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Karteikarten

Regression - Chapter Three

Cara Wales Englisch 18 Karten

Karten in diesem Stapel (18)

  1. 1Formula to remember when calculating the expectation of B0hat and B1hat: a) E(a) = ? b) E(aX) = ? c) E(X+Y) = ?a) E(a) = a where a is a constant
    b) E(aX) = aE(X) where a is a constant
    c) E(X+Y) = E(X) + E(Y)
  2. 2E(Yi) = ? E(Ybar) = ? PROOF ON SEPARATE SHEET!!E(Yi)= B0 + B1xi
    E(Ybar) = B0 + B1xbar
  3. 3E(B1hat) = ? E(B0hat) = ? PROOF ON SEPARATE SHEET!!E(B1hat) = B1
    E(B0hat) = B0
  4. 4Var(a) = ? Var(aX) = ? Var(X+Y) = ? Var(X-Y) = ? If X and Y are independent, then Var(X+Y)=? and Var(X-Y)=?Var(a) = 0
    Var(aX) = a^2Var(X)
    Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)
    Var(X-Y) = Var(X) + Var(Y) - 2Cov(X,Y)
    Var(X+Y) = Var(X)+Var(Y)
    Var(X-Y) = Var(X) + Var(Y)
  5. 5Cov(X,X) = ? Cov(aX,bY) = ? Cov(X+Y,Z) = ? Cov(X-Y,Z) = ? Cov(SIGMA(i=1,n)[aiXi],SIGMA(i=1,n)[biXi) =? If the Xi are uncorrelated random variables i.e. Cov(Xi,Xj) = ? for i=/jCov(X,X) = Var(X)
    Cov(aX,bY) = abCov(X,Y)
    Cov(X+Y,Z) = Cov(X,Z) + Cov(Y,Z)
    Cov(X-Y,Z) = Cov(X,Z) - Cov(Y,Z)
    Cov(SIGMA(i=1,n)[aiXi],SIGMA(i=1,n)[biXi) = SIGMA(i=1,n)[aibiVar(Xi)
    Cov(Xi,Xj) = 0 for i=/j
  6. 6Var(Yi) = ? Cov(Yi,Yj) = ? Var(Ybar) = ? PROOF ON SEPARATE SHEET!!Var(Yi) = sigma squared
    Cov(Yi,Yj) = 0 for i=/j
    Var(Ybar) = sigma squared/n
  7. 7Var(B1hat) = ? Cov(Ybar,B1hat) = ? Var(B0hat) = ? PROOF ON SEPARATE SHEET!!Var(B1hat) = sigma squared/Sxx
    Cov(Ybar,B1hat) = 0
    Var(B0hat) =
    sigma squared[1/n + xbar^2/Sxx]
  8. 8Since B0hat and B1hat are linear combinations of Normal random variables, the Yi's must then have ... Hence: B0hat ~N(?,?) B1hat ~ N(?,?)... a normal distribution
    B0hat ~N(B0,sigmasquared(1/n + xbar^2/Sxx))
    B1hat ~N(B1, sigmasquared/Sxx)
  9. 9Using standard covariance rules we have that Cov(B0hat,B1hat) = ? So B0hat and B1 hat are not ... variables, because ... So if B1hat is overestimated then ...Cov(B0hat,B1hat) = [xbar*simga^2]/Sxx
    independent random ... sigma^2 and Sxx are always positive, thus if xbar>0 which is normally the case the covariance will be negative
    ... B0hat is underestimated and vice versa
  10. 10If Bhat = (B0hat B1hat)^T then the variance-covariance matrix Var(Bhat) = ? Variance-covariance matrices are always what 3 things?Var(Bhat) =Var(B0hat) Cov(B0hat,B1hat)
    Cov(B0hat,B1hat) Var(B1hat)
    = sigma^2[(1/n + xbar^2/Sxx) -xbar/Sxx
    -xbar/Sxx 1/Sxx]
    1) symmetric 2) have non-negative diagonal elements 3) are positive definite (i.e. all eigenvalues are positive)
  11. 11A 95% confidence interval for B1 is used to? If we know sigma the 95% confidence interval for B1 is? If we dont know sigma we need to put in the formula sigmahat = ? Then we have to use what instead of 1.96?To find the range that we would expect B1 to lie in.
    (B1hat - 1.96[sigma/sqrt(Sxx)],B1hat +1.96[sigma/sqrt(Sxx)])
    sigmahat = s = sqrt(RSS/[n-2])
    t-tables as a normal distribution is not assumed
  12. 12So our new 95% confidence interval is? What does it mean if 0 is not in the confidence interval?(B1hat - t(0.975,n-2)[s/sqrt(Sxx)],B1hat + t(0.975,n-2)[s/sqrt(Sxx)]
    If 0 is not in the confidence interval this implies that 0 is not a plausible value and can be suggested that there is a relationship between x and y
  13. 13We can also test the null hypothesis which we denote as? If the null hypothesis that B1=0 is true then ... If we estimate sigma by sigmahat = s then if the null hypothesis is true then ..H0: B1=0 versus H1: B1 =/0
    ... B1hat/[sigma/sqrt(Sxx)]~N(0,1)
    ... B1hat/[s/sqrt(Sxx)]~t(n-2)
  14. 14If we want to use our regression equation to estimate the mean value of Y at a particular x-value xp, we need to change the equation to what?yphat = B0hat +B1hatxp
  15. 15So E(Yphat) = ? Var(Yphat) = ? PROOF ON SEPARATE SHEET!!E(Yphat) = B0+B1xp
    Var(Yphat) = sigma^2[1/n + (xp-xbar)^2/Sxx)
  16. 16The 95% confidence interval for Yphat becomes ?(Yphat - t(0.975,n-2)[sqrt(Var(Yphat))],Yphat + t(0.975,n-2)[sqrt(Var(Yphat))])
  17. 17If we want to get an idea of what possible values y could take at a particulat x-value we change our regression line to what? Where epsilonp ~N(?,?) so E(epsilonp) = ? E(Y*) = ? Var(Y*) = ?Y* = B0 + B1xp + epsilonp
    where epsilonp ~N(0,sigmasquared) so E(epsilonp) = 0
    E(Y*) = B0 + B1xp
    Var(Y*) = Var(Yphat) + Var(epsilonp) = sigma^2[1/n + (xp-xbar)^2/Sxx] + sigma^2 = sigma^2[1 + 1/n + (xp-xbar)^2/Sxx]
  18. 18So our 95% confidence interval for Y* is?(Yphat - t(0.975,n-2)[sqrt(Var(Y*hat))], Yphat + t(0.975,n-2)[sqrt(Var(Y*hat))]

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Karteikarten am Regression - Chapter Three, erstellt von Cara Wales am 24/05/2016.
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