Resampling Methods in CFA Level I Quantitative Methods

Welcome back, CFA candidates! Today, we’re going to explore the fascinating world of resampling methods, specifically the bootstrap method and the jackknife method. These alternative methods help us estimate the sampling distribution of a statistic, adding a little twist to your conventional statistical knowledge. So, let’s dive in!

Quick Recap: Central Limit Theorem and Confidence Intervals

Before we jump into resampling methods, let’s have a quick revision of the Central Limit Theorem (CLT) and how it applies to confidence interval estimation. In a nutshell, the CLT states that:

• For a population with mean µ and standard deviation σ, we can estimate the population mean using random sampling.
• If the sample size n > 30, the sampling distribution of the mean will be approximately normal with mean µ.
• The standard error of the sample mean is σ / √N.

This allows us to determine the level of confidence that the true population mean lies within an interval, using either the z-statistic or t-statistic.

Bootstrap Method: Resampling with a Twist

Imagine you’re a researcher with no idea about the population, and all you have is a sample. That’s where the bootstrap method comes in handy! It’s a statistical process that mimics random sampling from a population to construct the sampling distribution. The catch? Instead of sampling from the population, it resamples from the sample itself, treating it as the estimated population.

During the bootstrap process, an observation can be drawn multiple times from the population for each sampling process, meaning it may appear several times in the same sample. The confidence interval for the population mean is determined directly from the sampling distribution, and the estimator doesn’t have to be a mean (it could be a median, for example).

Bootstrap Process in a Nutshell

1. Draw random samples from the sample itself (estimated population).
2. Resampling is allowed, so an observation may appear multiple times in the same sample.
3. Repeat the process thousands of times to build an actual sampling distribution.
4. Calculate the mean and standard errors of the estimator.
5. Determine the confidence interval for the population mean directly from the sampling distribution.

Jackknife Method: A Different Resampling Technique

The jackknife method is another resampling technique that was popular in the past when computers were less advanced. Unlike bootstrap, which repeatedly draws samples of the same size with replacement, jackknife samples are selected by taking the original observed data sample and leaving out one observation at a time without replacement.

For an estimated population of size M, jackknife usually requires M repetitions, whereas bootstrap can have many more resamples as repetition is allowed. Like bootstrap, the statistic for each resample is calculated, and the data forms the sampling distribution, which determines the confidence intervals for the population statistic.

Jackknife vs. Bootstrap

• Jackknife is less computationally intensive, making it popular in the past when computers were less advanced.
• Bootstrap allows for more resamples as repetition is allowed, resulting in a more robust sampling distribution.
• Jackknife produces the same results for every run, as the resampling process is exhaustive.
• Bootstrap usually gives different results, as resamples are randomly drawn.

Key Takeaways: Resampling Methods

As we wrap up our discussion of resampling methods, here are some important points to remember:

• Resampling techniques like bootstrap and jackknife differ from conventional statistical methods.
• Bootstrap resamples from the sample itself, treating it as the estimated population.
• Jackknife samples by leaving out one observation at a time without replacement.
• Both methods aim to estimate the sampling distribution of a statistic and determine confidence intervals for the population statistic.

And there you have it! A friendly introduction to resampling methods in the CFA Level I Quantitative Methods curriculum. Remember, you shouldn’t have to make detailed calculations in the exam, but understanding the differences between resampling techniques and conventional statistical methods, as well as the differences between bootstrap and jackknife, will serve you well. Good luck, and see you in the next lesson!