A Law of Likelihood for Composite Hypotheses
This paper proposes a generalization of the law oflikelihood for composite hypotheses.
A Law of Likelihood for Composite Hypotheses  …
The Lévy flight hypothesis is that the distribution of movements has a powerlaw tail with . This is the PL model (1), and the hypothesis is not directly concerned with data that are . The PL model with corresponds to the inversesquare powerlaw that found close resemblance to for five species. The exponential distribution (2) represents the simple hypothesis that each movement step terminates with a constant probability per unit time , . Ref. found an exponential distribution to be supported for only two species (catshark and elephant seal).
Bickel
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A general function to quantify the weightof evidence in a sample of data for one hypothesis over another isderived from the law of likelihood and from a statisticalformalization of inference to the best explanation.
[0901.0463] A Law of Likelihood for Composite Hypotheses
A model for evaluating the economic implication of disease in terms of the economic loss of a person succumbing to morbidity or mortality at some specified age. Often such a model has two components, the direct cost of disease, for example, medical management and treatment, and the indirect cost of disease, namely the loss of economic productivity due to a person being removed from the labour force. [Berichte uber Landwirtschaft, 1996, 74, 16585.]
A general term for the procedure of assessing whether sample data is consistent or otherwise with statements made about the population. See also null hypothesis, alternative hypothesis, composite hypothesis, significance test, significance level, type I and type II error.
Likelihoodratio test  Wikipedia
We demonstrate methodological issues in two recent studies that reached similar conclusions concerning movements of marine animals (Nature 4511098; Science 3321551). The first study analysed vertical movement data to conclude that diverse marine predators (Atlantic cod, basking sharks, bigeye tuna, leatherback turtles and Magellanic penguins) exhibited “Lévywalklike behaviour”, close to a hypothesised optimal foraging strategy. By reproducing the original results for the bigeye tuna data, we show that the likelihood of tested models was calculated from residuals of regression fits (an incorrect method), rather than from the likelihood equations of the actual probability distributions being tested. This resulted in erroneous Akaike Information Criteria, and the testing of models that do not correspond to valid probability distributions. We demonstrate how this led to overwhelming support for a model that has no biological justification and that is statistically spurious because its probability density function goes negative. Reanalysis of the bigeye tuna data, using standard likelihood methods, overturns the original result and conclusion for that data set. The second study observed Lévy walk movement patterns by mussels. We demonstrate several issues concerning the likelihood calculations (including the aforementioned residuals issue). Reanalysis of the data rejects the original Lévy walk conclusion.
These results suggest that a stepwise likelihood ratio statistic can be useful in a sequential study for testing a composite alternative versus a simple null hypothesis.
A likelihood ratio test is any test with critical ..

Hypotheses for the likelihood ratio test
@MISC{Zhang901alaw, author = {Zhiwei Zhang}, title = {A Law of Likelihood for Composite Hypotheses}, year = }

Uncertainty analysis and composite hypothesis under …
You have a composite hypothesis of the form where the subset of the parameter space under the null is of the ..

Uncertainty analysis and composite hypothesis under the likelihood ..
Composite hypotheses ..
Hamidreza Amindavar  Amirkabir University of …
The first step to identify Lévy movement patterns involves correctly testing whether the movement data are consistent with coming from a distribution with a heavy powerlaw tail (here, ‘heavy’ means that the distribution has infinite variance). This testing has long been done using regressionbased techniques, though these have been shown to be inaccurate and problematic –; for a geological context see , , and for a general context see , . Likelihood methods, a cornerstone of modern statistical ecology , have recently been shown to correctly infer exponents of powerlaw distributions in ecological contexts , .
The Basques are genetically distinctive  Gene Expression
Recent work reanalysed 17 data sets from 7 other studies, which had all concluded that the foragers being studied exhibited Lévy flight movement patterns. The foragers ranged in size from microzooplankton to fishermen , . The reanalysis, using likelihood methods, overwhelmingly rejected the originally concluded powerlaw Lévy flight model for 16 out of the 17 data sets when tested against three other simple models. For only one data set (a single grey seal in the North Atlantic Ocean ), the data were found to be consistent with coming from a bounded powerlaw (or truncated Pareto) distribution, which is consistent with a truncated Lévy flight model. However, this does not necessarily then mean that the animal is using a Lévy flight search strategy, and the data set (distances moved in a day) had a sample size of only 71 and only spanned one order of magnitude (7.5 km to 78 km), which limits any interpretation of movement on multiples scales. For further background on the use of Lévy walks/flights in ecology, see a recent book (reviewed in ) and review paper .
S v Mamabolo (CCT 44/00) [2001] ZACC 17; 2001 (3) SA …
We consider a stepwise likelihood ratio statistic which, for each new observation, is updated by cumulatively multiplying the ratio of the conditional likelihoods for the composite alternative hypothesis evaluated at an estimate of the parameter obtained from the preceding observations versus the simple null hypothesis.
The Evolutionary Arms Race  Examples Among Trilobites
Using correct likelihood methods we first reanalyse an example data set from – vertical movements of bigeye tuna. We find no support for a powerlaw (Pareto) distribution when compared to a simple exponential distribution. This is in contrast to the original finding of close resemblance to an inversesquare power law. This demonstrates that the methodological issues we describe are not just minor technicalities but can yield the opposite biological conclusions to standard methods.