Lessons from Statistics

This is prolly going to be the longest post (over time). I will add whatever clicks to me as I learn stuff while studying actuarial science.

  • MSE is just the variance in the modelling perspective #fact-check . When you read "standard error", it is just standard deviation.
  • We often come across statements like "Using more parameters gives an advantage in fitting but a disadvantage in prediction". The reason for that is for every parameter that we wish to estimate we bring in a certain amount for variance (inherent to the estimation process). So, the more the number of parameters that we wish to estimate, the more flexible our estimates will be but the flexibility leads to more variability.
    • The variance is not just added up, but have a nested dependency. For example, if the \(X\) follows a Poisson distribution and the \(\mu = E[X]\) follows another distribution (say Weibull) which has its own parameters to be estimated from the data, \(\theta\) and \(\omega\). The variance of the these parameters will exaggerate the variance of the parameter \(\mu\). Nested Variance!

  • What is \(\alpha\) in hypothesis testing? It is the false-positive rate. Say, we were trying to find out if correlation exists? So we set the null hypothesis like so:

    \[ \begin{align} H_{0}&: \text{Correlation doesn't exist} \\ H_{1}&: \text{It exists!} \end{align} \]
    • We then take a sample from the population which is assumed to have \(H_{0}\) and we calculate a test statistic based on that assumption.
    • There are two ways in which this can go. Say we set our \(\alpha = 10\%\)
      • Population actually has \(H_{1}\) (in which case we hope that the sample gives evidence against \(H_{0}\))
      • Population actually has \(H_{0}\). But there is a \(10\%\) chance that the sample drawn from the population will betray the population and show evidence against \(H_{0}\). And if we believe it, its a false positive.
    • To sum it up. \(\alpha\) is just the probability that we will draw a sample from the population that will make use falsely reject \(H_{0}\). And we have to be ready to take that degree of risk. That is why it is suggested to take multiple samples if possible to perform these tests (think about it, if you perform the hypothesis on \(k\) samples, your \(\alpha_{k} = \alpha^k\))

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