All Questions
1,933 questions
1
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When Monte Carlo simulation can't be used to simulate a statistical system?
My question is simple. Which are the general conditions for which a Monte Carlo simulation can be used to represent a statistical system? Or conversely, which are the statistical system that cannot be ...
- 2,098
4
votes
2
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6k
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Minimizing relative error (or mean square error) and maximizing likelihood
I'm not a statistician, so I would appreciate an answer in the simplest possible words.
I've read that, in some sense, when we minimize the mean square error, we are maximizing the likelihood.
This ...
- 83
7
votes
1
answer
768
views
Hamiltonian Monte Carlo: why is reparameterizing needed?
In the Stan user's manual (Version 2.0.1, page 157), it says
A hierarchical model such as the above will suffer from the same kind of inefficiencies... [for a Hamiltonian Monte Carlo method] ...
- 11.8k
5
votes
3
answers
309
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Estimating the minimum of a finite probability mass function
Suppose we are given a discrete r.v. $X$, distributed according to some unknown, finite probability mass function $p(x)$. We can assume that $p(x)>0$ for every $x$ in its domain. We can sample $X$ ...
- 51
1
vote
1
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2k
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Accept-reject algorithm for Beta(1,$\beta$)
Consider the pdf
$$f(x)= \begin{cases} \beta x^{\beta -1 }\quad 0<x<1 \\ 0\quad \text{elsewhere} \end{cases} $$
for $\beta >1 $
Use the accept-reject algorithm to generate an observation ...
- 21.1k
1
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0
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165
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Find trendline for minimum (not mean) values in distribution
I would like to perform something like a linear regression on my distribution of data, but I'm interested in a trendline that estimates the minimum, not mean, value for each time bin. I'd like to do ...
- 125
9
votes
1
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9k
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Expected value of minimum order statistic from a normal sample
UPDATE Jan 25th 2014: the mistake is now corrected. Please ignore the calculated values of the Expected Value in the image uploaded - they are wrong- I don't delete the image because it has generated ...
- 60.8k
3
votes
1
answer
870
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$\mathbb{E}$ and Variance of the maximum of independent $\mathcal{N}(\mu_i, \sigma_i^2)$
I am interested in the expectation and the variance of the maximum of several independent, normal distributed variances. That is, given a set of $I$ different RVs with $X_i \sim \mathcal{N}(\mu_i, \...
- 14k
2
votes
1
answer
219
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Distribution of logically constrained parameters in Monte Carlo simulation
Papers like Briggs et al. 2002 say that logical constraints on inputs such as probability parameters exclude the the Normal distribution from consideration due to its unboundedness. In this example, ...
- 431
5
votes
1
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169
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Distribution of Extreme Spread for n, sigma
Simple form provided by WHuber: What is the distribution of the diameter of n points in the plane drawn iid from a bivariate Normal distribution? (Diameter is the greatest distance among any pair of ...
- 1,176
1
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0
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79
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determining probability of spatial pattern metric greater than some instance
Let's say I have a grid of cells from a satellite image that take on the value of 0 or 1, with 1 being forest.
There are many spatial pattern metrics that measure things about this pattern such as ...
- 371
2
votes
0
answers
589
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Convergence in Probability of the minimum
This is a homework question. I think I have the correct answer, but I am not sure. Also, the wording sounds very awkward. Is there a better way to show this (or better way to word this)?
Let $X_1,\...
- 10.2k
2
votes
1
answer
418
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Rate of convergence for SLLN
I am interested in writing a non-asymptotic rate of convergence for SLLN as a function of number of samples.
From the literature I've read so far, CLT provides an asymptotic convergence rate of $(1/...
- 55
1
vote
2
answers
779
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Joint cdf of extreme values
A die is rolled twice,
$X_1$ : the minimum value to appear in the two rolls
$X_2$ : the maximum
I would like to derive $\ F_{X_1,X_2}(x_1,x_2)$.
I know that that the CDF of $\ X_1 $ = $\ 1- [1-...
- 13
4
votes
1
answer
1k
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Convergence in probability of minimum
This is a homework problem. Suppose we have a random sample $X_1,\ldots,X_n \overset{iid}{\sim} F$ with density $f(x) = 2(x-\theta)$ for $x\in (\theta,\theta+1)$. Let $X_{(1)} = \min{\{X_1,\ldots,X_n\}...
- 10.2k
8
votes
1
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227
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What method is simulating pvalues from re sampling from the data
A while back I asked a question about correlating times between time stamps and received a response from Peter Ellis that said I could calculate mean distances between codes...
This already will give ...
