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ex-3: review

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Michele Guerini Rocco 2020-05-28 20:46:49 +02:00
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@ -388,40 +388,47 @@ Likelihood results:
---------------------------- ----------------------------
par $p$-value par $p$-value
------------ --------------- ------------ ---------------
$\alpha_L$ 0.18 $α_L$ 0.18
$\beta_L$ 0.55 $β_L$ 0.55
$\gamma_L$ 0.023 $γ_L$ 0.023
---------------------------- ----------------------------
Table: Likelihood results compatibility. Table: Likelihood results compatibility.
$\chi^2$ results: $\chi^2$ results:
--------------------------------- ----------------------------
par $p$-value par $p$-value
----------------- --------------- ------------ ---------------
$\alpha_{\chi}$ 0.22 $α_χ$ 0.22
$\beta_{\chi}$ 0.89 $β_χ$ 0.89
$\gamma_{\chi}$ 0.0001 $γ_χ$ 0.0001
--------------------------------- ----------------------------
Table: $\chi^2$ results compatibility. Table: $\chi^2$ results compatibility.
It can be concluded that only the third parameter, $\gamma$ is not compatible It can be concluded that, with both methods, the parameters $\alpha$ and $\beta$
with the expected one in both cases. An in-depth analysis of the algebraic were recovered succefully, while $\gamma$ is incompatible. However, the
arrangement of $F$ would be required in order to justify this outcome. covariance was estimated using the Cramér-Rao bound, so the errors may be
underestimated, which must be the case for $\gamma$.
Since two different methods similarly underestimated the true value of
$\gamma$, it was suspected the Monte Carlo simulation was faulty. This
phenomenon was observed frequently when generating multiple samples, so it
can't be attributed to statistical fluctuations in that particular sample.
The issue remains unsolved as no explanation was found.
\vspace{30pt}
## Isotropic hypothesis testing ## Isotropic hypothesis testing
What if the probability distribution function was isotropic? Could it be What if the probability distribution function were isotropic?
compatible with the found results? Is this hypothesys compatible with the observation?
If $F$ was isotropic, then $\alpha_I$, $\beta_I$ and $\gamma_I$ would be $1/3$
, 0, and 0 respectively, since this gives $F_I = 1/{4 \pi}$. The t-test gives a If $F$ is isotropic, $\alpha_I$, $\beta_I$ and $\gamma_I$ would be $1/3$ , 0,
$p$-value approximately zero for all the three parameters, meaning that there is and 0 respectively, since this gives $F_I = 1/{4 \pi}$. The t-test gives a
no compatibility at all with this hypothesis. $p$-value approximately zero for all the three parameters, meaning that there
is no compatibility at all with this hypothesis.