Questions tagged [risk]

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56
votes
9answers
20k views

Is it wrong to rephrase “1 in 80 deaths is caused by a car accident” as “1 in 80 people die as a result of a car accident?”

Statement One (S1): "One in 80 deaths is caused by a car accident." Statement Two (S2): "One in 80 people dies as a result of a car accident." Now, I personally don't see very much difference at all ...
12
votes
1answer
166 views

Example Of Strict von Neumann Inequality

Let $r(\pi, \delta)$ denote the Bayes risk of an estimator $\delta$ with respect to a prior $\pi$, let $\Pi$ denote the set of all priors on the parameter space $\Theta$, and let $\Delta$ denote the ...
12
votes
0answers
1k views

Implementation of CoVaR (a systemic risk measure) in R

I'm trying to estimate CoVaR using bivariate DCC GARCH in R. The concept of CoVaR is the dependence adjusted of VaR, which was first introduced by Adrian and Brunnermeier (2011). However, this ...
10
votes
1answer
732 views

How does an estimator that minimizes a weighted sum of squared bias and variance fit into decision theory?

Okay--my original message failed to elicit a response; so, let me put the question a differently. I will start by explaining my understanding of estimation from a decision theoretic perspective. I ...
7
votes
2answers
171 views

Is there a word for the phenomenon that the old are generally less affected by risk factors?

In epidemiology, this occurs often: Old people are less prone to the influence of risk factors. For example, the Framingham risk score, which tries to estimate cardiovascular risk, gives 8 or 9 points ...
7
votes
1answer
2k views

Computing VaR with AR-GARCH

I have the following AR(1)-GARCH(1,1) model for the daily returns $r_t$ $$r_t=\theta r_{t-1}+u_t\;\;\;u_t=\sigma_t\epsilon_t\;\;\;\sigma_t^2=\omega+\alpha u_{t-1}^2+\beta \sigma_{t-1}^2 $$ where $-1&...
6
votes
3answers
2k views

Why is empirical risk minimization prone to overfitting?

According to Chapter 8 of the book Deep Learning, "..empirical risk minimization is prone to overfitting. models with high capacity can simply memorize the training se." My question why is it so? ...
6
votes
1answer
1k views

Different definitions of Bayes risk

I'm having trouble understanding the proper definition of Bayes risk. Let the data/variate $x \sim P(X|\theta)$, $\theta\in \Theta$, $\pi$ be a distribution on $\Theta$ (prior), $\hat \theta(x)$ be ...
6
votes
5answers
5k views

Case-mix adjustment versus risk adjustment, what are their differences in practice and objective?

I have encountered in swathes of medical literature the use of the terms "case-mix" and "risk" adjustment without any citations or explanations of their exact usage and motivation from a modeling ...
6
votes
2answers
278 views

Why is the risk function defined to be the expectation of loss function?

In decision theory, we define the risk associated with a particular predictor function as the expected value of the loss function. Since the input and output are considered random variables therefore ...
6
votes
2answers
484 views

Credit Risk and Concentration

I am working with a UK credit-union and we are looking to build a model to assess our credit risk and changes to this over time. We have a number of loans to borrowers who each have a credit rating (...
5
votes
1answer
127 views

Comparing estimators of equal risk

I'm attending a course in mathematical statistics and it seems the lecturer tacitly assumes that given estimators $T_1,T_2 : \Omega \to \Lambda$ of a parameter $g : \Theta \to \Lambda$, a loss ...
5
votes
1answer
351 views

Intuitive interpretation of Bayes risk $R(\delta, \lambda) = \int_{\Omega}R(\theta, \delta) \lambda(\theta) d\theta$

Consider the risk function R of an estimator (statistic) $\delta(X)$ trying to estimate parameter $\theta$: $$R(\theta, \delta) = E_{X \sim P_{\theta}}[Loss(\theta,\delta(X)]$$ Which can be ...
5
votes
1answer
120 views

Mapping Frequentist Risk Notation to Regression

The frequentist risk in literature is defined as follows: $R(\theta, \delta) = E_{X|\theta} L(\theta,\delta(x)) = \int L(\theta,\delta(x)) p(x|\theta) dx$ This risk is focused on the quality of the ...
5
votes
2answers
460 views

Binary Classification : Prove that $\mathbb{E}_{\mathcal{D}_n}\left[R_e(h)\right] = R(h)$

this is my first question here :) Problem Statement Let $h \in \mathcal{H}$ be a hypothesis to some class of binary classifiers $\mathcal{H}$. Show that $$\mathbb{E}_{\mathcal{D}_n}\left[R_e(h)\...
5
votes
1answer
159 views

Inconsistent Empirical Risk Minimization procedure, but why?

