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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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Understanding a simple math example about Doubly Robust method of moments

I want help understanding this example on doubly robust moment conditions. No following text helps explain this example as it is independent of the earlier text. The things I don't understand is the ...
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12 views

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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M-estimators: which function is more resistant to outliers? Huber or Bisquare?

I'm studying about M-estimators. The question is: Which function is more resistant to outliers, Huber or Bisquare? In the article of John Fox & Sanford Weisberg, October 8 2013, I read that "The ...
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12 views

Minimum-volume confidence ellipsoid for regression with nuisance parameters

I have a linear regression problem with Gaussian errors, nuisance variables and the parameter of interest, $\alpha$. I want to find the smallest-volume region containing the true value of $\alpha$ ...
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What is the meaning of stochastic sampling?

I came across this term in the context of Kernel Methods for Supervised Learning. Subsampling is the selection of a subset from the training set. But what is stochastic subsampling, I understand that ...
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33 views

Integral of Huber piecewise function.

The below $\psi$-fuction is the derivetive of Huber function, how can we find the integral of it? the answer is given. Can anybody prove it.][1] $$ \psi(u_i) = \[ \begin{cases} u_i & |u_i|...
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1answer
45 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 ...
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91 views

What is the interpretation of (y bar square/variance) in 'Nominal is Best' method of Taguchi Design?

If y bar is the mean and s is the standard deviation then what is the interpretation of their ratio? Why they are squared?
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433 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
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3answers
258 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 ...
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1answer
88 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 ...
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1answer
42 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 ...
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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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6k 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 ...
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1answer
52 views

Notation in this imprecise Markov chain upper transition operator definition

In Example 11.6 on p. 270 of Hermans and Škulj's "Stochastic Processes" in Augustin et al.'s Introduction to Imprecise Probabilities, there is a definition of an upper transition operator as $$\...
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68 views

Least absolute deviation line fitting

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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1answer
314 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)...
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1answer
48 views

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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687 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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2answers
133 views

Simple calculation in imprecise probability urn example

In Miranda and de Cooman's chapter 3, "Structural judgements", in Augustin et al.'s Introduction to Imprecise Probability, example 3.4 on p. 65 shows that independence in the selection (type-2 ...
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1answer
504 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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1answer
134 views

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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50 views

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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64 views

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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177 views

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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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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1answer
74 views

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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624 views

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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1answer
182 views

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, im also looking for how we arrived at these ...
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1answer
36 views

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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1answer
3k views

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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1answer
84 views

Why does the natural extension of a lower prevision dominate it?

On p. 48 in Troffaes and de Cooman's Lower Previsions there's a claim that "It is clear that" the natural extension $\underline{E}_\underline{P}$ of lower prevision $\underline{P}$ dominates (42) it, ...
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63 views

Computation of average number of hops for an atom reach a certain distance

I apologize if this has been asked before. I tried searching all previous posts to look at different forms of this problem (such as random 2d walks on a lattice, relation to Isling's work, etc.). I ...
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1answer
48 views

Binomial Probabilities

Firstly, I am confused as to how to calculate the following binomial probability: $P(3\leq X\leq5)$ when $n=7$ and $p=.6$. I think I used the wrong formula because I set it equal to $P(X\leq 5)-P(...
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143 views

Divergence based robust inference

The term 'divergence' means a function $D$ which takes two probability distributions $g,f$ as input and puts out a non-negative real number $D(g,f)$. I have learnt that the inference based on ...
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1answer
4k views

Odds of winning a two part drawing

There is a local drawing that involves being selected out of an estimated 6000 entries, and then correctly selecting 1 of 3 numbers in order to win. The numbers have are actually cards in a deck that ...
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2answers
2k views

What does saying that an estimator is robust mean?

In statistics, how can you tell whether an estimator is robust or not? I need to discuss whether the maximum likelihood estimators (MLE) of the normal distribution are robust or not. The MLE are $$\...
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2answers
1k views

How to tell if two samples come from the same probability distribution?

I have two distributions (generated from binned data) and wish to answer the question: Do they come from the same underlying distribution? I don't have the form of the underlying distribution. Edit: ...
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1answer
385 views

Transform t-stat into skewness-adjusted t-stat

I'm trying to calculate a one-sample skewness-adjusted t-stat (the null hypothesis is a mean of 1) as proposed by Johnson (1978): $$ J = t + \frac{gt^2}{3\sqrt{n}} + \frac{g}{6\sqrt{n}}, $$ $t$ is ...
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1answer
131 views

How to find Influence function?

Derive $IF(x;T,F)$ when $$\displaystyle T(F)=\int_{F^{-1}(\alpha)}^{F^{-1}(1-\alpha)}x ~dF(x)$$ Here $IF$ stands for Influence function. Trial: Here $$\begin{align}IF(x;T,F) &=\lim_{t\to 0}\...
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1answer
81 views

What are problems an $M$ - estimator is trying to solve?

Anyone here have any experience with $M$ - estimators and do you think you can give a brief explanation that the problem an $M$ - estimator is trying to solve ? Thanks.