exestudiary

1. IntroductionBootstrap methods are a class of nonparametric Monte Carlo methods that estimate the distribution of a population by resampling.Resampling methods treat an observed sample as a finite population, and random samples are generated (resampled) from it to estimate population characteristics and make inferences about the sampled population. 2. Understanding by Example Let $X_1, X_2, \d..

Gibbs samplerThe Gibbs sampler is another special case of the Metropolis-Hastings sampler.The Gibbs sampler is often applied when the target is a multivariate distribution, i.e. we are interested in generating sample from $$f(x) = f(x_1, x_2, ..., x_d)$$Let $X = (X_1, ... , X_d)$ be a d-demensional random vector.Define $$X_{(-j)} = (X_1, .... X_{(j-1)}, X_{(j+1)}, ..., X_d)$$Denote the conditi..

Let X be a random variable with pdf (or pmf) f(x). Our goal is to generate a sample of X, but it is impossible to draw a sample from $f(x)$ directly. Let Y be a random variable with pdf(or pmf) $g(y)$ such that $f(t) / g(t) \leq c$ for some constant c. The AR method proceeds as follows:Generate y from the distribution with g.Generate u from Uniform(0,1). If $u otherwise reject y go to step1. ..