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In general, if $(S, \mathcal{S}, \nu)$ is a measure space, $\mu$ is another measure on $(S, \mathcal{S})$, and $f : S \to [0,\infty]$ is an $\mathcal{S}$-measurable function, we say that $f$ is the density of $\mu$ with respect to $\nu$ if, for every set $A \in \mathcal{S}$, we have $$\mu(A) = \int_A f\,d\nu.$$ (If such $f$ exists then it is unique up to $\...


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$\sigma$-finite is a particular kind of measure that is the countable union of measurable sets with finite measure. The other part of the density is about the Radon-Nikodym Theorem. One definition of a probability density function is as the Radon-Nikodym derivative of the induced measure with respect to a base measure, which is what is talked about in your ...


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The naive bootstrap method (where the original data is resampled with replacement) does indeed systematically underestimate the error of correlated data as discussed for example in Remark 2.1 on page 1192 in A. Singh, The Annals of Statistics 9, 1187-1195. I'm not at all an expert in this field, and came across the same problem as the original poster, also ...


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I will denote the function by $\Phi$ and make some remarks. 1) It is better to go for $u\in(0,1)$. This because $\Phi(u)\in\mathbb R$ for every $u\in(0,1)$ which is not necessarily true if $u=0$ or $u=1$. 2) It is not necessary to demand that $F$ is continuous. 3) Characteristic is the relation:$$F(x)\geq u\iff x\geq \Phi(u)$$Based on that relation it ...


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You are indeed right; what you exhibit there is the basis of many Monte Carlo methods. For instance, in importance sampling, the importance weights are used to compensate for the unknown normalizing constant. In particular, what you show there is exactly what is used in particle Markov chain Monte-Carlo methods to compute the normalization constant (the ...


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Fixing $a, b, c$, you should be evaluating points at u_x and u_y rather than x and y. Formula for u_x and u_y are unif(10000). Edit: func_to_int<-function(a,b,c,xval,yval){ exp(-a*(sqrt((xval-b)^2+(yval-c)^2))) } means_vec<-c() for(j in 1:5){ set.seed(j) u_y<-runif(10000) u_x<-runif(10000) sim_vec<-c() for(i in 1:length(u_x)){ ...


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