Stat 5101 Lecture Notes - School of Statistics

7.4. SAMPLING DISTRIBUTIONS OF SAMPLE QUANTILES 20519 19 9Out[4]= 1 - BetaRegularized[--, 1, --, -]37 2 2In[5]:= N[%]Out[5]= 0.0974132(The last command tells Mathematica to evaluate the immediately precedingexpression giving a numerical result). This can be done more concisely if lessintelligibly asIn[6]:= N[1 - CDF[FRatioDistribution[9, 19], 2]]Out[6]= 0.09741327.4 Sampling Distributions of Sample QuantilesThe sample quantiles are the quantiles of the empirical distribution associatedwith the data vector X =(X 1 ,...,X n ). They are mostly of interest onlyfor continuous population distributions. A sample quantile can always be takento be an order statistic by Theorem 7.5. Hence the exact sampling distributionsof the empirical quantiles are given by the exact sampling distributions for orderstatistics, which are given by equation (5) on p. 217 of Lindgrenn!f X(k) (y) =(k−1)!(n − k)! F (y)k−1 [1 − F (y)] n−k f(y) (7.35)when the population distribution is continuous, (where, as usual, F is the c. d. f.of the X i and f is their p. d. f.). Although this is a nice formula, it is fairlyuseless. We can’t calculate any moments or other useful quantities, except in thespecial case where the X i have a U(0, 1) distribution, so F (y) =yand f(y) =1for all y and we recognizen!f X(k) (y) =(k−1)!(n − k)! yk−1 (1 − y) n−k (7.36)as a Beta(k, n − k + 1) distribution.Much more useful is the asymptotic distribution of the sample quantilesgiven by the following. We will delay the proof of the theorem until the followingchapter, where we will develop the tools of multivariate convergence indistribution used in the proof.Theorem 7.27. Suppose X 1 , X 2 , ... are continuous random variables that areindependent and identically distributed with density f that is nonzero at the p-thquantile x p , and suppose√ n(knn − p )→ 0, as n →∞, (7.37)