Search Results

No, you would still use the supremum, so in that case the p-value would be: $$p(x)=\sup_{\theta \in \Theta_0} P_{\theta}(W(X) \leqslant W(x))$$

You can find a closed-form matrix method to evaluate the Kolmogorov distribution in Marsaglia et al (2003) and an improved algorithm in Carvalho (2015). This method writes the CDF of the Kolmogorov d …

Yes, it is meaningful Under the stated hypotheses, your p-value function is: $$p_0(x) = \mathbb{P}(X \geqslant x | f_0).$$ If you were to reverse the hypotheses, then your p-value function would be: $ …

The correct interpretation of a p-value is the conditional probability of an outcome at least as conductive to the alternative hypothesis as the observed value (at least as "extreme"), assuming the nu …

To get the p-value, we have to either know or estimate/approximate the null distribution of the test statistic. Different types of hypothesis tests use different test statistics and have different nu …

The p-value is computed with the sampling distribution Given a parameter $\theta \in \Theta$, an observed data vector $\mathbf{x}$ and a test statistic $T$ that is increasing with respect to evidence …

The CI can have any limits, but it is centered exactly around zero For a two-sample T-test (testing for a difference in the means of two populations), a p-value of exactly one corresponds to the case …

I do not see any error in your calculations and I am getting the same results as you when I run the various tests at issue (see below for closest result to their reported result). I think the most li …

There is a simple formula for the posterior probability of incorrect/correct inference when rejecting the null One can obtain a simple and elegant posterior result by conditioning on the result of the …

The answer to this question is what defines the particular test But how do we define what is more extreme? This choice is really the essence of what defines the particular hypothesis test under use. …

To see if the test is "good" you need to analyse the properties of the power function As you point out, it is possible to derive a test statistic ---and thereby derive the "evidentiary ordering" defin …

That is the correct definition for a test with a simple null hypothesis. For a test with a composite null hypothesis (i.e., more than one possible parameter value in the null space) things are compli …

There is no contradiction between the observation in the QQ plot and the result of the test. Your stock returns have excess kurtosis compared with the normal distribution and this is why the test is …

These values are order statistics, so they follow standard distributional results for those types of statistics. In the case where you have random vectors $\mathbf{X}_1, ..., \mathbf{X}_n \sim \text{ …

There are an awful lot of issues raised in your question, so I will try to give answers on each of the issues you raise. To frame some of these issues clearly, it is important to note at the outset t …