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Which quantiles of a continuous distribution can one estimate with more precision?

The median is an important quantity in data analysis. It represents the middle value of the data distribution. Estimates of the median, however, have a degree of uncertainty because (a) the estimates are calculated from a finite sample and (b) the data distribution of the underlying data is generally unknown. One important roles of a data scientist is to quantify and to communicate the degree of uncertainty in his or her data analysis.

Assignment

In this assignment, you will write a blog post to answer a series of questions related to the variation of the median (and a range of other quantiles). You will use analytic methods to answer the questions.

The audience of your blog post is a Senior Data Scientist who you hope to work with in the future. Your goal is to document how you wrote flexible functions for the distribution of order statistics.

Questions

Q: Begin with the median from a sample of N = 200 from the standard normal distribution. Write an R function that is the density function for the median in this sample. Note that the 100th order statistic is approximately the median, and use the order statistic formula discussed in class. Generate a plot of the function.

dorder <- function(x){
  
}

Q: Write an R function that is the probability function for the median in this sample. Use the order statistic formula discussed in class. Generate a plot of the function.

porder <- function(x){
  
}

Q: Write an R function that is the quantile function for the median in this sample. (You have several options for how to write this function.) Generate a plot of the function.

qorder <- function(p){
  
}

Q: Simulate the sampling distribution for the median. Create a plot of the empirical CDF (ECDF). Overlay the plot of the ECDF with a plot of the CDF.

Q: Using the simulated sampling distribution from the previous question, create a histogram (on the density scale). Overlay the histogram with a plot of the density function.

Q: One very common way to compare a random sample to a theoretical candidate distribution is the QQ plot. It is created by ploting quantiles of the theoretical distribution on the x-axis and empirical quantiles from the sample on the y-axis.

If sample and theoretical quantiles come from the same distribution, then the plotted points will fall along the line y = x, approximately. Here are two examples when the sample and theoretical quantiles came from the same distribution.

random_sample <- rexp(200)
q_candidate <- qexp

x <- q_candidate((1:200)/200)
y <- quantile(random_sample, probs = (1:200)/200)

svg("./assets/exponential-qq.svg", width = 5, height = 3)
tgsify::plotstyle(style = upright)
plot(x,y, asp = 1)
abline(0,1)
dev.off()
random_sample <- rnorm(200)
q_candidate <- qnorm

x <- q_candidate((1:200)/200)
y <- quantile(random_sample, probs = (1:200)/200)

svg("./assets/normal-qq.svg", width = 5, height = 3)
tgsify::plotstyle(style = upright)
plot(x,y, asp = 1, xlab = "Theoretical quantile", ylab = "Sample quantile")
abline(0,1)
dev.off()

Here is an example when the sample distribution does not match with the theoretical distribution. The sample distribution is t3 where as the theoretical distribution is N(0, 1). Notice the deviation from y = x.

random_sample <- rt(200, df = 3)
q_candidate <- qnorm

x <- q_candidate((1:200)/200)
y <- quantile(random_sample, probs = (1:200)/200)

svg("./assets/t-normal-qq.svg", width = 5, height = 3)
tgsify::plotstyle(style = upright)
plot(x,y, asp = 1, xlab = "Theoretical quantile", ylab = "Sample quantile")
abline(0,1)
dev.off()

The deviation occurs despite the fact that the two distributions are similar.

For the assigment, generate a QQ plot for the simulated data of the median relative to the known sampling distribution of the median.

Does the simulated data agree with the theoretical sampling distribution?

Q: Modify the dorder, porder, and qorder functions so that the functions take a new parameter k (for the kt**h order statistic) so that the functions will work for any order statistic and not just the median.

Q: Generate the QQ plot for simulated data from the sampling distribution of the sample max and the theoretical largest order statistic distribution.

Q: Modify the dorder, porder, and qorder functions so that the functions take new parameters dist and ... so that the functions will work for any continuous distribution that has d and p functions defined in R.

Q: Use the newly modified functions to plot the probability and density functions for the sample min (N = 200).

Submission instructions

  1. Within the repo Probability and Inference Portfolio Lastname Firstname, create a folder called 06-order-statistics
  2. Within the folder, create an .html for your blog post
  3. The name of the blog post file must be writeup.html
  4. Within the folder, include code scripts or .rmd or some other document that will successfully generate the output of the simulation when executed from within the folder. (Do not use absolute file paths.)
  5. Edit the README to be an index for the portfolio.
  6. Be prepared to share your blog post with the class when the deliverable is due.

Other instructions

  1. The deliverable should be your own work. You may discuss concepts with classmates, but you may not share code or text.