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9 Testing Hypotheses and Assessing Goodness of Fit (Generalized Likelihood…
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är inte är composite eftersom den inte specifierar en hel fördelning (utan en stor mängd möjliga fördelningar)
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\[\color{gray}{=\min(\Lambda^{*},1)}\]
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dvs\[-2\log\Lambda\rightarrow\chi_{\textrm{dim }\Omega\,-\,\textrm{dim }\omega_{0}}^{2}\]
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If, under \(H_0\), the probabilities \(p_i(\hat{\theta})\) depend on a \(k\)-dimensional parameter \(\theta\) that has been estimated from the data, dim \(\omega_0\,=k\).
Pearson's chi-square statistic är\[X^{2}=\sum_{i=1}^{m}\frac{(O_{i}-E_{i})^{2}}{E_{i}}\]
- \(X^2\) = Pearson's cumulative test statistic, which asymptotically approaches a \(\chi ^{2}\) distribution.
- \(O_i\) = # observations of type \(i\)
- \(N\) = total number of observations
- \(E_i=Np_i\) = the expected (theoretical) frequency of type i, asserted by the null hypothesis that the fraction of type i in the population is \(p_{i}\)
- \(n\) = antalet celler i tabellen
antalet frihetsgrader är \(n\), minus reduktionen i grader av frihet, \(p\)
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