Mean Square Error Monte Carlo
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Monte Carlo Simulation Regression
it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Mean Square Error of Monte Carlo up vote 2 down vote favorite Trying to develop the expression for the Mean Square Error (MSE) of Monte Carlo, I found myself monte carlo simulation in r code a bit lost when going through a simple proof in the literature. I am working in the context of mathematical finance, where the aim is to find an approximation to the "true value" of the function $V$. Let's define $V$ as $V = \mathbb E[f]$, whose discrete approximation is $\hat{V} = \mathbb E[\hat{f}]$ and its Monte Carlo estimate is formulated as $$ \hat{Y} = \frac{1}{N}\sum_{n=1}^{N}\hat{f}^{\,(n)} $$ The expression for the MSE in the aforementioned proof is obtained by doing the following: \begin{eqnarray*} \mathbb E\left[\left(\hat{Y} - V \right)^2 \right]&=&\mathbb E\left[\left(\hat{Y} - \mathbb E\left[\,\hat{f}\,\right] + \mathbb E\left[\,\hat{f}\,\right] - \mathbb E\left[\,f\,\right] \right)^2 \right] = \\ &=& \mathbb E[(\hat{Y} - \mathbb E\left[\,\hat{f}\,\right])^2] + \left(\mathbb E[\hat{f}] - \mathbb E[f] \right)^2 + \mathbb E\left[2\left(\hat{Y} - \mathbb E [\,\hat{f}]\right)\left(\mathbb E\left[\,\hat{f}\,\right] - \mathbb E\left[\,f\,\right]\right) \right] = \\ &=& \frac{1}{N}\mathrm{Var}(\,\hat{f}\,) + \left(\mathbb E[\hat{f}] - \mathbb E[f] \right)^2 \end{eqnarray*} It is the last line that flusters me. Particularly, my concerns are: I am assuming that the third element of the second line on the right side tends to $0$ since $\mathbb E [
which is a common measure of estimator quality, of the fitted values of a dependent variable. In the Bayesian setting, the term MMSE more specifically refers to monte carlo variance estimation with quadratic cost function. In such case, the MMSE estimator is
Monte Carlo Simulation Sample Size
given by the posterior mean of the parameter to be estimated. Since the posterior mean is cumbersome to calculate,
Simulation Study
the form of the MMSE estimator is usually constrained to be within a certain class of functions. Linear MMSE estimators are a popular choice since they are easy to use, calculate, http://math.stackexchange.com/questions/1330833/mean-square-error-of-monte-carlo and very versatile. It has given rise to many popular estimators such as the Wiener-Kolmogorov filter and Kalman filter. Contents 1 Motivation 2 Definition 3 Properties 4 Linear MMSE estimator 4.1 Computation 5 Linear MMSE estimator for linear observation process 5.1 Alternative form 6 Sequential linear MMSE estimation 6.1 Special Case: Scalar Observations 7 Examples 7.1 Example 1 7.2 Example 2 7.3 https://en.wikipedia.org/wiki/Minimum_mean_square_error Example 3 7.4 Example 4 8 See also 9 Notes 10 Further reading Motivation[edit] The term MMSE more specifically refers to estimation in a Bayesian setting with quadratic cost function. The basic idea behind the Bayesian approach to estimation stems from practical situations where we often have some prior information about the parameter to be estimated. For instance, we may have prior information about the range that the parameter can assume; or we may have an old estimate of the parameter that we want to modify when a new observation is made available; or the statistics of an actual random signal such as speech. This is in contrast to the non-Bayesian approach like minimum-variance unbiased estimator (MVUE) where absolutely nothing is assumed to be known about the parameter in advance and which does not account for such situations. In the Bayesian approach, such prior information is captured by the prior probability density function of the parameters; and based directly on Bayes theorem, it allows us to make better posterior estimates as more observations become available. Thus unlike non-Bayesian approach where parameters of interest are
Contents Simulating Random Variables in R The MSE of an http://www.stat.umn.edu/geyer/old03/5102/examp/sim.html Estimator The Monte Carlo Error of a Monte Carlo Estimator Theory Practice Importance Sampling Theory Practice Simulating Random Variables in R R has many functions for simulating random variables with specified distributions. We will just use these without explaining how they work. These functions are very sophisticated and complicated in the algorithms they use. monte carlo Anything simple enough to explain is too simple to use. So we will just take these functions as given. .Random.seed Random Number Generation RNG Random Number Generation RNGkind Random Number Generation set.seed Random Number Generation rbeta The Beta Distribution rbinom The Binomial Distribution rcauchy The Cauchy Distribution rchisq The (non-central) Chi-Squared Distribution rexp The monte carlo simulation Exponential Distribution rf The F Distribution rgamma The Gamma Distribution rgeom The Geometric Distribution rhyper The Hypergeometric Distribution rlnorm The Log Normal Distribution rlogis The Logistic Distribution rnbinom The Negative Binomial Distribution rnorm The Normal Distribution rpois The Poisson Distribution rsignrank Distribution of the Wilcoxon Signed Rank Statistic rt The Student t Distribution runif The Uniform Distribution rweibull The Weibull Distribution rwilcox Distribution of the Wilcoxon Rank Sum Statistic mvrnorm Simulate from a Multivariate Normal Distribution sample Random Samples and Permutations Just a few comments. The first four items in the table (the R object .Random.seed and the R functions RNG, RNGkind, and set.seed) are used for control of the random number generator system as a whole. The only ones of these we will use are .Random.seed and set.seed which tell the random number generator system the place to start and guarantee repeatability. For example, set.seed(1234) produces a valid state of the random number generator, and every time set.seed(1234) the r
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