# hypothesis testing

## Neyman-Pearson and some other Uniformly Most Powerful Tests

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 $$\theta$$.