Mathematical Statistics with Applications
Mathematical Statistics with Applications
7th Edition
ISBN: 9780495110811
Author: Dennis Wackerly, William Mendenhall, Richard L. Scheaffer
Publisher: Cengage Learning
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Chapter 12, Problem 37SE
To determine

Find the fractions that should be assigned to each value of x to minimize V(β^2).

Expert Solution & Answer
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Answer to Problem 37SE

The fractions of total number of observations that can be assigned at x=1,0and 1 to minimize V(β^2) are 0.25, 0.50, and 0.25, respectively.

Explanation of Solution

Calculation:

Consider that k1,k2,k3 are the fractions of total number of observations that can be assigned at x=1,0and 1, respectively.

Thus, it can be aid that if total number of observation is n that is large enough, then there are nk1 points at x=1, there are nk2 points at x=0, and there are nk3 points at x=1.

Now, the Design matrix X can be written as,

X=[111111......111100100......100111111..1..1..1]n×n

Now,

XX=[nn(k3k1)n(k1+k3)n(k3k1)n(k1+k3)n(k3k1)n(k1+k3)n(k3k1)n(k1+k3)]=n[1babababa]=nA, where a=k1+k3,b=k3k1.

Now, it is needed to minimize V(β^2)=σ2c22, where c33 is the (3×3) element of (XX)1.

The determination of matrix A is obtained as,

A=[1babababa]|A|=1|abba|b|bbaa|+a|baab|=a2b2b(abba)+a(b2a2)=a2b20+ab2a3=(k1+k3)2(k3k1)2+(k1+k3)(k3+k1)2(k1+k3)3=k12+k32+2k1k3k32k12+2k1k3+k13+k1k32+2k12k3+k3k12+k33+2k1k32k133k12k33k1k32k33=4k1k2k3

Now, the inverse of A is obtained as,

A1=1n|A|[(1)1+1|abba|(1)1+2|bbaa|(1)1+3|baab|(1)2+1|baba|(1)2+2|bbaa|(1)2+3|1bab|(1)3+1|baab|(1)3+2|1bab|(1)3+3|1bba|]=14nk1k2k3[a2b20b2a20aa2abbb2a2abbab2]

Now, the c22 element of A1 matrix is ab24nk1k2k3.

Hence, the V(β^2) can be written as,

V(β^2)=σ2ab24nk1k2k3=σ2n[k1+k3(k3k1)24k1k2k3]=σ2n[k1+k3[(k3+k1)24k1k3]4k1k2k3]=σ2n[(k1+k3)[1k1k3]4k1k2k34k1k34k1k2k3]=σ2n[(k1+k3)4k1k31k2]=σ2n[(k1+k3)4k1k311k1k3][as k1+k2+k3=1]

Now, it is needed to partial differentiate V(β^2) with respect to k1and k3 and equate to .

That is,

(V(β^2))k1=0(σ2n[(k1+k3)4k1k311k1k3])k1=04k12=(1k1k3)2............(1),

And

(V(β^2))k3=0(σ2n[(k1+k3)4k1k311k1k3])k3=04k32=(1k1k3)2..........(2).

As k1,k2,k3 are positive constants and by symmetry of k1=k3, the equation (1) can be written as,

4k12=(12k1)2k1=0.25

Similarly, k3=0.25.

Thus,

1k1k3=10.250.25=0.50.

Thus, the fractions of total number of observations that can be assigned at x=1,0and 1 to minimize V(β^2) are 0.25, 0.50, and 0.25, respectively.

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