- 670
20
votes
1
answer
12k
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MCMC on a bounded parameter space?
I am trying to apply MCMC on a problem, but my priors(in my case they are $\alpha\in[0,1],\beta\in[0,1]$)) are restricted to an area? Can I use normal MCMC and ignore the samples that fall outside of ...
- 1,615
2
votes
3
answers
190
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MCMC for an explicitly uncomputable prior?
I am trying to sample from a posterior distribution and I only have an explicit formula for likelihood but I can sample from the prior distribution. How can I sample from the posterior distribution ...
- 1,615
9
votes
3
answers
2k
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Extreme value theory for count data
I am aware of extreme value theory for continuous distributions. I need to fit an extreme value distribution to the maximum observation of number of events on a day, per month. This seems to be the ...
- 205
1
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0
answers
55
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Approximately sampling $(X, Y)$ when sampling $X$ is easy
Suppose I am interested in sampling many pairs $(\mathbf X, Y)$ from some distribution $f(\mathbf x, y)$ where $\mathbf x \in \mathbb R^p$, $p$ large ; I am interested in both exact and approximate ...
- 9,102
5
votes
2
answers
2k
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Expected maximum given population size, mean, and variance
How would one estimate the maximum given population size, a few moments, and perhaps some additional assumption on the distribution?
Something like "I'm going to do $N_s≫1$ measurements out of ...
- 205
8
votes
2
answers
309
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Generating causally dependent random variables
I'm trying to generate sets of causally connected random variables and started off doing this with a monte carlo approach.
The baseline is a 2-dimensional measured histogram from which I draw random ...
- 131
5
votes
1
answer
14k
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Implementing a Metropolis Hastings Algorithm in R
Consider a univariate normal model with mean $µ$ and variance $τ$ . Suppose we use a Beta(2,2) prior for $µ$ (somehow we know µ is between zero and one) and a $log-normal(1,10)$ prior for $τ$ (recall ...
- 103
0
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0
answers
169
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Error Bars in a Monte Carlo Coin Experiment
An excel program runs a Monte Carlo coin experiment. 10,000 coins are tossed 1000 times, with the number of times ...
9
votes
1
answer
5k
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Monte Carlo estimation of probabilities
I would appreciate some advice on how to use Monte Carlo for estimating probabilities.
Generally speaking the problem I have involves running an experiment and counting the frequency of output (which ...
- 359
3
votes
2
answers
3k
views
Improved Monte-Carlo method vs. hit-and-miss method
I do not understand Which is more accurate, the hit-and-miss method or the improved Monte-Carlo method?
Here it is written that that the hit-and-miss has a higher variance but they showed ...
- 253
5
votes
1
answer
733
views
OLS robust to outliers
I am facing the following problem: I have a training sample and estimate a model on that training sample. My model is simply OLS: $y_t = a + \beta x_t + \varepsilon_t$. The model is estimated on ...
- 853
1
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1
answer
1k
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What is loc parameter in GPD distribution in POT package for R?
I fitted the Generalized Pareto distribution (GPD) using the POT package in R.
The fitted object provides shape and scale ...
- 131
8
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2
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1k
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Are there any alternatives to simulation for determining the distribution of number of events from two dependent non-homogeneous Poisson processes?
A "state of the art" model for the distribution of goals scored in a soccer match is that of Dixon
and Robinson (1998) "A Birth Process Model for Association Football Matches" which accounts for two ...
- 2,615
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1
answer
1k
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Simulating Monte Carlo with different standard deviations and interval confidence
I have a question regarding Monte Carlo simulation (direct simulation), applied to propagation of uncertainties.
From what I understand Monte Carlo accepts random numbers of each input variable of ...
- 79
0
votes
1
answer
609
views
random sampling in a polygon
I would like to sample a uniformly random point in a polygon...
If sample a large number they'd be equally likely to fall into two regions if they have the same area.
This would be quite simple if ...
- 111
7
votes
1
answer
2k
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Two player dice game probability
$A$ and $B$ play a dice game where a player wins if their score is higher. $B$ wins if their score is equal. What is the probability of $A$ winning if both the players roll their dice $n$ times and ...
- 73
7
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0
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764
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Endogeneity in spatially lagged regression model
The standard convention in Spatial Statistic is that the spatial lag term in a regression model will be biased due to simultaneity. Looking at the following model, it would be difficult to argue with ...
- 71
1
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0
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447
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Are direct Monte Carlo and Markov Chain Monte Carlo equivalent?
I'm not quite sure where to start... I'm trying to wrap my head around two methods for drawing samples from a parameter distribution, given a forward model and some distribution of the observable. I'...
