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Sampling Distribution Theory

Sampling Distribution Theory. ch6. 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]

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Sampling Distribution Theory

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  1. Sampling Distribution Theory ch6

  2. F Distribution: F(r1, r2) • From two indep. random samples of size n1 & n2 from N(μ1,σ12) & N(μ2,σ22),some comparisons can be performed, such as: • Fα(r1,r2)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 c=F0.99(4,9) If F is F(6,9):

  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 0.98-0.31=0.67. • Ex6.9-2: Let Y1<Y2<Y3<Y4<Y5 be the order statistics for X1, X2, X3, X4, X5, each from the p.d.f. f(x)=2x, 0<x<1. • Consider P(Y4<1/2) ≡at least 4 of Xi’s must be less than 1/2: 4 successes.

  4. General Cases • The event that the rth order statistic Yr is at most y, {Yr≤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 gr(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) Y1<Y2<Y3<Y4<Y5 are the order statistics for X1, X2, X3, X4, X5, each from the p.d.f. f(x)=2x, 0<x<1. On a single trial

  6. More Examples • Ex: 4 indep. Trials(Y1 ~ Y4) from a distribution with f(x)=1, 0<x<1. • Find the p.d.f. of Y3. • Ex: 7 indep. trials(Y1 ~ Y7) from a distribution f(x)=3(1-x)2, 0<x<1. • Find the p.d.f. of the sample median, i.e. Y4, is less than • Method 1: find g4(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(X1),F(X2),…,F(Xn)} ⇒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 Wr=F(Yr) 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 πp is defined s.t. the area under f(x) to the left of πp is p. • Therefore, Yr is the estimator of πp, where r=(n+1)p. • In case (n+1)p is not an integer, a (weighted) average of Yr and Yr+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, 1047. • ∵n=12, the sample median is • ∵(n+1)(0.25)=3.25, the 25th percentile or first quartile is • ∵(n+1)(0.75)=9.75, the 75th percentile or third quartile is • ∵(n+1)(0.6)=7.8, the 60th percentile

  9. Another Example • Ex5.6-7: The order statistics of 13 indep. Trials(Y1<Y2< …< Y13) from a continuous type distribution with the 35th percentile π0.35. • Find P(Y3< π0.35< Y7) • The event {Y3< π0.35< Y7} happens iff there are at least 3 but less than 7 “successes”, where the success probability is p=0.35. Success By Table II on p.677~681.

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