Frequentist statistics

In frequentist statistics, probability is instead a long-run relative occurrence of some event out of an infinite number of repeated trials, where the event is a possible outcome of the trial. A hypothesis or parameter that expresses a state of nature cannot have a probability in frequentist statistics, because after an infinite number of experiments there can be no uncertainty in the parameter left. A hypothesis or parameter value is either... 

In frequentist statistics, by contrast, nuisance parameters are usually treated with point estimates, and inference on the parameter of interest is based on calculations with the nuisance parameter as a constant. This can result in large errors, because there may be considerable uncertainty in the value of the nuisance parameter. 

For the normal distribution there are analytical solutions allowing the assessment of both FA and HC using frequentist statistics. In contrast, Bayesian solutions are numerical. This highlights the flexibility of the Bayesian approach since it can easily deal with any distribution, which is not always possible with the frequentist approach. 

Aldenberg and Jaworska (2000) demonstrate that frequentist statistics and the Bayesian approach with noninformative prior results in identical confidence intervals for the normal distribution. Generally speaking, this is more the exception than the rule. 

Data are a random, representative sample of the scenario and situation of interest. In this case, the analyst can use frequentist statistical methods. 

As Morgan Henrion (1990) point out, for many quantities of interest in models used for decision-making, there may not be a relevant population of trials of similar events upon which to perform frequentist statistical inference. For example, some events may be unique or in the future, for which it is not possible to obtain empirical sample data. Thus, frequentist statistics are powerful with regard to their domain of applicability, but the domain of applicability is limited compared with the needs of analysts attempting to perform studies relevant to the needs of decision-makers. 

Monte Carlo simulation is based on random sampling. Thus, it is possible to use frequentist statistical methods to estimate confidence intervals for the simulated mean of a model output, taking into account the sample variance and the sample size. Therefore, one can use frequentist methods to establish criteria for how many samples to simulate. For example, one may wish to estimate the mean of the model output with a specified precision. The number of... 

Frequentist statistics (Also frequently and classically referred to as classical statistics.) An approach to statistics commonly encountered in the analysis of clinical trials and having the following features. (1) An interpretation of probabilities is made which relates them to the long-run relative frequency of events in a series of (hypothetical) trials is made. (2) It is denied that hypotheses or parameters can have a probability the former are either true or false and the latter are either equal to some value or not. (3) A decision to accept or reject a hypothesis (or presumed parameter value) is made indirectly using the probability of the evidence given the hypothesis (or presumed parameter value) rather than vice versa. (4) The probability of more extreme evidence must also be taken into account. (5) The experimenter s intentions in designing the trial have a bearing on the interpretation of the results. 

Null hypothesis. A term used in frequentist statistics in connection with hypothesis testing. The null hypothesis is a hypothesis which is assumed (for formal testing purposes) to hold true until evidence has been accrued which leads to its rejection. For example, one might assume, for the time being, that a drug was ineffective but attempt to show that this null hypothesis was untrue. 

Sequential trial. (A somewhat unfortunate term since nearly all patients are recruited sequentially and in this sense nearly all trials would be sequential.) A clinical trial in which the results are analysed at various intervals with the intention of stopping the trial when a conclusive result has been reached. A stopping rule is usually defined in advance. In frequentist statistics it is necessary to make an adjustment to the results of an otherwise standard analysis to take the stopping rule into account. Such an adjustment is not necessary in Bayesian statistics, although the stopping rule itself ought to be at least partly dependent on the prior distribution so that, for this reason alone, the posterior distribution will vary with the stopping rule. 

Type II error. A concept used in frequentist statistics. It is an error committed in rejecting the alternative hypothesis when it is, in fact, true. 

We assume that plants of a certain type may be considered as a homogeneous group (a far reaching assumption but there is no better one) and that events in this group are registered over a certain period of time as in [23]. We can then make statements about the expected frequency of occurrence for an event which has not occurred during the period of time of registration. For the upper bound of the 95 % confidence interval, hgs, we find from frequentist statistics (cf. Appendix C)... 

The use of the methods of subjective statistics has often been regarded as opposed to the objective frequentist statistics. In reality there is not such a great difference. The extent, to which the evaluation with frequentist statistics requires engineering judgment, as explained above, must be borne in mind. 

A further important difference between the two notions of statistics must be mentioned. In frequentist statistics a fixed but unknown parameter, for example the failure rate k, is estimated from component lifetime observations. The result is not exact but only obtained with a certain level of confidence. This level indicates how often the result would lie within the confidence interval, if the measurement were repeated many times. For example, if the measurement is repeated 100 times it... 

This can be explained on the one hand by the fact that practitioners are more familiar with frequentist statistics, more largely taught in engineering courses, on the other hand by a persisting old-fashioned and rather negative vision of Bayesian statistics. 

In frequentist statistics, the parameter is considered a fixed but unknown value. The sample space is the set of all possible observation values. Probability is interpreted as long-run relative frequency over all values in the sample space given the unknown parameter. The performance of any statistical procedure is determined by averaging... 

Frequentist statistics have problems dealing with nuisance parameters, unless an ancillary statistic exists. 

Frequentist statistics gives prior measures of precision, calculated by sample space averaging. These may have no relevance in the post-data setting. 

It is common in frequentist statistics to consider a point-null hypothesis that 0 takes exactly a specific value 0. When 0 is continuous, such a hypothesis has zero probability of being true and is ill posed in the Bayesian framework. We can approximate such a point-null hypothesis as Hq [e o-0 II < s, as long as the prior is proper. 

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