Then, you’ll recall that the sample average, x* is the most efficient unbiased estimator of μ. Suppose that the underlying population is N, where σ 2 is known, and we obtain a sample of n observations using simple random sampling. Let’s consider a simple, specific, example. Finally, we have to be able to precisely specify the null and alternative hypotheses ( in advance of seeing the sample data), and these hypotheses have to relate to some parameter(s) in the population.If we mess up with either of these two things, then the sampling distribution of our test statistic will differ from what we think it is, and the probabilities that are associated with the implementation of Step 6 will be wrong. In reality, much (most) of the data that we use have been obtained using “complex surveys”, involving multi-level cluster sampling and stratified sampling. Usually, simple random sampling is assumed. Second, the type of sampling that we use is crucial.(Yes, we can think about our test’s “robustness” to these assumptions, but that’s a different issue.) For instance, we might assume that the population values follow a Normal distribution, with a known variance, but a mean that is unknown and whose value we’re testing. First, notice that there’s this idea that we know (or can reasonably assume) quite a lot about the underlying population.However, there are some key aspects to this hypothesis testing story that I’ve highlighted above, and which are worth remembering when we consider (below) a different possible way of proceeding. Now, you probably didn’t need to be reminded of all of this. The answer to this question is, of course, the familiar p-value. Or – by asking, “if the null hypothesis really were true, how likely is it that I’d see a value for my test statistic that’s as extreme (or more extreme) than what I’ve actually observed here (again, taking into account the assumptions about the underlying population and the sampling method that was used). If not, we would not reject the null hypothesis. If it is surprising, we would reject the null hypothesis.Then, we ask – “if in fact the null hypothesis were actually true, is the value of our test statistic “surprising”, or not?.Typically, this estimator would have to be transformed ( e.g., “standardized”) to make it “pivotal” – that is, having a sampling distribution that does not depend on any other unknown parameters. Usually, this would be a statistic that had already been found to be a “good” estimator of the parameter under test. We combine the sample values into a single statistic.For example, we might use simple random sampling, so that all sample values are mutually independent of each other. Then we take a carefully constructed sample from the population of interest. We call this the “alternative hypothesis”. We also state clearly what situation will prevail if the hypothesis to be tested is not true.We want to test the validity of statement (“null hypothesis”) about a parameter associated with a well-defined underlying population.The whole procedure would have been presented, more or less, along the following lines: It would have just been “statistical hypothesis testing”. It probably wasn’t described to you in so many words. I’m sure that the first exposure that you had to this was actually in terms of “classical”, Neyman-Pearson, testing. When you took your first course in economic statistics, or econometrics, no doubt you encountered some of the basic concepts associated with testing hypotheses. This might seem a bit redundant, but it will help us to see how permutation tests differ from the sort of tests that we usually use in econometrics. Let’s begin with some background discussion to set the scene.
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