All Questions
1,933 questions
0
votes
0
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13
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Importance sampling with relative weight from two histograms
I have two datasets of real values, $X = (x_1, \dots, x_N)$ and $Y = (y_1, \dots, y_M)$. Here $Y$ is a subset of $X$. These data points can be regarded as samples from some unknown densities, $x\sim p(...
- 4,552
4
votes
2
answers
115
views
How to generate N-dimensional multivariate-normal sample from N-2 marginals [closed]
I am facing a problem with my "calculator" which uses samples generated through a N-dimensional multivariate normal. I've included below a code snippet to illustrate the issue.
From sample_1 ...
- 41
2
votes
2
answers
44
views
Mathematical Reference for Metropolis-Within-Gibbs Algorithm
Is there a MATHEMATICAL reference for the Metropolis-Within-Gibbs algorithm with proves the algorithm mathematically ? (Presumably, the reference shall use facts in Markov Chain Theory, the fact that ...
- 639
0
votes
0
answers
39
views
Probability that one expectation is larger than another expectation using MC estimate
Consider an expectation $a=\int f(\boldsymbol{x})p(\boldsymbol{x})d\boldsymbol{x}$ and an expectation $b=\int g(\boldsymbol{x})q(\boldsymbol{x})d\boldsymbol{x}$.
For a given small number of samples $f(...
- 51
0
votes
0
answers
14
views
Under which conditions does PCA consistently estimate latent factors in a Dynamic Factor Model?
Consider a dataset of N time series and T observation periods.
Assume each series $x_t$ is generated from a single (unobserved) common factor $f_t$ following this model:
$$
X_t = \Lambda f_t + \...
3
votes
0
answers
46
views
Is there an analytical solution to the distribution of a sum of observations drawn from a Frechet distribution?
Let $X_i$ be an iid draw from a Frechet distribution. Let $\alpha_i \in \mathbb{R}$.
Is there an analytical expression of the distribution of $\alpha_1X_1 + \alpha_2X_2 + \alpha_3X_3$? That is, can I ...
- 31
1
vote
0
answers
22
views
Convert units, get different results when fitting extreme value distribution with extRemes
I am using the fevd() and lr.test() functions to examine precipitation using the extRemes R ...
- 11
1
vote
0
answers
33
views
Importance sampling weights when sampling without replacement
Suppose we have a function $f(x_i)$ for a discrete variable $x_i$ (indexed 1 to N). We would like to calculate $\sum_{i=1}^N(f(x_i))$. Instead of the full sum, we sample some subset of size M from $...
- 663
0
votes
0
answers
21
views
What are the alternative summary measures of a maximum order statistic when the expectation of the underlying distribution is not finite?
Suppose, $n$ units are placed on a life test. The time-to-failure follows a continuous probability distribution with non-existing finite moments(like a lower-truncated Cauchy or inverse Lomax). Let, $...
- 305
1
vote
1
answer
78
views
Expectation of the minimum of random variables (Exponential + Erlang)
Consider the following random variable
$$
Z=\min_i\{X_i+Y_i\}
$$
for $-n\leq i\leq n$, where $X_i\overset{\mathrm{iid}}{\sim}\text{Exp}(\lambda)$, $Y_i\overset{\mathrm{iid}}{\sim}\text{Erlang}(|i|,\...
- 150
3
votes
1
answer
85
views
Maximum of two independent gamma variables
Let $X_1$, $X_2$ be two independent random variables with different gamma distributions, and $X = \max\{X_1, X_2\}$.
Are there known results for the distribution of $X$? Actually I only need to know $\...
- 1,191
0
votes
0
answers
25
views
Linearity of and pointwise equality in expectation of min() function
Consider the expressions $f = c + s*E[min(a/s, X)]$ and $g = E[min(c + a, c+sX)]$ where
c >= 0
0 < s <= 1
a >= 0
X ~ Poisson($\lambda$/s)
I'd like to think that $f = g$, reasoning as ...
