Let the joint distribution of (X, Y) be bivariate normal with mean vector ( <mtable r

orlovskihmw 2022-07-03 Answered
Let the joint distribution of (X, Y) be bivariate normal with mean vector ( 0 0 ) and variance-covariance matrix
( 1 𝝆 𝝆 1 ) , where βˆ’ 𝟏 < 𝝆 < 𝟏 . Let 𝚽 𝝆 ( 𝟎 , 𝟎 ) = 𝑷 ( 𝑿 ≀ 𝟎 , 𝒀 ≀ 𝟎 ) . Then what will be Kendall’s Ο„ coefficient between X and Y equal to?
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Answers (1)

tilsjaskak6
Answered 2022-07-04 Author has 14 answers
Step 1
Originally, Kendall's tau, also called rank correlation, is a statistical measure that can be applied to a discrete set of observed data.
In the more recent literature about dependency modelling with Copulas which became popular in mathematical finance the following definition of Kendall's tau is given.
Let Φ ρ ( x , y ) , Φ ( x ) , Φ ( y ) be the bivariate and the univariate CDFs of the standard normal distribution. Then the Gaussian Copula is defined as
C ρ ( x , y ) = Ξ¦ ρ ( Ξ¦ βˆ’ 1 ( x ) , Ξ¦ βˆ’ 1 ( y ) )
Kendall's tau is then defined as
ρ Ο„ = E [ s i g n [ ( X βˆ’ X ~ ) ( Y βˆ’ Y ~ ) ] ] = P [ ( X βˆ’ X ~ ) ( Y βˆ’ Y ~ ) > 0 ] βˆ’ P [ ( X βˆ’ X ~ ) ( Y βˆ’ Y ~ ) < 0 ] .
where (X, Y) is bivariate standard normal, and ( X ~ , Y ~ ) has the same distribution but is independent of (X, Y). It can be shown (see (1) and duplicate) that
ρ Ο„ = 4 ∫ 0 1 ∫ 0 1 C ρ ( x , y ) d C ρ ( x , y ) βˆ’ 1 = 2 Ο€ arcsin ⁑ ρ .

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New questions

Question on designing a state observer for discrete time system
I came through this problem while studying for an exam in control systems:
Consider the following discrete time system
x β†’ ( k + 1 ) = A x β†’ ( k ) + b u β†’ ( k ) , y β†’ ( k ) = c x β†’ ( k )
where b = ( 0 , 1 ) T , c = ( 1 , 0 ) , A = [ 2 1 0 βˆ’ g ] for some g ∈ R
Find a feedback regulation (if there is any) of the form u ( k ) = βˆ’ K x ^ ( k ) where x ^ ( k ) is the country estimation vector that is produced via a linear complete-order state observer such that the nation of the system and the estimation blunders e ( k ) = x β†’ ( k ) βˆ’ x ^ ( k ) go to zero after a few finite time. layout the kingdom observer and the block diagram.
My method
it is clean that the eigenvalues of the machine are Ξ» 1 = 2 , Ξ» 2 = βˆ’ g (consequently it is not BIBO solid) and that the pair (A,b) is controllable for every fee of g, as nicely a the pair (A,c) is observable for all values of g. consequently we will shift the eigenvalues with the aid of deciding on a benefit matrix okay such that our device is strong, i.e. it has its eigenvalues inside the unit circle | z | = 1.
The state observer equation is
[ x β†’ ( k + 1 ) e β†’ ( k + 1 ) ] T = [ A βˆ’ b K B k O A βˆ’ L C ] [ x β†’ ( k ) e β†’ ( k ) ] T
With characteristic equation
Ο‡ ( z ) = | z I βˆ’ A + b K | | z I βˆ’ A + L C | = Ο‡ K ( z ) Ο‡ L ( z )
Also consider
K = [ k 1 k 2 k 3 k 4 ]
and let a = k 1 + k 3 , Ξ² = k 2 + k 4
Then Ο‡ K ( z ) = ( z βˆ’ 2 ) ( z + g + Ξ² ) + a.
So we can select some eigenvalues inside the unit circle and determine a , Ξ² in terms of g. Choosing e.g. Ξ» 1 , 2 = Β± 1 / 2 we get a = 3 g + 33 / 8 , Ξ² = 9 / 4 βˆ’ g , g ∈ R
Questions
I want to ask the following:
Is my approach correct? Should I select the eigenvalues myself since I am asked to design the observer or should I just solve the characteristic equation and impose | Ξ» 1 , 2 | < 1?
Should I determine L matrix as well since the error must also vanish? (because it is not asked)