I am learning Statistics in a foreign language, and I must admit I do not understand a certain subject and want to search and learn it in English but I do not know this method's name officially. Directly translated it is called the method of ratio values. This is what's about:
We are first given two hypothesis which are not trivial. For example, we know that our $X: \mathcal{N}(m,\mathfrak{S}_o^2)$ and we have two hypothesis' that are not trivial in this case: $H_0(m\leq m_o)$ and $H_1(m>m_0)$ .We will use the likelihood function:$L(m,x_1,x_2,...)$, and we have as
$m^*$- value of parameter $m$ for which the function $L(m,x_1,x_2,...), m\in\Lambda_0$ reaches it's maximum value.
$m^{**}$- value of parameter $m$ for which the function $L(m,x_1,x_2,...), m\in\Lambda$ reaches it's maximum value.
$\Lambda_0 \subset \Lambda$, in this case $\Lambda_0=[- \infty,m_0]$ this is determined by the first hypothesis:$H_0$
Then it goes on to say that the critical space is $$C=\{\frac{L(m^*,x_1,x_2,...)}{L(m^{**},x_1,x_2,...)}\leq c_\alpha:0<c_\alpha<1\}$$
And in this example it goes on to estimate $m^{**}=\overline{X_n}$ And say that
$m^*=m_0 \text{ for } m_0\leq \overline{X_n}$
and that
$m^*=\min\{m_0,\overline{X_n}\} \text{ for } m_0> \overline{X_n}$
then is goes on to say: $$\frac{L(m^*,x_1,x_2,...)}{L(m^{**},x_1,x_2,...)}=\begin{cases}e^{-\frac{1}{2\mathfrak{S}_o^2}[\sum(x_k-m_0)^2- \sum(x_k-\overline{X_n})^2] \text{ for } m_0< \overline{X_n} } \\ 1 \text{ for }m_0> \overline{X_n} \end{cases}$$
and using the above definition for critical space (or incorporating $\leq c$ ) ...the example goes on...
What is the name of this method??? (I just cannot find it) Using this method we find a test for these hypothesis'.
It's the likelihood ratio test.
See here