- 215
2
votes
0
answers
310
views
How to include GAM response error in a Monte Carlo simulation
I am running a Monte Carlo simulation, using the results of a GAM (response) as the basis for my overall model. I would like to incorporate the error in the GAM into the final result. Since predict....
- 21
-1
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2
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252
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A question on Monte Carlo
We simulate $E(\exp(z))$ by Monte Carlo method,where $z\sim{}N(0,1)$. For sample sizes $2^{16}$ and $2^{17}$, the variance errors are 0.00531 and 0.00364, respectively. The ratio of these two errors ...
- 1
1
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0
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2k
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Estimating p-value of a very rare event using Monte Carlo
I am doing a Monte Carlo sampling from the null hypothesis in order to estimate a p-value of my data. Normally I would do, say, 1000 Monte Carlo runs, count the number of runs where I get values of my ...
- 107k
12
votes
1
answer
1k
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HMC: How many dimensions is too many?
From what I have read Hamiltonian Monte Carlo is the "goto" MCMC method when your problem is high dimensional.
Practically speaking, how many dimensions 10's, 100'...
- 961
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votes
1
answer
3k
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How can I sample from a correlated multivariate Bernoulli distribution with known covariances?
I have $N$ Bernoulli random variables $X_1, ..., X_{N}$ with known parameters $p_1, ..., p_{N}$. They are dependent with known covariances.
How can I sample from the joint distribution of the $X_i$?
...
- 397
2
votes
1
answer
2k
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Convergence in distribution of the maximum of a sequence of random variables
How to find the following:
Let $X_1$, $X_2$, $X_3$,..., $X_n$, be i.i.d with chi-square distribution with one-degree of freedom. Find $a_n$ and $b_n$ such that $ a_n(\max_i X_i - b_n)$ converges in ...
- 535
2
votes
1
answer
2k
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Monte Carlo simulation of low probability events with impact
I wonder if I'm lacking the terminology to phrase this question: I would like to model, in Excel, the risk of a fairly rare event, E, happening over time, where time is divided into chunks called ...
- 23
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2
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14k
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What is the difference between the Monte Carlo (MC) and Monte Carlo Markov Chain (MCMC) method?
The goal of both methods seems to be to derive an estimate of a posterior/target distribution. If a process model exists which links some input parameters (which are themselves uncertain and can be ...
- 133
2
votes
1
answer
539
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Sampling from marginal using integrated conditional
I would like to sample from:
$$
p(\theta_2|x)=\int p(\theta_2|\theta_1,x) . p(\theta_1|x) . d\theta_1
$$
knowing that I can easily sample from $p(\theta_1|x)$ and (less easily) from $p(\theta_2|\...
- 5,093
6
votes
1
answer
4k
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What is the current 'standard' for modern statistical computing hardware?
I am in the market for a new system (probably a laptop) that would be be used primarily for Bayesian/MCMC analyses. If I had unlimited funds I would obviously buy very high end hardware and be done ...
user25658
3
votes
1
answer
84
views
How can posterior be persisted and reconstituted as future prior?
Suppose I model a data generating process as a hierarchal model and have made some training observation from the process. To learn about the process, with the observations I run the bayesian ...
- 93
11
votes
1
answer
18k
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Rules to apply Monte Carlo simulation of p-values for chi-squared test
I'd like to understand the use of Monte Carlo simulation in the chisq.test() function in R.
I have a qualitative variable which has 128 levels / classes. My sample ...
- 351
10
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1
answer
667
views
Monte Carlo Integration for non-square integrable functions
I hope this is the right place to ask, if not feel free to move it to a more appropriate forum.
I've been wondering for quite a while now how to treat non-square integrable functions with Monte Carlo ...
- 203
2
votes
0
answers
235
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left-censored dependent variables and prediction
I'm coding up a monte-carlo analysis; I've got a deterministic model that depends on parameters that are uncertain. One of those uncertain parameters is a partially-observed vector of prices by ...
- 13.7k
5
votes
3
answers
1k
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Approximation of partition function (normalizer)
Say we have a nasty probability distribution like,
$$
P(x) = \frac{P^*(x)}{Z}
$$
where we can easily compute $P^*(x)$ for a given $x$ but not $P(x)$ because partition function $Z$ is expensive to ...
- 111
1
vote
2
answers
76
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Non-parametric tests to compare numeric profiles
I am trying to compare two different numerical profiles and determine whether they are the same or not, computing a p-value.
These profiles are composed of 200 values each (each of which corresponds ...
- 135