0
votes
0
answers
20
views
Interpreting parameter distributions and 95% confidence intervals from Monte Carlo sampling
I have fit two datasets with a model (multiparametric biochemical network models) and these fits give estimates of many parameters, including one that I'll call A. The best-fit values for parameter A ...
- 1
1
vote
0
answers
18
views
Do better than P=1/N in Monte Carlo analysis?
There is a function (say a force) $F(\underline{x})$ that is dependent on a 3-dimensional vector $\underline{x}$ of (e.g.) wind speed, turbulence and wind shear. The probability distributions (pdfs) ...
Dick Veldkamp
0
votes
0
answers
29
views
Monte Carlo simulation with Importance Sampling - variance of estimator vs weighted variance
I am using Monte Carlo simulations associated with Importance Sampling and I have some difficulties interpretating the variance estimator:
Using a dummy example extracted from here, I use Monte Carlo ...
- 1
0
votes
0
answers
29
views
Why subsample correlations change if I partition the sample in tails and body even if the full sample distribution is a bivariate gaussian?
I generated a data sample through a Monte Carlo Simulation where the underlying distribution is a “Bivariate Normal” with a correlation coefficient of 0.5, a mean of 0% for both series, and a standard ...
- 77
0
votes
0
answers
18
views
Monte Carlo Sampling for Optimal Replacement Time - Confidence Intervals
Barlow et al. (1960) described a function for optimal replacement time (ORT) estimation
$T^*(t) = \frac{C_{P} \cdot S(t)+C_{C}\cdot (1-S(t))}{\int^t_0S(\tau)d\tau}$
where $C_{P}$ is the preventive ...
- 51
2
votes
1
answer
32
views
Verifying consistency of heteroscedasticity and autocorrelation robust SEs with a Monte Carlo Experiment
I'm trying to demonstrate the consistency of the default HAC standard errors given in R's sandwich package via a Monte Carlo experiment. I'm using a linear model ...
- 33
2
votes
1
answer
68
views
Generate Quasi Random Numbers for a Multivariate DIstribution
Algorithms like Sobol or Holton provide quasi random numbers (that is, the numbers "look" random in the sense of a uniform distribution, but they are deterministic) in the hypercube $[0,1]^d$...
- 312
1
vote
0
answers
37
views
Monte Carlo integration methods utilizing a set of representative points given by a black box [closed]
Consider the task of integrating a function with respect to a multimodal distribution. Suppose I am given a set of "black box samples" from the modes of the target distribution and no other ...
- 2,540
1
vote
0
answers
37
views
How can I measure Monte Carlo convergence in distribution with heavy tails?
I'm performing a Monte Carlo study on a simple agent based simulation, and I'm trying to formulate a heuristic for the number of MC samples to use. I'm able to measure convergence of statistics like ...
0
votes
0
answers
36
views
Fitting a regression line which passes through the anchor point
In our setting, we have data $X_1, \ldots, X_n$, which can be ordered as $X_{1,n}\leq X_{2,n}\leq \ldots \leq X_{n,n}$ and we have the points $(-\log (1-\frac{i}{n+1}), X_{i,n})$ for $i=1,\ldots,n$.
...
- 656
4
votes
2
answers
83
views
How can one estimate the average part size of a partition of a set with random sampling?
I am sure this is a simple question, but I have no experience with statistics. Say we have some set $X$ whose size $n = |X|$ is known. $X$ is divided into $j$ (disjoint) parts $p_i$, but we do not ...
- 93
1
vote
0
answers
48
views
Efficient Methods for Approximating High-Dimensional Integrals with Gaussian-Like Factors
I'm seeking a computationally efficient method to approximately evaluate high-dimensional integrals of the form:
$$\int f(\textbf{x}) \prod_i g_i(x_i) \, d\textbf{x}$$
where $f(\mathbf{x}) = (\mathbf{...
