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May 3, 2020

How to Pursue Mathematics after High School?

For Students who are passionate for Mathematics and want to pursue it for higher studies in India and abroad.

This is a problem from ISI MStat Examination, 2019. This tests one's familiarity with the simple and multiple linear regression model and estimation of model parameters and is based on the Invariant Regression Coefficient.

The Problem- Invariant Regression Coefficient

Suppose \( \{ (x_i,y_i,z_i):i=1,2,…,n \} \) is a set of trivariate observations on three variables:\(X,Y,Z \),, where \(z_i=0 \) for \(i=1,2,…,n-1 \) and \(z_n=1 \).Suppose the least squares linear regression equation of \(Y \) on \(X\) based on the first \(n-1 \) observations is \( y=\hat{\alpha_0}+\hat{\alpha_1}x \) and the least squares linear regression equation of \(Y \) on \( X \) and \(Z \) based on all the \( n \) observations is \(y=\hat{\beta_0}+\hat{\beta_1}x+\hat{\beta_2}z \) . Show that $\hat{\alpha_1}=\hat{\beta_1}$.

Prerequisites

1.Knowing how to estimate the parameters in a linear regression model (Least Square sense)

2. Brief idea about multiple linear regression.

Solution

Based on the first \( n-1 \) observations, as \(z_i=0 \), so, we consider a typical linear regression model of \( Y \) on \( X \).

Thus,the least square estimate is given by \( \hat{\alpha_1}=\frac{\sum_{i=1}^{n-1} (x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{n-1} (x_i-\bar{x})^2} \)

And in the second case, we have:

\( y_1=\beta_0+\beta_1 x_1+\epsilon_1 \)

\( y_2=\beta_0+\beta_1 x_2+ \epsilon_2 \)

\( \vdots \)

\( y_n=\beta_{0}+\beta_1 x_n+\beta_2+ \epsilon_n \)

Thus, the error sum of squares for this model is given by:

\( SSE=\sum_{i=1}^{n-1} (y_i-\beta_0-\beta_1 x_i)^2+(y_n-\beta_1 x_n -\beta_0 -\beta_2)^2 \) , as \( z_n=1 \).

By differentiating SSE with respect to \( \beta_2 \), at the optimal value, we must have:

\( \hat{\beta}_2 = y_n -\hat{\beta_1}x_n-\hat{\beta_0} \)

That is, the last term of SSE must vanish to attain optimality.

So, it is again equivalent to minimize

\( \sum_{i=1}^{n-1} (y_i-\beta_0-\beta_1 x_i)^2 \) with respect to \( \beta_{0} ,\beta_{1} \)

This, is nothing but the simple linear regression model again and thus, \( \hat{\beta_1}=\hat{\alpha_1} \) and furthermore, \( \hat{\beta_0}=\hat{\alpha_0} \).

Food For Thought

Suppose you have two sets of independent samples. Let they be \( \{ (y_1,x_1), ...(y_{n_1},x_{n_1}) \} \) and \( \{ (y_{n_1 +1},x_{n_1 +1} ) ,...,(y_{n_1 + n_2} ,x_{n_1 + n_2} ) \} \).

Now you want to fit 2 models to these samples:

\(y_i=\beta_0 + \beta_1 x_i + \epsilon_i \) for \( i=1,2,..,n_1 \)

and

\(y_i=\gamma_0 + \gamma_1 x_i + \epsilon_i \) for \( i=n_1 +1 ,.. ,n_1 + n_2 \)

Can you write these two models as a single model?

After that ,considering all assumptions for linear regression to be true (If you are not aware of these assumptions you may browse through any regression book or search the internet), is it justifiable to infer \( \beta_1 = \gamma_1 \) ?

What to do to shape your Career in Mathematics after 12th? 

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

  • What are some of the best colleges for Mathematics that you can aim to apply for after high school?
  • How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
  • What are the best universities for MS, MMath, and Ph.D. Programs in India?
  • What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
  • How can you pursue a Ph.D. in Mathematics outside India?
  • What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Want to Explore Advanced Mathematics at Cheenta?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

To Explore and Experience Advanced Mathematics at Cheenta
Register here

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