Thread: Consistency using the median

How can you use the sample median Y' to prove consistency?

If 2n+1 random observations are drawn from a continuous and symmetric pdf with mean (mu) and if f_Y(mu;mu) does not equal 0, then the sample median Y'_(n+1), is unbiased for (mu) and Var( Y'_(n+1) ) = 1/8[ f_Y(mu;mu)^2 *n ]. Show that (mu) hat = Y'_(n+1) is consistent for (mu).

Originally Posted by lteece89

How can you use the sample median Y' to prove consistency?

If 2n+1 random observations are drawn from a continuous and symmetric pdf with mean (mu) and if f_Y(mu;mu) does not equal 0, then the sample median Y'_(n+1), is unbiased for (mu) and Var( Y'_(n+1) ) = 1/8[ f_Y(mu;mu)^2 *n ]. Show that (mu) hat = Y'_(n+1) is consistent for (mu).

When the mean and variance exist it suffices to show that the mean converges to the thing you want and the variance goes to 0 (for all values of mu).