- 730
2
votes
1
answer
73
views
Monte Carlo simulations and Jensen’s inequality in cost-effectiveness analysis: point estimates vs. expected values of simulations
In cost-effectiveness analysis, we use a cost-effectiveness ratio: effect/cost. Because it is a ratio, calculating and representing uncertainty around it is not straightforward. We can use Monte Carlo ...
- 63
1
vote
1
answer
58
views
Identify maximum in quadratic regression
I am looking for a way to find the maximum in a quadratic regression.
Specifically, I have two variables X and Y. Y is a discrete and commonly used scale representing the severity of a disease, ...
- 433
0
votes
1
answer
48
views
Monte Carlo method for likelihoods ratio density estimation
I recently started reading Stephen Kay's Fundamentals of Statistical Signal Processing - Detection Theory (Volume II) and there is something I do not fully understand about likelihoods and hypothesis ...
- 103
5
votes
2
answers
346
views
What is the median of the minimum or maximum of multiple samples?
Suppose I have a variable with a known distribution, and suppose I sample that variable k times and record the minimum. If I repeat this many times, will the median of the minimum converge to a ...
- 55
3
votes
1
answer
60
views
Subtraction of Monte Carlo integrals - Catastrophic cancellation
I am attempting to estimate a quantity $Q$ which is given by the difference between two functions of Monte Carlo integrals over some set of points $\{x_i\}_{i=1}^N$, call the estimator $\hat{Q}$:
$$ \...
- 404
2
votes
1
answer
39
views
analytical asymptotic approximation of the expected maximum, mean, and minimum distance of nearest neighbours in unit ball
Say I uniformly at random distribute $x = n^3$ (independent identically distributed) points in a ball of radius $r=1$ in $\mathbb{R}^3$.
What can be said about the expected maximum, minimum, and mean ...
- 277
0
votes
0
answers
45
views
Bootstrap sampling to get monthly statistic from daily data
I have daily (iid) data for historic winter seasons: $d:$ (price, value, temperature, etc).
The "value" is actually a concave up function of "price" and the other covariates.
I'm ...
- 11
5
votes
1
answer
216
views
In a sum of high-variance lognormals, what fraction comes from the first term?
If $X_i \overset{\textrm{iid}}{\sim} \text{Lognormal}(0, \sigma^2)$ for $i=1,\ldots,n$ and $Y_1 = X_1 / \sum_{j=1}^n X_j$, then I would expect that a particular* limiting distribution of $Y_1$, ...
- 195
2
votes
2
answers
232
views
Is it possible for some p-values to be impossible? (because statistic generated by parametric bootstrap is mostly the same value.)
I am using a parametric bootstrap/monte carlo hypothesis testing method to generate the null distribution of the log likelihood ratio statistic. However, I am worried I might be doing it wrong ...
- 425
0
votes
0
answers
25
views
Unbiased test for homogeneity of means of exponenential samples
Given $K$ independent samples of $Y_{i1},\dots,Y_{in_i} \ \text{i.i.d.} \ \sim Exp(\lambda_i)$ with $i=1,\dots, K$ and $n_i$ the size of the $i$-th sample, is there any statistics with analytically ...
2
votes
1
answer
58
views
Forecasting time series using simulations
Suppose we have a stationary time series $x_{1}, x_{2}, ..., x_{T}$. Goal is to forecast up to $T+h$, i.e., forecast $x_{T+1}, x_{T+2}, ..., x_{T+h}$.
Forecasting methodology:
Using econometric ...
- 557
0
votes
0
answers
17
views
Declustering impact, stationarity and discretization
I have a seasonal time series, and I am considering declustering (before any other preprocessing steps) it using runs declustering. If I observe an extremal index of 1, can I claim that my data is i.i....
- 1
2
votes
0
answers
40
views
What makes a statistic valid for monte carlo simulation?
A while back I was reading Garland et. al. (1993) about studying whether two groups of animals, say herbivores and carnivores differ in their mean value for some trait, like the amount of territory ...
