Yes, it is possible to estimate the model parameters through the available observations provided that the model conditions are provided

I wish you success

@ Mohamed Abdalmoaty ,

The following is the reasonable assumption regarding the error term:

" Errors ei , (i = 1 , 2 , 3 , ......... . n) are independently and identically distributed following normal distribution with zero expectation and unknown variance."

Dear Dhritikesh Chakrabarty

I assumed that you want to minimize the errors e_{i}=Y_{i} - A - BX_{i}.

Let S(A,B)=∑(e_{i})²=∑(Y_{i}-A-BX_{i})²

Then Solve

((∂S)/(∂A))=0,((∂S)/(∂B))=0

We obtain ∑(Y_{i}-A-BX_{i})=0...(1)

∑X_{i}(Y_{i}-A-BX_{i})=0..(2)

implies

∑(Y_{i})-nA-B∑X_{i}=0..(1)

∑X_{i}Y_{i} - A∑X_{i} - B∑(X_{i})² = 0..(2)

Now, we solve 2 equations in 2 unknowns A and B.

All summations run from 1 to n.

Best regards

@ Issam Kaddoura ,

I do not want to minimize the errors . But I want the actual value of the parameter.

In other wards, I want that estimate of the parameter which is free from error.

Dhritikesh Chakrabarty

Free of errors means ei equals zero for all i. And the question has a positive answer only for exactly two observations

(X1 ,Y1 )and (X2 ,Y2 ), where A and B can be determined by solving two equations in two unknowns.

If the number of observations is more or less than two, with different points data, then a solution for exact A and B cease to exist.

@ Issam Kaddoura ,

I have mentioned that the estimates of A & B are to be free from errors.

This does not mean that ei equals zero for all i.

Dhritikesh Chakrabarty

This is not consistent. Free of errors means ei =0 for all i.

The meaning of (estimates of A and B) is ( not exact values)

So the proposed question is in question!!

@ Issam Kaddoura ,

Estimate of a parameter does not mean the actual value or the true value of the parameter.

Estimate of a parameter, usually, is a value which is not equal to the true value of the parameter.

In other words, estimate of a parameter, usually, is a value which suffers from some error.

If this error can be made zero then estimate will be equal to the true value of the parameter.

I want this error-free estimate.

Thus, the error-free estimate ( i.e. the estimate free from error) does not mean that ei =0 for all i.

It means that the error involved in the estimate is zero.

@ Issam Kaddoura , As I know, if the number of observations is two then there is unique mathematical solution.

On the other hand If the number of observations is either more than or less than two then there is no mathematical solution.

Thus, I agree with your statement

"" If the number of observations is more or less than two, with different points data, then a solution for exact A and B cease to exist. "

That is why it is a challenging problem to determine the actual values of the parameters, from a set of observations on (X , Y) on which I am interested.

1) Observations are not error-free in the real world

2) Actual value has no meaning in the real world. You can get to a physical parameter x through a measurement m(x), with a precision dm(x).

Your problem has two parameters A and B.

In mathematical terms, there is a one to one mapping between (A, B) and the line (geometrical object) y=Ax+B.

What you are describing is a question of "best fit" of such line to cloud of points (xi, yi). It's an approximation with a precision (sum of squares of errors, if you use "least square fitting").

You find all details of above in Wikipedia.

@ Renaud Di Francesco ,

I am of the same view that the observations are not error-free in the real world.

However, it is difficult for me to understand that actual value has no meaning in the real world. I think, though may not be correct, that actual value exists and the endeavour/attempt of a researcher is to find out that actual value. Objective of research, though may not be completely successful, is to go for searching it.

The presence of random error arises due to uncontrolled fluctuations in factor levels of those process variables which are uncontrolled/uncontrollable during the experimentation process, therefore to eliminate this error, it is essential the during the experiment, both variables of interest and so called all other noise variables be at precise fixed levels, under such condition it is possible collect data without random error contribution and estimate model parameters free of their contribution.

@ Debopam Ghosh,

The cause/source of random error, in practice, is uncontrollable and the random error is not measurable.

I like to point out one more reality that the cause/source of random error is unknown. If the the cause and/or the source of an error is known, then this error no longer remains as random error. It, in this case, automatically becomes assignable error since the scope becomes available in this case for controlling the cause/source of the error.

Since the cause/source of random error is unknown therefore it is not possible to collect data which are free from random error irrespective of the methods used in collection of data. Note that the collection work also may suffer from random error.

Do not forget physics, and at the root of all, quantum physics, and the Heisenberg uncertainty principle. Measurements cannot be error-free, that's what physics says.

Errors or uncertainty can be studied as random variables. Another equivalent formulation is that statistical observations can be modeled by random variables, if you prefer the language of statistics.

Dera Renaud Di Francesco ,

It is a reality that measurements cannot be error-free .

I like to point out that tool used in measurement, experimenter, experimenting material, environment etc. cannot be free from error. These are the reasons why the data are not free from error in reality.

If it is not possible to determine the value of the parameters A and B accurately , then is it possible to determine the values of A and B, in this situation, which are almost accurate ?

In the statistical world, it is not possible to determine the values of A and B which are accurate. We can only estimate the values of A and B using various methods and criteria

Dear Dreamlee Sharma ,

You have mentioned, in the current light of statistics, that it is not possible to determine the values of A and B which are accurate and that one can only estimate the values of A and B using various methods and criteria.

Can you think, in the light beyond the statistical world, of some solution of the problem.

It is not binding that the solution of a problem must always in the light of statistical philosophy.