Project 3. (30%)
Computer-simulation based empirical methods are becoming tools of choice for problems in statistics. Because of the difficulty of computing by hand, these methods did not gain much popularity. However, with the advent of high-quality pseudo random number generators and high-speed computers, these numerical techniques became more and more popular and practical. Modern statistics is increasingly being equipped with theoretical concepts complemented with effective computational tools to handle the challenges that arise in diverse areas such as Economics, Finance, Statistics, Mathematics, Computer Science, Physical and Life Sciences, etc..
• Explain the basic procedures or algorithms of some popular empirical methods as follows:
• Bootstrap methods to estimate the confidence intervals;
• Expectation-Maximisation (EM) algorithm to estimate iteratively parameters with missing or incomplete data or in situations where we can’t solve Maximum Likelihood Estimation (MLE) analytically;
• Metropolis and Metroplis-Hastings MCMC algorithm to generate dependent samples from the target distribution for solving the complicated integration problems;
• Gibbs sampler to sample from a multivariate distribution by sampling from the marginal distributions of the target distribution.
• Write your own R commands to solve the following problems:
• Use the Metropolis-Hastings sampler to generate a sample from beta (2,2) distribution using a uniform(0,1) candidate distribution. Make a comparison between the quantiles of the targeted beta(2,2) distribution with the quantiles of the generated chain in a quantile-quantile plot.
• compare the above method with the following two alternative methods : ①. Accept-reject algorithm to simulate the beta(2,2)distribution; ②. the transformation method to simulate the beta (2,2) using gamma distributions.
• Check the similarity of the above three different beta generators using:
• Histograms.
• overlaid density curves.
• Kolmogorov-Smirnov test.