Remark that we can estimate the first term using only available sample. However, the second one depe

The histograms are the simplest and, sometimes, the most effective tool to describe quickly a dataset density. Suppose you have an independent and identically distributed random sample from some unknown continuous distribution called . Recall in a past post , we explained the histogram construction. We found the following histogram estimator for ,
where the are a uniform partition of size of the real line. Then, continuing with the histogram series ( II and III ), we explored its asymptotic properties. Finally, we illustrated the theory with some simulated data. Specifically in this post , we adjust the binwidth minimizing the mean integrated squared error (MISE) and getting
Notice bazar the bindwidth depends of the unknown quantity , thus the main problem remains unsolved and we cannot use it in practice. Yeah I know!, I cheated before minimizing the MISE and forgetting constant influence constant. In fact, all worked well because my example bazar was a normal distributed with a relative large sample. Of course, those conditions are rare in statistics and we need improve the choice of . In this post, we will find a fully data-driven estimator for using a technique called cross-validation . Define the integrated squared error as follows, bazar
Remark that we can estimate the first term using only available sample. However, the second one depends on the unknown function bazar . The first thought that comes in mind to approximate is to use the empirical estimator
Here, is the leave-one-out estimator. You guessed right! We have removed the sample in each evaluation to ensure the independence between and the ‘s. In fact, we can prove that (e.g., \cite{tsybakov2008introduction}). Then, the general criterion to find by cross-validation is,
The last equation looks ugly and trying to minimize it in this state, seems futile. Given that we are working—for now—with the histogram case, we can simplify it even further and find something easier to estimate. First, denote by the number of observations belonging to the interval . The random variable has the form
We have achieved our main goal on finding a statistic (a formula which depends only in the data) that we can minimize it numerically to find the optimal bindwidth bazar . Related articles Calculating histograms (gnuplotting.org) Calibration Affirmation (r-bloggers.com) bazar How many bins? (luisospina.wordpress.com) Python: Numerical Descriptions of the Data (statsblogs.com)
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