Given a random variable $Y$ and the typical squared loss function: $$L(Y,\hat{Y}) = (Y-\hat{Y})^2$$ the minimizer for expected loss $E[L(Y,\hat{Y})]$ is know to be the mean, $\hat{Y} = E[Y] = \mu$. ...
5
votes
1answer
401 views

The Kelly Criterion — Where Did I Go Wrong?

Suppose the following simple/basic investment scenario: I have $100$USD in my bank account as a starting point (will increase/decrease as I invest). There are $1,000$ different investments that I'm ...
5
votes
1answer
967 views

VaR in case of ARMA-GARCH?

How do I calculate the VaR in case of using an ARMA-GARCH approach? I am not good at time series, so I am more or less confused with the different possible notations of an ARMA-GARCH process. I hope ...
5
votes
1answer
187 views

How can I estimate the likelihood of being in a traffic accident on the US101 by day of week and time of day?

My plan is this: Find source of data about when and where accidents occurred on the US 101 Find a source of data about traffic volume on the same road Subset the data to include only accidents that ...
5
votes
1answer
950 views

Question about computing Bayes Error - with or without loss function?

I am new to Bayesian Decision Theory and don't understand the following concept: So from what I understood, the Bayes error is used to report the performance of a Bayes classifier in terms of the ...
4
votes
2answers
872 views

why do we calculate risk when we already have loss functions?

If we already have let's say mean squared error as a loss function which can tell how good our algorithm is, then why we calculate the expectation of loss function as Risk? Apologies, if this a naive ...
4
votes
1answer
128 views

Does an optimal linear classifier perform no better then chance iff class distributions have the same mean?

Assuming there is some nice $P(x, y)$ over $\mathbb R^n \times \{0, 1\}$, can we claim that: The expected accuracy of the optimal linear classifier trained on a large sample from $P(x, y)$ would not ...
4
votes
2answers
114 views

A good resource to learn about the intuition and optimization models behind Markovitz modern portfolio theorem

I would like to learn about the Markovitz theory because I am going to interview at a financial firm that uses variants of it extensively. I am looking for a resource to learn about the intuition ...
4
votes
3answers
754 views

Yearly Aggregated Loss Distribution (operational risk)

Firstly I should mention I am quite unfamiliar with the subject (operational risk). And I am also beginner in risk management. It is also worth to mention that this task is on academic level . I ...
4
votes
0answers
1k views

Copula-based Value-at-risk in R

I'm working on a value-at-risk calculation using copulas on different stock market indices. I know how to fit the copula, but I can't figure out how to apply the VaR approach in the next step. The ...
4
votes
0answers
1k views

How does the RMS package's nomogram calculate points for continuous variables?

I have been reading a number of papers where researchers have created risk scores based on logistic regression models. Often they refer to "Sullivan's method" but I have no access to this paper and ...
3
votes
2answers
313 views

Calculating the risk of an estimator using zero-one loss

Consider two observations where $$P_\theta(x=\theta+1)=P_\theta(x=\theta-1)=0.5,\ \ \theta\in\mathbb{R}$$Let $\mathbb{D}=\Theta=\mathbb{R}$ the decision space. Suppose that the associated loss ...
3
votes
1answer
1k views

Bayes risk of Normal-Normal model

Consider $x\sim N(\theta,1)$ and $\theta\sim N(0,n)$. Show that the Bayes risk is equal to $\frac{n}{n+1}$. I know that $$r(\theta,\delta)=\int_\chi\int_\Theta L(\theta,\delta(x))\pi(\theta|x)d\...
3
votes
1answer
622 views

Explaining Odds Ratio and Relative Risk to the statistically challenged

I'm peer-reviewing a manuscript for a psychology journal in which I believe the authors have mixed up odds-ratio and risk-ratio. They are being so stubborn in their insistence that they have not mixed ...
3
votes
2answers
215 views

Interpretation of standard deviation if data is not normally distributed

This is very basic question. But I want to know how one can interpret the standard deviation if data is not normally distributed. My concern is regarding financial market. Investors generally use ...
3
votes
1answer
101 views

Convergence rate: $E\|\hat f - f\|^2 = O(\psi_n)$ vs $\|\hat f - f\| = O_p(\psi_n^{1/2})$

I have seen two types of results on convergence rates for some estimator $\hat f$: $E\|\hat f - f\|^2 = O(\psi_n)$ and $\|\hat f - f\| = O_p(\sqrt{\psi_n})$. The first result seems to be stronger, ...
3
votes
1answer
1k views

How to estimate the probability that the mean of an unknown distribution is over a threshold given small sample size

I am trying to quantify my concerns regarding a proposed incinerator in our community. The company is basing its potential to emit dioxins (a class of chlorinated organic compounds with a reference ...
3
votes
1answer
558 views

Where did this risk exposure 'estimation-formula' come from?

