Bernoulli Sequence where it follows the pattern of: Tails, Tails, Tails, Tails, Heads, Tails, Tails, Tails, ...
Arthur's user avatar
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What is the CDF, pdf and expectation of $\theta^*$ of $\theta$?

I have that $x_1, x_2,...,x_n$ are from a rv $X$ that has the density function $f_X(x)=\frac{2x}{\theta^2} \quad$ for $0 \le x \le \theta \quad$ and $f_X(x)=0 \quad$ otherwise. I have determined that ...
Maria Fernandez's user avatar
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1 answer
257 views

What is the MLE $\theta^*$ of $\theta$? [duplicate]

I have that $x_1, x_2,...,x_n$ are from a rv $X$ that has the density function $f_X(x)=\frac{2x}{\theta^2} \quad$ for $0 \le x \le \theta \quad$ and $f_X(x)=0 \quad$ otherwise. Ihave to determine the ...
Maria Fernandez's user avatar
0 votes
1 answer
21 views

Superior limit of sequence of random variables.

Let $X_{n}$ be a sequence of real random variables and let $X$ a real random variable defined over the same p.s. $(\Omega,\mathcal{A},\mathbb{P})$ and such that $X_{n}(\omega)\searrow X(\omega)$ for ...
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A question about real random variables

Let $X_{n}$ be a sequence of real random variables and let $X$ a real random variable defined over the same probability space $(\Omega,\mathcal{A},\mathbb{P})$ and such that $X_{n}(\omega)\searrow_{n} ...
user13761697's user avatar
1 vote
1 answer
79 views

A possible characterization for the median of a r.r.v.

Let $X$ be a real random variable. It is clear that any median $m\in\mathbb{R}$ of $X$ satisfies that $$\text{E}[|X - m|] = \min_{x\in\mathbb{R}}\text{E}[|X - x|]$$. My question is the following. If $...
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I need help or some advice to solve this exercise about functions random variables.

I've thought about it for a while using the cumulative distribution, but I've not concluded. Three people $A$, $B$ and $C$ arrive at the same time at a telephone booth that has two telephone sets. ...
novatoEST's user avatar
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41 views

KL Divergence estimation

I am trying to construct a certain bound for the KL divergence between two two numbers. I want to show that $-n D((a+s/\sqrt{n}|| a)\le -\frac{s}{2a(1-a)}+Error(1/\sqrt{n})$ $(a\in(0,1))$ \begin{align*...
user593295's user avatar
3 votes
1 answer
114 views

Infimum of the set of medians.

Let $\{F_{n}\}_{n}$ be a sequence of cumulative distribution functions such that converge to $F$, in the sense that $F_{n}(x)\rightarrow F(x)$ for all $x\in\mathbb{R}$. We define the function infimum ...
user13761697's user avatar
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49 views

Limit of the Median of Random Variables

Let $X_{1},X_{2},\cdots$ be real random variables identically distributed. We consider the sequence $m_{n} := Med(X_{n})$, where $Med(\cdot)$ denotes the Median of a random variable. My question is ...
user13761697's user avatar
2 votes
1 answer
299 views

Robust Optimization: Using Bertsimas and Sim approach for linear model formulation (maximization problem)

I tried to use Bertsimas and Sim approach for an uncertain linear model, but the thing is the answer I got for the Bertsimas and Sim linear model when Γ = 2 is different from Soyster model's result, ...
Aisa.Imn's user avatar
1 vote
1 answer
361 views

Proximal Operator / Proximal Mapping of the Huber Loss Function

Given the Scalar Huber Loss Function: $$ {L}_{\delta} \left( x \right) = \begin{cases} \frac{1}{2} {x}^{2} & \text{for} \; \left| x \right| \leq \delta \\ \delta (\left| ...
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1 vote
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Multivariate robust estimation for high dimension

Is there a known way to extend (at least some) robust location estimators to multidimensional case, possibly in an efficient manner? For example, I know that, given a list of scalars $x_1, ..., x_n$, ...
SpiderRico's user avatar
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Estimate scale parameter from 5% contaminated mean-zero normal sample

I have a normal sample but that is contaminated in the left and the right tails (no more than 2.5% each). The contamination gives rise to high and low values. I wanted to know what methods I have in ...
user3236841's user avatar
1 vote
1 answer
158 views

How to prove that the optimal point for a quasilinear function lies in its extreme points

I was reading an article about the robust optimization of the MNL choice model,and in one of its proofs it uses the point that if we're tring to solve the minimun of a quasilinear function ,which is ...
Nancy Zhang's user avatar
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1 answer
2k views

Stationary of Moving Average Process [closed]

I have studied about moving average process $MA$ of first and second orders, and I need the values of parameters that make the process $MA(1)$ and $MA(2)$ are stationary. Thanks
AliSami's user avatar
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3 votes
3 answers
895 views

Robust line segment fitting to a digital path

Robust line fitting to a set of 2D points is a well studied problem for which several approaches are known. They usually consider the point cloud as unstructured. I call a digital path a sequence of ...
user avatar
0 votes
1 answer
402 views

Proofs for consistency of estimating equations / M-estimators without a compact parameter space?

Most proofs for the consistency of parameters obtained from estimating equations depend on a compact parameter space. However, I have almost never worked with parameter spaces that are compact (they ...
Guillaume F.'s user avatar
1 vote
1 answer
188 views

Examples when Bootstrap-t-test should be used

I am currently learning about robust methods for comparing means, and read about the Bootstrap-t-test and its implementation in R. However, I found that this test tends to give results similar to the ...
Llarian's user avatar
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1 vote
1 answer
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Robustness of estimators

I have a question regarding the robustness of estimators. I have 4 estimators and I have been asked to consider which estimator is most robust to mis-specification. What is mis-specification? What are ...
user12321's user avatar
1 vote
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10k views

Calculating Median of grouped frequency distribution using (N+1) rather than (N)

The mathematical expression describing the rank of the median of a distribution of N observations is: for a list of raw data values, (N+1)/2; for an ungrouped frequency distribution, typically ...
Tries Hard's user avatar
4 votes
1 answer
1k views

Least Absolute Deviation (LAD) Line Fitting / Regression

I want to implement robust line fitting over a set of $n$ points $(x_i,y_i)$ by means of the Least Absolute Deviation method, which minimizes the sum $$\sum_{i=1}^n |y_i-a-bx_i|.$$ As described for ...
user avatar
1 vote
1 answer
2k views

Consistency of an asymptotically linear estimator

An estimator $\hat{\boldsymbol{\gamma}}\triangleq \hat{\boldsymbol{\gamma}}(\mathbf{x}_1,\ldots,\mathbf{x}_M)$ of the $q$-dimensional vector $\boldsymbol{\gamma}_0$ is called asymptotically linear (AL)...
Vuk's user avatar
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1 answer
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Source for claim by Rousseeuw & Verboven regarding robust Newton-Raphson

Classic Newton-Raphson estimation converges to $x$ using the relationship: $$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$ In their paper on robust estimators of location and scale for small sample sizes,...
Avraham's user avatar
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5 votes
2 answers
2k views

Robust orientation of a point cloud

I have 2D point clouds which are 4-way symmetrical (invariant by 90° rotation). The points are usually arranged on the nodes of a square grid, densely populated, but some cases can be more complicated....
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