Assignment I Questions - Applied Regression Analysis | STAT 333 | Assignments Statistics

Statistics 333 Assignment 1 Due Sept. 19, 2003

1. For the simple linear regression model Yi=

1Xi+

i,i=1, . . . , n,consider the least

squares fits ˆ

Yi=b0+b1Xiand residuals givenbye

i=Y

i−ˆ

i=Y

i−(b

0+b

),i=1, . . . , n.

Since b0=Y−b1X,itiseasy to see (1/n)Σn

i=1ˆ

Yi=Yand hence also that Σn

i=1(Yi−ˆ

Yi)=0.

Similarly,also showthat the residuals eisatisfy the additional ‘orthogonality’ constraint

Σn

i=1Xi(Yi−ˆ

Yi)=0.

2. For the simple linear regression model in Exercise 1, under the standard model assump-

tions for the random errors

i,showthat the least squares estimator b1=Sxy /Sxx and

Y=(1/n)Σn

i=1Yihave zero covariance, i.e., Cov( b1,Y)=0,where Sxy =Σn

i=1(Xi−X)Yi.

3. For the simple linear regression model, the estimator of the error variance

2=Var(

i)is

givenby

2=1

n−2

i=1

Σ(Yi−ˆ

Yi)2≡1

n−2SSE .

Showthat this estimator S2is unbiased for

2,i.e., prove that E(S2)=

Note: Toprove this result, use the following approach. First, verify the identity

Yi−(

1Xi)=[Yi−(b0+b1Xi)]+[Y−(

1X)]+[(b1−

1)(X

i−X)].

Then verify that the sum of squares of elements on the left-hand side,

Σn

i=1[Yi−(

1Xi)]2,isequal to the sum of the 3 sums of squares of individual elements

on the right-hand side, due to ‘orthogonality’ in cross-terms (Exer.1). Hence, verify that

Σn

i=1[Yi−(

1Xi)]2=SSE +n[Y−(

1X)]2+Sxx (b1−

1)2.

Finally,equate the expected values of both sides of the sums of squares relation, to get

2=E[SSE ] +nE[Y−(

1X)]2+Sxx E[(b1−

1)2], ‘evaluate’ the other expected

value terms, and solvefor E[SSE ].Also use definitions and known results for Var( b1)

etc., for instance, E{[Yi−(

1Xi)]2}=Var(Yi)≡

2by definition of variance.

4. Consider the zero or no intercept model givenbyY

,i=1, . . . , n,with the errors

ibeing independent, normal r.v.’ s with mean 0 and variance

i) Derive the least squares estimator b1of

1,and also derive the variance of this estimator.

ii) For an arbitrary fixed value X0of X,establish that a 100(1 −

)%confidence interval for

the mean response value E(Y|X0)=

1X0is givenby

0±t

(

/2)

n−1S√  

0/Σn

i=1X2

where S2=Σn

i=1(Yi−b1Xi)2/(n−1) provides an unbiased estimator of

2with n−1df.

Note: You can conclude that (b1−

1)/se( b1)has the tn−1distribution.

5. An experiment was conducted to study the mass of a tracer material exchanged between

the main flowofanopen channel and the "dead zone" caused by a sudden open channel

expansion. Researchers need this information to improve the water quality modeling capa-

bility of a river. Itisimportant to determine the exchange constant Kfor varying flow

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Statistics 333 Assignment 1 Due Sept. 19, 2003

1. For the simple linear regression model Y

= β 0

β 1

X

ε i

, i = 1,... , n , consider the least

squares fits

Y

= b 0

X

and residuals given by e

= Y

Y

= Y

− ( b 0

X

), i = 1,... , n.

Since b

= Y − b 1

X , it is easy to see (1/ n )

i = 1

Y

= Y and hence also that

i = 1

( Y

Y

Similarly, also show that the residuals e

satisfy the additional ‘orthogonality’ constraint

i = 1

X

( Y

Y

2. For the simple linear regression model in Exercise 1, under the standard model assump-

tions for the random errors ε

, show that the least squares estimator b

= S

/ S

and

Y = (1/ n )

i = 1

Y

have zero covariance, i.e., Cov( b

, Y ) = 0 , where S

i = 1

( X

− X ) Y

3. For the simple linear regression model, the estimator of the error variance σ

= Var( ε i

) is

given by

S

n − 2

i = 1

( Y

Y

≡

n − 2

SSE.

Show that this estimator S

is unbiased for σ

, i.e., prove that E ( S

) = σ

Note: To prove this result, use the following approach. First, verify the identity

Y

− ( β 0

β 1

X

) = [ Y

− ( b 0

X

) ] + [ Y − ( β

β 1

X ) ] + [ ( b

− β 1

) ( X

− X ) ].

