Question

Determine a method to generate random observations for the extreme-valued pdf that is given by $$ f(x)=\exp \left\\{x-e^{x}\right\\}, \quad-\infty

Short answer

Create an R function to generate random sample observations from the extreme-valued distribution using inverse transform sampling method. After generating 10,000 observations, plot the histogram and overlay the histogram with the theoretical pdf for comparison. The theoretical pdf can be plotted using the curve function in R. Note that the histogram and the theoretical pdf should be similar, which verifies the correctness of the generated samples.

Step-by-step solution

Understand the Extreme-Value Distribution

The given probability density function \(f(x) = e^{x - e^x}\) is the pdf of an extreme-value distribution. Extreme-value distributions are often used to model the smallest or largest value among a large set of independent, identically distributed random variables.

Generate Random Sample of Observations

In R, we can create a function to simulate a random sample from a given distribution using the inverse transform sampling method. Let's use function r for generating the random samples. Here is the R function: ```r r_extreme_valued <- function(n){ u <- runif(n) x <- -log(-log(u)) return(x) } ```In this function, 'n' represents the number of observations to be produced (10000 in this case), 'u' is a randomly generated number between 0 and 1, and 'x' is a random variable following our desired extreme-value distribution. The function 'runif' in R generates random deviates of the uniform distribution, and -log(-log(u)) is the quantile function of the extreme-value distribution.

Generate 10,000 Observations and Plot the Histogram

Next, we use the defined function to generate 10,000 observations and plot the histogram. ```r n <- 10000 x <- r_extreme_valued(n) hist(x, prob = TRUE, main = '', xlab = 'Observations', ylab = 'Density', border = 'blue') ```The function 'hist' in R is used to create histograms. The argument 'prob = TRUE' makes the histogram present relative frequencies instead of the counts, giving an approximation of the probability density function.

Overlay a Plot of the PDF on the Histogram

Finally, we overlay the actual pdf on the histogram to compare the random sample with the theoretical distribution. ```r curve(exp(x - exp(x)), add = TRUE, col = 'red', lwd = 2) ```The function 'curve' in R is used to create a plot of a function. With the argument 'add = TRUE', the plot is added to the current plot (i.e., the histogram), 'col = 'red'' sets the color of the plot to red, and 'lwd = 2' sets the line width to 2. The function \(exp(x - exp(x))\) is the pdf of the extreme-value distribution.

Key concepts

Probability Density Function (PDF)

The Probability Density Function (PDF) is a function that describes the likelihood of a continuous random variable taking on a particular value. The PDF for an extreme-value distribution is mathematically expressed as \( f(x) = \text{exp}\big\{ x - e^x \big\} \). This function peaks at the point representing the most likely value of the random variable, and the area under the whole curve represents the total probability, which is always equal to 1.

For a given continuous random variable, the PDF is crucial because it gives us an understanding of how the data is distributed across different values. It can tell us where the majority of data points fall, which values are less likely and which are more likely. Importantly, by integrating this function over a specific interval, we can determine the probability that the variable falls within that range.

Random Sample Generation

Random Sample Generation refers to creating a set of random observations that are drawn from a specified probability distribution. In the context of an extreme-value PDF, we aim to generate data that mimics real observations from the extreme ends of a dataset, such as maximum temperatures or financial losses.

To generate a random sample effectively, it's essential to have a sound statistical understanding of the distribution you're sampling from and to use a reliable method that accurately reflects the theoretical distribution. The quality of your random sample can significantly impact the validity of subsequent statistical analysis and conclusions.

Inverse Transform Sampling

Inverse Transform Sampling is a technique used to generate observations from a probability distribution. It involves taking random numbers from a uniform distribution and transforming them through the inverse of the desired distribution's cumulative distribution function (CDF).

In the case of the extreme-value PDF, the inverse CDF (also known as the quantile function) can be derived analytically, leading to the transformation \(-\text{log}(-\text{log}(u))\), where \(u\) is a uniformly distributed random number between 0 and 1. This method allows us to convert simple random numbers into a random sample from the extreme-value distribution, preserving its statistical properties.

R Programming for Statistics

R is a programming language and an environment specifically designed for statistical computing and graphics. It is widely used by statisticians and data scientists for data analysis, visualization, and statistical modeling.

The ability to write custom functions in R, like one for generating a random sample from an extreme-value distribution, is a powerful feature that extends the software's capabilities. R's comprehensive collection of packages and built-in functions streamline the process of complex statistical computing tasks, making it an indispensable tool for modern data analysis.

Histogram Overlay with PDF

Overlaying a histogram with a Probability Density Function (PDF) is a graphical technique used in statistics to compare empirical data with a theoretical distribution. When you overlay a PDF onto a histogram of a sample, you can visually assess how well the sample matches the theoretical distribution.

Using R, creating a histogram from a sample and overlaying it with a PDF can be accomplished with a few lines of code. The histogram provides a representation of the sample data, and the overlay of the PDF shows the expected distribution pattern. This comparison can be useful for verifying the fit of the sample data to the assumed distribution and for identifying any potential discrepancies between the observed data and theoretical expectations.