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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Infinitesimal Robustness, influence function of $T$ at $F$.

This text is taken from Introduction to robust estimation and hypothesis testing. Wilcox R. First I will write down the description that leads to definition of relative influence on $T(F)$ and then I ...
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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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Sample mean of contaminated normal distributions

Consider the statistical structure $(\mathbb R, \mathcal{B}, \{N(\mu,1): \mu\in \mathbb{R}\}$ and suppose the problem of estimating $\mu$ with quadratic loss function. Consider the estimate $\bar x$ (...
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Risk function for sample mean and sample median (robust statistics)

Let's consider a statistical model \begin{align*} M_0=(\mathbb R, \mathcal{B}, N(0,1)) \end{align*} a sample of size $2n+1$ and estimates: mean $\bar x$ and median $t_{me}$ with risk functions $\...
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Huber Loss with Errors

I am performing linear regression, where I have data $y$ with errors $\sigma_{y}$ as a function of x. I would like to fit a function $f(x) = mx + c$ in a robust manner, while still weighting to the ...
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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, ...
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2 votes
2 answers
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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 ...
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Can we find sample quantiles in higher dimensions?

How can i extend the sample quantile to higher dimensions? For example, we can extend the median by $$ \text{Median} = \arg\min_c\sum_{j=1}^N\|x_j-c\|_p$$ Since the median is a sample quantile, I was ...
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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 ...
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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, ...
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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 ...
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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 ...
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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} ...
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1 vote
1 answer
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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. ...
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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*...
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3 votes
1 answer
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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 ...
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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 ...
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2 votes
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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, ...
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1 vote
1 answer
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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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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 $x_1, ..., x_n$, ...
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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 ...
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1 vote
1 answer
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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 ...
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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
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3 answers
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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 ...
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1 answer
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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 ...
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1 vote
1 answer
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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 ...
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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 ...
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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 ...
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3 votes
1 answer
557 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 ...
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1 vote
1 answer
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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)...
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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,...
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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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1 vote
1 answer
639 views

Proper loss function for this robust regression problem

For classification problems, we may use cross entropy as our loss function, while for regression problems, we may choose mean square error. However, here I'd like to find a proper loss function for ...
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1 vote
1 answer
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Why is an M-estimator from statistics not necessarily measurable?

I read somewhere that an M-estimator, defined as estimators that maximize a criterion of the form: $$ \theta \mapsto M_n(\theta) = \sum_{i=1}^{n}m_\theta(Y_i) $$ for some functions $\{m_\theta: \...
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About trimmed inner product

Recently, I have read a paper titled as "Robust Sparse Regression under Adversarial Corruption" appeared in ICML 2013, which is one of the most famous and top conference in machine learning(actually, ...
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Estimation of the eigenvectors of a covariance matrix in the presence of misspecified eigenvalues.

I have the following objective function: $$f(L) = E[y'L^{\phantom{'}}D^{-1}L'y],$$ where $E[\,\cdot\,]$ is the expectation operator; $D$ is a $k$-dimensional diagonal matrix with strictly decreasing ...
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2 votes
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optimal orthogonal matrix in L1 sense

I want to find an orthogonal matrix $O\in SO(n)$ such that $\|Y - OX \|_1$ is minimized, where X and Y are matrices (of appropriate sizes). I know that there is a solution to this problem using SVD ...
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Best optimization method for minimizing error of estimated default probability

The credit risk modelling problem can be stated as: $$ \min_{a \in \mathbb{R}^{1 \times n}} \sum_i \Big( d_i -\frac{\exp(a x_i) }{1+\exp(a x_i)}\Big)^2 $$ Where $ d_i \in \mathbb{N} \in [0,1] $ is the ...
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3 votes
1 answer
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Time series determined by other time series

Intuitive Question Suppose I'm given a set of $k$ time-series $\{X_t^1,\dots, X_t^k\}$. Is there a way to determine how much of each series is dependent on the others. Formal Question More ...
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2 votes
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Notation for minimum

I'm reading Huber's Robust Statistics right now, and at the beginning of Chapter 3, he writes the following notation: $$\sum \rho(x_i;T_n) = \min!$$ Similarly, a few lines down, he writes: $$\sum \...
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What is the difference between the infinity norm of a transfer function and the infinity norm of a matrix

I am studying robust control system, and get confused with the following two definitions of infinity norm. ($G(j\omega)$ is the transfer function of a MIMO system) [1] $$\left \| G \right \| _\infty =...
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3 votes
1 answer
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Explaining the standard deviation formula

I'm revisiting standard deviation for the first time years, and I can't for the life of me recall the difference between two formulas. In particular, I'm also looking for how we arrived at these ...
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Calculating 5 different ranges for people resource management

I am working on a project for my company. My team is building a project charter template. In this template needs to be a drop down that estimates how many full-time employee days(FTE) will be ...
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Proximal Operator of the Huber Loss Function

I want to solve the following problem: $$ \arg\min_x |x|_\mu + \frac{1}{2\sigma} |x-x^k|^2 $$ Where the Huber Loss Function is given by: $$|x|_\mu = \begin{cases} \frac{|x|^2}{2}, & |x| \leq \...
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2 votes
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Huber loss vs l1 loss

From a robust statistics perspective are there any advantages of the Huber loss vs. L1 loss (apart from differentiability at the origin) ? Specifically, if I don't care about gradients (for e.g. when ...
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