Introduction
Hierarchical composite endpoints (HCE) are complex endpoints combining events of different clinical importance into an ordinal outcome that prioritize the most severe event of a patient. Up to now, one of the difficulties in interpreting HCEs has been the lack of proper tools for visualizing the treatment effect captured by HCE.
Introduction
Suppose data consisting of i.i.d. observations \(X^n=(X_1,X_2,\cdots,X_n)\) are available from a distribution \(F(x,\theta),\,\theta\in\Theta\subset\mathbf{R}.\) The exact value \(\theta\) corresponding to the distribution that generated the observations is unknown. The problem is, using the available data \(X^n,\) construct tests for making decisions on the possible value of unknown parameter \(\theta\).
CEO Salary Estimation Problem
Consider the following problem. An investigative reporter wants to figure out how much salary makes the CEO of an investment bank X. For this he conducts interviews with some of the employees of that bank and writes down their salaries, which forms the following sample
Continuous distributions
Mann-Whitney estimate for the win probability
Consider two independent, continuous RVs (random variables) \(\xi\) and \(\eta\). The following probability is called the win probability of RV \(\eta\) against the RV \(\xi\)
Introduction
This is a short reminder of some simple properties of exponential distributions.
The continuous random variable (RV) \(\xi\) has an exponential distribution with the rate \(\lambda>0\) if its CMD (cumulative distribution function) has the following form
Consistent Estimators
In the estimation problem of a one-dimensional parameter \(\theta\) from an i.i.d. sample \[X^n=(X_1,\cdots,X_n),\ \ X_i\sim F(x,\theta),\,\theta\in\Theta\subset R\]
we introduced the notion of consistency of an estimator.
This means that whatever the unknown value of \(\theta\) is, this estimator is going to be close to that value in higher probability as \(n\) increases
\[\hat\theta_n\approx\theta, \text{ for large } n.