I have been spending the last few months trying to think of a math example to show the importance of correlation.
Specifically, I want to show that : Given some random observations from some correlated data generating process (e.g. coin flips, dice rolls, etc.) - estimating the parameters of this data generating process without accounting for the correlation results in worse parameter estimates (e.g. probability of heads, probability of rolling a 3) compared to estimation when accounting for the correlation.
Ideally, I want to able to show this theoretically and then empirically via random simulation.
The main idea I had was to consider a Stochastic Process - in a Stochastic Process, the probability of a random variable assuming a certain value is not independent (i.e. correlated) of the previous value. Thus, I thought that if I could think of such an example, I could show that estimating parameters using random data from this stochastic process without taking into consideration correlation will be worse than had correlation been taken into consideration (e.g. when taking into consideration correlation, mean of estimated parameters is closer to the actual value ... confidence intervals i.e. variance estimates have better coverage of the actual parameter values).
Over the past few months, here are some examples that I have thought of (I am still in the process of thinking of one more problem involving correlated coin flips):
- Average Result of Dividing Two Two Dice Rolls?
- Designing a Proof by Simulation
- Variance for the Sum of 2 Random Variables
- Probability Distribution of the Sum of 2 Dependent Dice
- Creating a Simulation to Disprove the Law of Large Numbers
- When is the Law of Large Numbers not True?
- Creating an Example to Disprove the Central Limit Theorem?
But I have been told that this examples are not quite capturing the essence of my problem.
Can someone help me think of math problem to illustrate this concept?
Thanks!
