Simple Linear Regression 1. Let i, y2n be a set of response variable. A probabil
ID: 3067312 • Letter: S
Question
Simple Linear Regression 1. Let i, y2n be a set of response variable. A probabilistic linear model is defined as The estimated line of means is defined as The line that minimizes the sum of squres of the observed values of y from those predicted is the best fitting line. Sum of squares of deviation is defined as TL TL yia- (a) Differentiate SSE with respect to a and b respectively. (b) Solve the equations obtained in part a. (c) Verify that the least-square estimation of and A and ?? are b-sex and a-v-br.Explanation / Answer
Solution
Let for convenience of presentation, S represent SSE
= Sum(i = 1 to n)(yi - a – bxi)2 ……………………………………………………………(1)
The estimates, a and b will be least squares estimates, if S is minimum.
Now, partial derivative of S w.r.t.a
= Sum(i = 1 to n)(- 2)(yi - a – bxi).
Equating this to zero and solving for a,
na = Sum(i = 1 to n)yi - bSum(i = 1 to n)xi or by dividing both sides by n, a = ybar - bxbar …………………………………………………………………………….(2)
(2) in (1):
S = Sum(i = 1 to n)(yi - ybar + bxbar – bxi)2 = Sum(i = 1 to n){(yi – ybar) – b(xi – xbar)}2 Partial derivative of S w.r.t b
= Sum(i = 1 to n)(- 2){(yi – ybar) – b(xi – xbar)}(xi – xbar)
= (- 2)Sum(i = 1 to n){(yi – ybar)(xi – xbar)} +2b.Sum(i = 1 to n){(xi – xbar)2}…………(3)
Equating (3) to zero and solving for b, b = Sum(i = 1 to n){(yi – ybar)(xi – xbar)}/Sum(i = 1 to n){(xi – xbar)2}……………(4) Or,
If Sum(i = 1 to n){(yi – ybar)(xi – xbar)} = Sxy and Sxx = Sum(i = 1 to n){(xi – xbar)2
b = Sxy/Sxx ……………………………………………………………………………….(4)
Thus, least squares estimators of ? and ? are: a = ybar – bxbar and b = Sxy/Sxx
DONE
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