Let ${Y_1,…,Y_n}$ be independent random variables and $Y_i$~$N(\beta x_i, 1)$ where $x_1,…,x_n$ are fixed known constants, and $\beta$ is an unknown parameter. I’m looking to find the p-value or rejection region for the test $$H_0: \beta=0 \quad \text{vs} \quad H_1:\beta\ne0$$ The Likelihood Ratio Test statistic $\Lambda$ is $$e^{-1/2\left(\frac{\sum_{i=1}^n y_i x_i}{\sum_{i=1}^n x_i^2}\right)^2\sum_{i=1}^n x_i^2}$$ I have already […]

I’m having trouble to find a UMP test after finding a MP test. Consider one observation $X$ from CDF $F_\theta(x) = x^\theta$ where $x \in [0, 1]$ and $\theta > 0$. I found the MP test for testing $H_0: \theta = 1$ against $H_1: \theta = 2$ with significance level $\alpha=0.05$ using the Neyman Pearson […]

Particular questions such as Why is $\pi$ so close to $3$? or Why is $\pi^2$ so close to $10$? may be regarded as the first two cases of the question sequence Why is $\pi^k$ so close to its nearest integer? For instance, we may stare in awe in front of the almost-unit $$\frac{\pi}{31^\frac{1}{3}}=1.000067…$$ or, in […]

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