lecture 05_addendum_chisquare,t and f distributions

7/28/2019 Lecture 05_addendum_Chisquare,t and F Distributions

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Chi-square, t-, and F-Distributions

(and Their Interrelationship)

1 Some Genesis

Z1, Z2, . . . , Ziid N(0,1)X2Z1+Z2+. . .+Z2.

Specifically, if= 1,Z22 . The density function of chi-square distribution1

will not be pursued here. We only note that: Chi-square is aclassof distribu-

tion indexed by itsdegree of freedom, like thet-distribution. In fact, chi-squarehas a relation witht. We will show this later.

2 Mean and Variance

IfX22, we show that:

E{X2}=,

VAR{X2}= 2.

For the above example,Zj2 ,ifEZj= 1,VARZj= 2,j= 1,2. . . , , then1

E{X2}=E{Z1+. . .+Z}2 2

=EZ1+. . .+EZ

= 1 + 1 +. . .+ 1 =.

Similarly,VAR{X2}= 2. So it suffices to show

E{Z2}= 1,VAR{Z2}= 2.

SinceZis distributed as N(0,1),E{Z2}is calculated as

12

z2ez/2dz, 2

which can easily be shown to be 1. Furthermore,VAR{Z2}can be calculated

through the formula:VAR{Z2}=E{Z4} (E{Z2})2.


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3 Illustration: As an example for Law of Large Num-

ber(LLN) and Central Limit Theorem(CLT)

IfX22,X2can be written asX2=Z1+. . .+Z,for someZ1, . . . , Ziid

N(0,1). Then when ,

X2 EX22)

VAR(X

moreover,

X2=N(0,1) (CLT);

2

X221 (LLN).

Application: To Make Inference on2

IfX1, . . . , XniidN(, 2), thenZj(Xj)/N(0,1), j= 1, . . . , n. Weknow, from a previous context, thatnZj2, or equivalently,

1n

n

Xj 21(Xj

{}=

n

)22,

n

2j=1

ifisknown, or otherwise (ifis unknown)needs to be estimated (byX,

say,) such that (still needs to be proved):

n

1(XjX)2

21.(1)n

2

Usuallyis not known, so we use formula (1) to make inference on2. Denotes2as thep-th percentile from the rightfor the2-distribution, the two-

,p2

sided confidence interval (CI) ofcan be constructed as follows:

n

1(Xj

Pr{21,1/2


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That is, the 100(1)% CI for2isn 2 n 21(XjX)

(,

21,/2n

1(XjX))

21,1/2n

[QUESTION:How to use this formula with actual data?]

The Interrelationship Betweent-,2-, andF- Statistics

tversus2

IfX1, . . . , XniidN(, 2), then

XN(0,1).

/nWhenis unknown,

(XiX)2

tn1,where =n1

Note that

X=

/n

X 1

/n

1

=Z

Z

=

=

Combining (3) and (4) gives

tn1=

or, in general,

t=

3


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(XiX)2(n1)2

Z.(4)

21n

n1

Z,

21

nn1

Z.

2


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Fversus2

From some notes about ANOVA, we have learned that

2/aa

Fa,b2/b

b

(Sir R. A. Fisher). (5)

The original concern of Fisher is to construct astatisticwhich has a sampling

distribution, in some extent, free from the degrees of freedomaandbunder

the null hypothesis. With this concern, he presented his F-statistic in a way

that: Since2has expectationa, so the numerator2/ahas expectation 1;a asimilarly, the denominator also has expectation 1. As Fisher said,the value

of F-statistic will fluctuate near 1 under the null hypothesisH0:1=. . .=(if=a+ 1).From (5), we are able to express the2-distribution in terms ofF:

2/aa

Fa,=

limb(2/b)b

2/a=a,

1

from Section 3 (LLN). So2-table can be treated as a part ofF-tables.

5.3 tversusF

Zt=

2/

2 /11=

2/

=F1,.

Or, in other words,t2=F1,, a formulae useful in assessingpartial effectin

linear regression model.


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lecture 05_addendum_chisquare,t and f distributions

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