Write Matlab codes to compute the 1-D nonparametric probability density function for a given set…

Write Matlab codes to compute the 1-D nonparametric probability density function for a given set of samples of a random variable. Call this function “KernelDensity.m”. The conceptual background for this procedure is described in the lecture notes. Based on that description, implement your procedure according to the following specifications:

1) Use the Matlab function randn( ) to generate 10,000 samples of a Gaussian random variate with zero mean and unit standard deviation. These samples should be stored in a single column array, x.

2) Your program should scale the sample data internally to the unit interval, [0,1]. This will help to standardize the construction of a grid for the density function as well as the selection of a smoothing parameter, s.

3) Use a uniform grid array, v, of 101 points at which the density is to be estimated. You might find the Matlab function linspace( ) helpful here. For simplicity, confine the grid to the range of data in the scaled sample population.

4) Use a Gaussian-like kernal function,, to weight the contribution of each sample, xj, to the density at vi.

K(V,,.x;,0) = exp[-(x, – v;)? / 20?]

5) Ensure that your estimated density function is properly normalized. In Matlab, this is easily done with the trapz( ) function.

6) Your program should accept three argument: the sample data, x, the smoothing parameter, s, and the number of grid points, M.

(a)Test your program by computing the estimated density function for the 10,000 standard normal samples mentioned above. Try four different values of smoothing parameter: 0.10, 0.05, 0.025, and 0.0125. Plot the resulting density functions on a single plot for easy comparison and be sure to include a legend on the plot so I know which scorresponds to each curve. Comment on your results; in particular, select what you think is the best of the four values of s and say why picked this value.

(b)From your results in (a), what do you think would be the consequence of using a value of s that is too large? Too small?

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