Questions tagged [risk]

Risk has several meanings in different contexts within statistics

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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 ...
17
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0answers
2k 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 ...
12
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1answer
190 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 ...
11
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1answer
3k 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 ...
10
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1answer
241 views

Model fitting vs minimizing expected risk

I'm confused about the mechanics of model fitting vs minimizing risk in decision theory. There's numerous resources online, but I can't seem to find a straight answer regarding what I'm confused about....
10
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1answer
962 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
178 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
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5answers
6k 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 ...
7
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2answers
990 views

How to calculate 95% CI of vaccine with 90% efficacy?

A vaccine is reported in the news to have 90% efficacy. I'd like to know how much confidence there is in that efficacy measure. The protocol for this reports that a vaccine or placebo was administered ...
7
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1answer
3k 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
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1answer
2k views

Cross entropy vs KL divergence: What's minimized directly in practice?

My understanding is that in ML one can establish a connection between these quantities using the following line of reasoning: Assuming we plan to use ML to make decisions, we choose to minimize our ...
6
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3answers
3k 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
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2answers
990 views

How can I show that the average empirical risk is equal to the true risk for a binary classifier?

Suppose that $h \in \mathcal{H}$ is a hypothesis in some class of binary classifiers $\mathcal{H}$, $\mathcal{D}_n$ is a training dataset of size $n$, $\mathcal{L}$ is the loss function for the ...
6
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2answers
647 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
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1answer
138 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 ...
6
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3answers
818 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 ...
6
votes
2answers
575 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
162 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
494 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
147 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 ...
5
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1answer
204 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
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2answers
555 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
1k 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
188 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
72 views

How about evaluating an estimator using the VARIANCE of loss (instead of the expectation of loss)?

The risk of an estimator $\delta$ is defined as $$E_\theta[L(\theta,\delta(X))],$$ where, say, $L(\theta,\delta(X)) = (\theta-\delta(X))^2$, and $E_\theta(X)$ is defined as $\int XdP_\theta$, namely ...
5
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1answer
2k 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 ...
5
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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 ...
4
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2answers
1k 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
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1answer
887 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 ...
4
votes
2answers
126 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
2answers
163 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 ...
4
votes
1answer
123 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, ...
4
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0answers
36 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 ...
4
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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 ...
3
votes
2answers
738 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
3answers
401 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 ...
3
votes
1answer
2k 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
2answers
403 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
896 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 ...
3
votes
1answer
2k 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
567 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
1answer
90 views

How does probability of default evolve over time?

Say I have a probability of default of 0.02 (which is annual so over next year) for a certain client. Then say this client takes out a 180 day loan, how can I adjust my probability of default for this ...
3
votes
1answer
80 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) ...
3
votes
1answer
7k 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
1answer
41 views

Conditional correlation, copula, portfolio optimization and diversification

I have a data set which consists of > 500 hedge funds, their historical monthly returns, and their benchmark (index) monthly returns. The number of data points (# of monthly returns) differs from a ...
3
votes
0answers
423 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
505 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
3k 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
2answers
20 views

Combined relative risk in meta study

I am looking at a meta study, and I've never done one (or looked at it much) before. From what I've learned about it so far, the way you calculate the compounded effect is by first calculating this $\...
2
votes
1answer
79 views

What is the entropy of a riskless random variable?

Variance and standard deviation are often used as proxies for risk and volatility. I make the analogy to information theory as follows, correct if it's wrong: a random variable $x\in \mathbb{R}$ that ...