Kernel density estimation (KDE)

Question

Due to an assignment I need to implement a algorithm based on KDE to schedule an input data in different servers.

So far, I studied statistics in my bachelor but we did not go that far and they did not explain me about estimation functions and Kernel. So my questions are the next:

Assuming this unidimensional population:

1 2 5 5 3 4 5 6 7 8 1 3 9 6 7 5 3 1 2 3 5 7 8 2 1 3 4 5 1 7 6 3

1. What is a kernel and why is useful to estimate the trend in a population?
2. As far as I know, KDE can use different kernel functions, such as triangle, uniform, etc... How do I know which one should I choose?
3. What is the bandwidth smoothing parameter?

Answer 1

1. Basically, a kernel density estimation allows a non-parametric estimatation of the probability density function of a random variable. It gives a much smoother result than a histogram.

2. The best kernel depends on your application...

3. The kernel has to know domain on which it has to compute the density. The domain extent is specified by the bandwidth. There are ways to identified the optimal bandwidth (see Chapman & Hall, 1986 or Silverman, 1998)

Answer 2

http://en.wikipedia.org/wiki/Kernel_density_estimation gives a good introduction.

Your first question is about the trend in a population. The main sense of "trend" in statistics is that of an overall change in the mean or level of a variable with some other variable, most commonly time, but occasionally space or something else. In ordinary language "trend" is often used much more loosely to refer to any kind of pattern or relationship, but I'd advise against that in statistical discussions. Kernel density estimation just gives you a picture of the distribution of one or more variables, which is different from a trend as statistical people would use that word.

You may want to revise your question after some reading, but my main point is that this is a well-documented and standard technique with resources easily available to you.