| << Chapter < Page | Chapter >> Page > |
The random variable
is not a statistic, since it uses the unknown parameter μ . However, the following is a statistic.
It would appear that might be a reasonable estimate of the population variance. However, the following result shows that a slight modification is desirable.
The statistic
is an estimator for the population variance.
VERIFICATION
Consider the statistic
Noting that , we use the last expression to show
The quantity has a bias in the average. If we consider
The quantity V n with rather than is often called the sample variance to distinguish it from the population variance. If the set of numbers
represent the complete set of values in a population of N members, the variance for the population would be given by
Here we use rather than .
Since the statistic V n has mean value σ 2 , it seems a reasonable candidate for an estimator of the population variance. If we ask how good is it, we need to considerits variance. As a random variable, it has a variance. An evaluation similar to that for the mean, but more complicated in detail, shows that
For large n , is small, so that V n is a good large-sample estimator for σ 2 .
Consider a population random variable uniform [-1, 1]. Then and . We take 100 samples of size 100, and determine the sample sums. This gives a sample of size 100 of the sample sum random variable S 100 , which has mean zero and variance 100/3.For each observed value of the sample sum random variable, we plot the fraction of observed sums less than or equal to that value. Thisyields an experimental distribution function for S 100 , which is compared with the distribution function for a random variable .
rand('seed',0) % Seeds random number generator for later comparison
tappr % Approximation setupEnter matrix [a b] of x-range endpoints [-1 1]Enter number of x approximation points 100
Enter density as a function of t 0.5*(t<=1)
Use row matrices X and PX as in the simple case
qsample % Creates sample
Enter row matrix of VALUES XEnter row matrix of PROBABILITIES PX
Sample size n = 10000 % Master sample size 10,000Sample average ex = 0.003746
Approximate population mean E(X) = 1.561e-17Sample variance vx = 0.3344
Approximate population variance V(X) = 0.3333m = 100;
a = reshape(T,m,m); % Forms 100 samples of size 100A = sum(a); % Matrix A of sample sums
[t,f]= csort(A,ones(1,m)); % Sorts A and determines cumulative
p = cumsum(f)/m; % fraction of elements<= each value
pg = gaussian(0,100/3,t); % Gaussian dbn for sample sum valuesplot(t,p,'k-',t,pg,'k-.') % Comparative plot
% Plotting details (see
[link] ) ![Figure one is a graph of two plots, titled, Central limit theorem for sample sums. The horizontal axis is labeled, sample sum values, and the vertical axis is labeled, cumulative fraction. The values on the horizontal axis range from -15 to 20 in increments of 5. The values on the vertical axis range from 0 to 1 in increments of 0.1. There are two captions inside the graph. The first reads, X uniform on [-1 1], and the second reads, E[X] = 0 Var[X] = 1/3. The first plot is a smooth, dashed line, labeled gaussian. The second plot is a wavering, jagged solid line labeled experimental. Both plots follow generally the same shape. They begin in the bottom right at approximately (-12, 0) with a positive slope, and they move to the right, increasing at an increasing rate. At nearly the midpoint in the graph, approximately (0, 0.5), the graphs adjust and begin increasing at a decreasing rate, approaching the top-right corner of the graph while tapering off to a horizontal line. The gaussian, dashed line follows this path's description more accurately, while the solid experimental line seems to be closely fitted to the gaussian line's path with some imperfections causing it to waver jaggedly at a couple spots along the path.](/ocw/mirror/col10708_1.6_complete/m23496/fig13_4_1.png)
Notification Switch
Would you like to follow the 'Applied probability' conversation and receive update notifications?
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|