Question

Problem 4. The data {(1, 10),(2, 5.49),(3, 0.89),(4, −0.14),(5, −1.07),(6, 0.84)} comes from a model F(x) = (r/ x )+ sx. Use least squares to estimate the parameters r, s.

Answer 1

Given

Data Points: {(1, 10), (2, 5.49), (3, 0.89), (4, -0.14), (5, -1.07), (6, 0.84)}

In tabulated form:

i	1	2	3	4	5	6
	1	2	3	4	5	6
	10	5.49	0.89	-0.14	-1.07	0.84

Model:  

Solution

Let the fitted values be

Now consider the residues,

Then, using least squares method, our aim is to minimize

To get the least square estimate, i.e., the least value of Q, we should ensure that:

And

Now,

And,

Solving, equations (1) and (2), we get:

At these values of r and s, the partial second order derivatives should be positive to ensure that Q is in fact a minimum value.

Computing the three partial second order derivatives:

  

As required, all these are positive.

Hence, the values of r and s are: