0
can be written, where each Qi is a sum of squares of linear combinations of the Us. Further suppose that
where ri is the rank of Qi. Cochran's theorem states that the Qi are independent, and each Qi has a chi-squared distribution with ri degrees of freedom.[citation needed]
Here the rank of Qi should be interpreted as meaning the rank of the matrix B(i), with elements Bj,k(i), in the representation of Qi as a quadratic form:
Less formally, it is the number of linear combinations included in the sum of squares defining Qi, provided that these linear combinations are linearly independent.
ExamplesSample mean and sample varianceIf X1, ..., Xn are independent normally distributed random variables with mean μ and standard deviation σ thenis standard normal for each i. It is possible to write
(here, summation is from 1 to n, that is over the observations). To see this identity, multiply throughout by and note that
and expand to give
The third term is zero because it is equal to a constant times
and the second term has just n identical terms added together. Thus
and hence
Now the rank of Q2 is just 1 (it is the square of just one linear combination of the standard normal variables). The rank of Q1 can be shown to be n − 1, and thus the conditions for Cochran's theorem are met.
Cochran's theorem then states that Q1 and Q2 are independent, with chi-squared distributions with n − 1 and 1 degree of freedom respectively. This shows that the sample mean and sample variance are independent. This can also be shown by Basu's theorem, and in
Wiki User
∙ 11y ago
Add your answer:
Earn +20 pts
State and prove Arzela Ascoli theorem?
I will give a link that explains and proves the theorem.
How do you prove theorem 3.6.1?
Theorem 8.11 in what book?
Prove Hall's theorem from Tutte' theorem graph theory?
in this theorem we will neglect the given resistance and in next step mean as second step we will solve
State and prove stokes theorem?
You can find an introduction to Stokes' Theorem in the corresponding Wikipedia article - as well as a short explanation that makes it seem reasonable. Perhaps you can find a proof under the links at the bottom of the Wikipedia article ("Further reading").
Select the postulate or theorem that could be used to prove QRT STR?
Well, this will depend on the length of the sides of the triangle for what postulate or theorem you will be using.
State and prove Arzela Ascoli theorem?
I will give a link that explains and proves the theorem.
State and prove fundamental theorem of cyclic groups?
..?
Can a corollary be used to prove a theorem?
Yes, the corollary to one theorem can be used to prove another theorem.
How do you prove theorem 3.6.1?
Theorem 8.11 in what book?
How do you find the standard form after you have used the De Moivre's theorem?
(cos0 + i sin0) m = (cosm0 + i sinm0)
How do you solve a remainder theorem?
You cannot solve a theorem: you can prove the theorem or you can solve a question based on the remainder theorem.
What theorem can you use to prove that AEB is congruent to CED?
asa theorem
What elements are necessary for a geometric proof?
A theorem to prove. A series of logical statements. A series of reasons for the statements. answer theorem to prove
What theorem is used to prove a segment is a bisector?
A segment need not be a bisector. No theorem can be used to prove something that may not be true!
which congruence postulate or theorem would you use to prove MEX?
HL congruence theorem
Prove theorem 11.5 for rectangles?
Q.e.d.
State and prove convolution theorem for fourier transform?
Convolution TheoremsThe convolution theorem states that convolution in time domain corresponds to multiplication in frequency domain and vice versa:Proof of (a):Proof of (b):
