Does failure to reject the null in Neyman-Pearson approach mean that one should "accept" it? Based on Neyman-Pearson Hypothesis Testing Theory, will the failure to reject a null hypothesis imply that one should accept the null hypothesis?
(It would be absolutely absurd to have this claim if one adopts Fisher's null hypothesis testing theory, but would not it make sense if one chooses NPHT?)
 A: If "the Neymann–Pearson approach" is understood narrowly as extending Fisher's by introducing an alternative hypothesis in addition to the null hypothesis, then there's no motive to change the terminology. The alternative can influence only the choice of test statistic (by consideration of the test's power); once that choice has been made the distribution of the test statistic is calculated under the null. "Fail to reject" reflects this asymmetry between the null & alternative hypotheses†; you're provisionally assuming the null true till amassing enough evidence to the contrary. ("Retain" is an alternative way of putting it.) Pedagogues labour this nice semantic distinction in an attempt to avert the misconception that an "insignificant" result necessarily reflects a preponderance of evidence against the alternative. For example, consider a single observation from a Gaussian distribution with unit variance & unknown mean, $X\sim\mathcal{N}(\mu,1)$. With the point null & alternative $H_0: \mu=0$ vs $H_\mathrm{A}: \mu=1$, a test of size $0.05$ rejects the null only when $x > 1.64$, even though the alternative is better supported whenever $x > 0.5$.
If on the other hand you were to take the decision-theoretic framework of the NP approach seriously (see the excellent answer here), you just wouldn't bother to perform a test that was underpowered for your purposes; then talk of "accepting" the null hypothesis would seem a lot more sensible. Some notable expositors of testing theory from this viewpoint have apparently thought so. Lehmann & Romano (2005), Testing Statistical Hypotheses, unabashedly use "accept" & "reject" throughout. Casella & Berger (2002), Statistical Inference, use "accept" & "reject" too, even saying "We view a hypothesis testing problem as a problem in which one of two actions is going to be taken—the actions being the assertion of $H_0$ or $H_1$".&ddagger;
† An asymmetry exacerbated when the null's composite—either through specifying a range of parameter values, or because of a nuisance parameter that hasn't been removed through conditioning on an ancillary statistic— in which case it isn't rejected unless the test statistic is sufficiently extreme under any of its constituent simple nulls.
&ddagger; While Cox & Hinkley (1974) Theoretical Statistics, manage four chapters on testing without using "accept" or "reject", except once in scare quotes.