I was reading a book and the authors metioned that risk exposure can be estimated scientifically using this forumula: $risk(\$) = \frac{(a + 4m + b)}{6}$ and standard deviation $\sigma = \frac{b-a}{...
3
votes
2answers
133 views

I'm looking for a risk analysis book with citations from academic backgrounds that focuses on shocks and the unknown. Looking for a Taleb alternative [closed]

I'm less than ... enthusiastic about some of Taleb's claims regarding, say, the paleo diet. I make no comment on his political content, but I want someone more factual and hard, and less prone to ...
3
votes
1answer
4k views

Empirical Risk Minimization: empirical vs expected and true vs surrogate

In Tie-Yan Liu's book, he says that in a statistical learning theory for empirical risk minimization has to observe four risk functions: We also need to define the true loss of the learning problem, ...
3
votes
0answers
30 views

Bounds for the expected value of the Kolmogorov-Smirnoff loss function

Let $$ \mathcal{F}=\{F:\mathbb{R}\longrightarrow\mathbb{R}: \text{$F$ is the CDF of some probability measure on $\mathbb{R}$}\}. $$ Consider the loss function, $L:\mathcal F\times\mathcal F\to\mathbb ...
3
votes
0answers
384 views

Predict probability of rare event

Let's say I have a dataset about passages of cars on a road. The dataset contains information about time, driver, car, weather, and most importantly whether the car was involved in an accident. Of ...
3
votes
0answers
497 views

How to find a conditional probability using copula-based Markov process?

I have a monthly time series of a water quality parameter. I used copula-based Markov process of C(Y(t), Y(t-1) and I forecasted the mean behavior of Yt by following equation: Now, I need to find ...
3
votes
0answers
2k views

How can I convert annual standard deviation to a longer period?

Quicken provides annual standard deviation of returns for a given portfolio using analysis done by the Newport Group. I'd like to convert this to a longer term number--say 10, 20, or 30 years. ...
2
votes
1answer
205 views

Upper bound using Bayes risk

Bayes' risk is $L^*=0$ for a classification problem. $g_n(x)$ is a classification rule (plug-in) such that $g_n=0$ is $\eta_n(x)\leq 1/2$ and $g_n=1$ otherwise. The function $\eta$ is given by $\eta(x)...
2
votes
3answers
334 views

Bayesian Risk and Subjectivity

I am studying the differences in bayesian and frequentist approaches to point estimation. I understand that there are objective and subjective approaches to Bayesian and some people don't like the ...
2
votes
1answer
333 views

Risk and posterior expectation Bayesian Statistics

Consider $x\sim B(n,\theta)$ with $n$ known a)If $\pi(\theta)\sim Beta(\sqrt{n}/2,\sqrt{n}/2)$ give the associated posterior distribution and posterior expectation $\delta^\pi(x)$ b)Show ...
2
votes
1answer
105 views

Is this book excerpt an accurate statistical analysis of project risk?

I found this passage on the odds of a project succeeding in a book on risk management. I know this is open-ended, but is the author's math and logical reasoning correct here?
2
votes
1answer
431 views

Can you reduce the risk involved in an uncertain event?

I'm not sure if this is the right Stack Exchange site but I felt it came closest. Based on Knights 1971 definition of risk uncertainty is defined as a situation where factors exogenous to the ...
2
votes
1answer
786 views

Get distribution for aggregate loss using Monte Carlo

I am given two data sets containing dates and losses (in some currency). Given a distribution for the amount of losses and an (a,b,0) distribution for frequency of losses, how can I use Monte Carlo ...
2
votes
1answer
70 views

Convert classifier output for disease to probability using Bayes

Method 1 I am given a classifier for some disease that takes as input patient characteristics and has some sensitivity and specificity. Hence the classifier is a function c(patient characteristics) ...
2
votes
1answer
441 views

Recalibration by regressing on intercept only with offset

Consider a risk prediction model developed on one population. When transporting this model to another population, the calibration can be off. There are several strategies for recalibration. One is to ...
2
votes
2answers
755 views

Structural risk minimization and SVMs

I know what is SRM but I didn't understand the relation between SRM and SVMs. Can anyone explain me this? Why they say that SVMs rely on a SRM approach? Thank you so much!
2
votes
1answer
218 views

Question on Portfolio Risk & Covariance

Good evening, I was reading a stats textbook where I came across the following statement: In order to reduce the portfolio risk of a portfolio involving two risky investments, we should choose ...
2
votes
1answer
62 views

Relating lifetime risk to prevalence of a disease

I am somewhat confused about lifetime risk. The National Kidney Federation released data saying that the lifetime risk of a white American male needing dialysis was around 3%. However, considerably ...