**AMS597 2021:**

- (a) Use sample function to generate a random vector that follows a multinomial distribution with probability (0.1, 0.2, 0.4, 0.3).

(b) Using only random uniform generator (DO NOT use sample), generate a random vector that follows a multinomial distribution with probability (0.1, 0.2, 0.4, 0.3).

- Generate 100 exponentially distributed random variables with rate 2, and plot their empirical distribution function.
- Use the following dataset to answer the questions. http://www.ams.sunysb.edu/~pfkuan/Teaching/AMS597/Data/d_logret_6stocks.txt
- Perform a t-test for American Express with the null hypothesis that the mean of its log return is zero.
- Perform a Wilcoxon signed-rank test for American Express with the null hypothesis that the mean of its log return is zero.
- Perform a two-sample t-test to conclude if the mean log return of Pfizer and American Express are same or not.
- Compare the variance of log returns for Pfizer and American Express.
- Perform a two-sample Wilcoxon test to conclude if the mean log returns of Pfizer and American Express are same or not.

- Write your own function t.test which can perform both one and two sample t-test. For two sample t-test, it can perform both equal and unequal variance version. Your my.t.test will take the following argument (1) the vector x, (2) optional vector y if it is two sample t-test, (3) type of alternative hypothesis alternative, (4) the mean or mean difference that you are testing mu. Your function my.t.test should contain a routine to check for equal variance assumption using the F test. If the p-value of the F test is ≤ 0
*.*05, then it will perform two sample t-test with unequal variance assumption. Your function my.t.test should return the test statistic stat, degrees of freedom df and pvalue p.value. You may use var.test() or write your own F test, but you should not use t.test(). - Write your own function Wilcoxon rank sum test wilcox.test which can perform both exact and normal approximation test for two-sided alternative hypothesis. Your function will compute the p-value using normal approximation if
*n*_{1 }and*n*_{2 }≥ 12. Otherwise, it will

compute the p-value using exact test. Your function will return the test statistics W1 and W2, p-value p.value and a message indicating the type of test used (normal or exact). You may assume there is no ties in the data.

- A regression through the origin model may be used when specific knowledge about the problem at hand suggests that the response variable is zero if and only if the predictor variable is zero. For such problems, the model can be written as

where * _{i}*’s are iid

*N*(0

*,σ*

^{2}) random noise.

- Derive the least-squares estimate of
*β*. - Run the following R code:

set.seed(123) x <- rnorm(50) y <- 2*x+rnorm(50)

Use the formula you derived in part (a) above to estimate based on this data, and draw a scatterplot of the data with the fitted line overlaid.

- Perform the regression through origin using the R function lm on the same data.
- Write your own function kendall to compute Kendall’s
*τ*between two variables and apply your function to*x*and*y*generated above.

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