- 425
4
votes
1
answer
226
views
How to get standardized coefficients of Monte Carlo method for indirect effects in lavaan/semTools?
I've run a path analysis using semTools. I'm interested to test indirect effects. I would like to report the results based on Monte Carlo confidence interval. However, the code ...
- 171
7
votes
3
answers
803
views
P values non-significant but Monte Carlo confidence interval does not contain zero for indirect effects
I've run a path analysis using semTools. I'm interested to test indirect effects. The p values for all indirect effects were non-significant, but the Monte Carlo confidence interval for some of them ...
- 171
0
votes
0
answers
55
views
Does the mean of the maxima of a set of distributions converge?
This question is related to a recent one I posted. In that question I ask what statistic might best represent the central tendency of the true discrete distribution of a property for a sample for ...
- 165
3
votes
3
answers
125
views
What statistic best estimates the sample mean in case of missing data in a distribution?
I have samples of particles and am interested in the particle lengths. The problem is that the samples are assessed using image analysis. As the particles overlap, the measurements are incomplete and ...
- 165
2
votes
1
answer
109
views
Metropolis-Hastings algorithm doesn't converge to the global minimum
I calculated the total root mean squared error of 24 parameters that are estimated with metropolis hastings, I ran the algorithm for 100.000 iterations, and as the chain forward it reached a global ...
- 75
1
vote
0
answers
103
views
How to deal with outliers in panel data? [closed]
When we have cross-sectional data, we can easily detect and remove outliers. But how should one approach outliers when we are dealing with panel data? Since we have $i$ entities and $t$ times periods, ...
- 345
0
votes
0
answers
51
views
How to understand intuitively the CDF formula for the maximum statistic of three iid rv’s? [duplicate]
Given that all three iid rv’s ($X_1, X_2, X_3$) have CDF $F(x)$, the formula for the CDF $G(y)$ of the largest rv ($Y=X_i$) among the three is:
$G(y)=P(X_1 \leq y) \cdot P(X_2 \leq y) \cdot P(X_3 \leq ...
1
vote
1
answer
53
views
Derivation of a dynamical Generalized Pareto distribution
I'm currently reading a paper for my master thesis on the tail index estimation for asset returns using the peak over threshold method. In this paper the authors introduce the cumulative distribution ...
0
votes
1
answer
54
views
Are the p-values obtained on the same sample using synthetic AA tests (Monte Carlo) independent values?
Let's say we have the following procedure.
We take a fixed sample of size n and perform the procedure 1000 time:
we divide (split) it equally into 2 groups;
we calculate p value using the F function (...
3
votes
1
answer
40
views
Understanding how to evaluate the integral causal-effect expression
I have this expression
$$
p( Y \mid \text{do}(Z=z)) =
\int_{B, S, W, X} dBdSdWdX \ \ P(B | S) P(W | B, S) P(X | B, S, Z=z) \left[ \int_{Z'} dZ' P(Z'| B,S,W) P(Y | B, S, W, X, Z') P(S) \right]
$$
...
- 989
3
votes
1
answer
106
views
Convolution with a pathological distribution
Problem definition
Consider the following random bivariate vector
\begin{equation}
\begin{aligned}
y&=z+v \\
z&\sim p_z(z;c) \\
v&\sim p_v(v)
\end{aligned}
\end{equation}
where $p_z$ ...
- 473
0
votes
0
answers
14
views
Probability of chain of events over a finite set of event with no repetition
I'm trying to tackle a problem that I suspect resembles others I'm unfamiliar with. I would love pointers for further reading. The problem is as follows:
We have a finite set of actions we can take $\...
- 1
1
vote
2
answers
149
views
Distribution of a random variable conditional on its being a maximum or not
Consider the random variables $\epsilon_1,\dots, \epsilon_D$ defined on the probability space $(\Omega, \mathcal{F}, P)$. Assume they are continuous. Let
$$
Y=\sum_{d=1}^D d\times \mathbb{1}\{\...
- 935