• Find a pivotal quantity g(X,θ). For the needand understanding of asymptotic theory, we consider an example. conﬁdence interval is valid for any sample size. So the result gives the “asymptotic sampling distribution of the MLE”. This theorem states that the sum of a series of distributions converges to a normal distribution: a result that is independent of the parent distribution. Now a really interesting thing to note is that an estimator can be biased and consistent. 2. y x E Var i n. i ii i This is why in some use cases, even though your metric may not be perfect (and biased): you can actually get a pretty accurate answer with enough sample data. So if a parent distribution has a normal, or Bernoulli, or Chi-Squared, or any distribution for that matter: when enough estimators of over distributions are added together, the result is a normal. We know from the central limit theorem that the sample mean has a distribution ~N(0,1/N) and the sample median is ~N(0, π/2N). I'm working on a school assignment, where I am supposed to preform a non linear regression on y= 1-(1/(1+beta*X))+U, we generate Y with a given beta value, and then treat X and Y as our observations and try to find the estimate of beta. (a) Find the asymptotic joint distribution of (X(np),X(n(1−p))) when samplingfrom a Cauchy distributionC(µ,σ).You may assume 0

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