Then verify that the sum of squares of elements on the left-hand side,

i = 1

[ Y

− ( β 0

β 1

X

) ]

, is equal to the sum of the 3 sums of squares of individual elements

on the right-hand side, due to ‘orthogonality’ in cross-terms (Exer. 1). Hence, verify that

i = 1

[ Y

− ( β 0

β 1

X

) ]

= SSE + n [ Y − ( β

β 1

X ) ]

+ S

( b 1

− β 1

Finally, equate the expected values of both sides of the sums of squares relation, to get

n σ

= E [ SSE ] + n E [ Y − ( β

β 1

X ) ]

+ S

E [ ( b

− β 1

] , ‘evaluate’ the other expected

value terms, and solve for E [ SSE ]. Also use definitions and known results for Var( b

etc., for instance, E {[ Y

− ( β 0

β 1

X

) ]

} = Var( Y

) ≡ σ

by definition of variance.

4. Consider the zero or no intercept model given by Y

= β 1

X

ε i

, i = 1,... , n , with the errors

ε i

being independent, normal r.v.’s with mean 0 and variance σ

i) Derive the least squares estimator b

of β

, and also derive the variance of this estimator.

ii) For an arbitrary fixed value X

of X , establish that a 100(1 − α )% confidence interval for

the mean response value E ( Y | X

) = β 1

X

is given by

b 1

X

± t

( α /2)

n − 1

S

X

i = 1

X

where S

i = 1

( Y

− b 1

X

/( n − 1) provides an unbiased estimator of σ

with n − 1 df.

Note: You can conclude that ( b

− β 1

)/ se( b

) has the t

n − 1

distribution.

5. An experiment was conducted to study the mass of a tracer material exchanged between

the main flow of an open channel and the "dead zone" caused by a sudden open channel

expansion. Researchers need this information to improve the water quality modeling capa-

bility of a river. It is important to determine the exchange constant K for varying flow

conditions. The value of K describes the exchange process when a dead zone appears. In

a study, values of the Froude Numbers ( N

) were used to predict K. Numbers are func-

tions of upstream channel velocity and water depth. The data collected were as follows,

with the negative sign of the K values indicating "flushing", the direction of mass transfer

out of the dead zone:

Obs N

K

These data are stored in the file: /u/r/e/reinsel/stat333/exchange.dat ; also on webpage.

Perform and show calculations for (ii)-(v) below by ‘direct calculations’ on a calculator or

computer; then use regression in Minitab or other software to confirm calculations.

i) Construct a scatter plot of K versus N

, and provide some relevant comments.

ii) Use the least squares method to fit the model K

= β 0

β 1

( N

ε i

iii) Compute S

, R

, and obtain the basic analysis of variance (ANOVA) table. Provide

some brief interpretation for these results, e.g., in terms of the amount of variation of the K

values explained by the fitted regression (i.e., the variable N

iv) Obtain the standard errors for the least squares estimates b

and b

, and under the usual

normal theory model assumptions, give a 95% confidence interval for β

v) Determine the explicit form for the 95% confidence interval of a mean response

E ( K | N

) = β 0

β 1

Assignment I Questions - Applied Regression Analysis | STAT 333, Assignments of Statistics

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Partial preview of the text

Download Assignment I Questions - Applied Regression Analysis | STAT 333 and more Assignments Statistics in PDF only on Docsity!

Statistics 333 Assignment 1 Due Sept. 19, 2003

1. For the simple linear regression model Y

X

, i = 1,... , n , consider the least

squares fits

Y

X

and residuals given by e

= Y

Y

= Y

X

), i = 1,... , n.

Since b

X , it is easy to see (1/ n )

Y

= Y and hence also that

( Y

Y

Similarly, also show that the residuals e

satisfy the additional ‘orthogonality’ constraint

X

( Y

Y

2. For the simple linear regression model in Exercise 1, under the standard model assump-

tions for the random errors ε

, show that the least squares estimator b

= S

/ S

and

Y

have zero covariance, i.e., Cov( b

, Y ) = 0 , where S

( X

− X ) Y

3. For the simple linear regression model, the estimator of the error variance σ

) is

given by

S

( Y

Y

SSE.

Show that this estimator S

is unbiased for σ

, i.e., prove that E ( S

Note: To prove this result, use the following approach. First, verify the identity

Y

X

) = [ Y

X

) ] + [ Y − ( β

X ) ] + [ ( b

) ( X

− X ) ].

Then verify that the sum of squares of elements on the left-hand side,

[ Y

X

) ]

, is equal to the sum of the 3 sums of squares of individual elements

on the right-hand side, due to ‘orthogonality’ in cross-terms (Exer. 1). Hence, verify that

[ Y

X

) ]

= SSE + n [ Y − ( β

X ) ]

+ S

Finally, equate the expected values of both sides of the sums of squares relation, to get

= E [ SSE ] + n E [ Y − ( β

X ) ]

+ S

E [ ( b

] , ‘evaluate’ the other expected

value terms, and solve for E [ SSE ]. Also use definitions and known results for Var( b

etc., for instance, E {[ Y

X

) ]