1 Sampling Distribution Theory ch6

2 F Distribution: F(r 1, r 2 )  From two indep. random samples of size n 1 & n 2 from N(μ 1,σ 1 2 ) & N(μ 2,σ 2 2 ),some comparisons can be performed, such as:  F α (r 1,r 2 )is the upper 100α percent point. [Table VII, p.689~693]  Ex: Suppose F has a F(4,9) distribution.  Find constants c & d, such that P(F  c)=0.01, P(F  d)=0.05 If F is F(6,9): c=F 0.99 (4,9)

3 Order Statistics  The order statistics are the observations of the random sample arranged in magnitude from the smallest to the largest.  Assume there is no tie: identical observations.  Ex6.9-1: n=5 trials: {0.62, 0.98, 0.31, 0.81, 0.53} for the p.d.f. f(x)=2x, 0<x<1. The order statistics are {0.31, 0.53, 0.62, 0.81, 0.98}.  The sample median is 0.62, and the sample range is =0.67.  Ex6.9-2: Let Y 1 <Y 2 <Y 3 <Y 4 <Y 5 be the order statistics for X 1, X 2, X 3, X 4, X 5, each from the p.d.f. f(x)=2x, 0<x<1.  Consider P(Y 4 <1/2) ≡at least 4 of X i ’s must be less than 1/2: 4 successes.

4 General Cases  The event that the rth order statistic Y r is at most y, {Y r ≤y}, can occur iff at least r of the n observations are no more than y.  The probability of “success” on each trial is F(y).  We must have at least r successes. Thus,

5 Alternative Approach  A heuristic approach to obtain g r (y):  Within a short interval Δy:  There are (r-1) items fall less than y, and (n-r) items above y+Δy.  The multinomial probability with n trials is approximated as.  Ex6.9-3: (from Ex6.9-2) Y 1 <Y 2 <Y 3 <Y 4 <Y 5 are the order statistics for X 1, X 2, X 3, X 4, X 5, each from the p.d.f. f(x)=2x, 0<x<1. On a single trial

6 More Examples  Ex: 4 indep. Trials(Y 1 ~ Y 4 ) from a distribution with f(x)=1, 0<x<1.  Find the p.d.f. of Y 3.  Ex: 7 indep. trials(Y 1 ~ Y 7 ) from a distribution f(x)=3(1-x) 2, 0<x<1.  Find the p.d.f. of the sample median, i.e. Y 4, is less than  Method 1: find g 4 (y), then  Method 2: find then By Table II on p.677.

7 Order Statistics of Uniform Distributions  Thm3.5-2: if X has a distribution function F(X), which has U(0,1). {F(X 1 ),F(X 2 ),…,F(X n )} ⇒ Wi’s are the order statistics of n indep. observations from U(0,1).  The distribution function of U(0,1) is G(w)=w, 0<w<1.  The p.d.f. of the rth order statistic W r =F(Y r ) is ⇒ Y’s partition the support of X into n+1 parts, and thus n+1 areas under f(x) and above the x-axis.  Each area equals 1/(n+1) on the average. p.d.f. Beta

8 Percentiles  The (100p) th sample percentile π p is defined s.t. the area under f(x) to the left of π p is p.  Therefore, Y r is the estimator of π p, where r=(n+1)p.  In case (n+1)p is not an integer, a (weighted) average of Y r and Y r+1 can be used, where r=floor[(n+1)p].  The sample median is  Ex6.9-5: X is the weight of soap; n=12 observations of X is listed:  1013, 1019, 1021, 1024, 1026, 1028, 1033, 1035, 1039, 1040, 1043,  ∵ n=12, the sample median is  ∵ (n+1)(0.25)=3.25, the 25 th percentile or first quartile is  ∵ (n+1)(0.75)=9.75, the 75 th percentile or third quartile is  ∵ (n+1)(0.6)=7.8, the 60 th percentile

9 Another Example  Ex: The order statistics of 13 indep. Trials(Y 1 <Y 2 < …< Y 13 ) from a continuous type distribution with the 35 th percentile π  Ex5.6-7: The order statistics of 13 indep. Trials(Y 1 <Y 2 < …< Y 13 ) from a continuous type distribution with the 35 th percentile π  Find P(Y 3 < π 0.35 < Y 7 )  The event {Y 3 < π 0.35 < Y 7 } happens iff there are at least 3 but less than 7 “successes”, where the success probability is p=0.35. By Table II on p.677~681. Success