ex-3: review

notes/sections/3.md

@@ -388,40 +388,47 @@ Likelihood results:
 ----------------------------
  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.
 
 $\chi^2$ results:
 
----------------------------------
- par              $p$-value
------------------ ---------------
- $\alpha_{\chi}$  0.22
+----------------------------
+ par         $p$-value
+------------ ---------------
+ $α_χ$        0.22
  
- $\beta_{\chi}$   0.89
+ $β_χ$        0.89
  
- $\gamma_{\chi}$  0.0001
----------------------------------
+ $γ_χ$        0.0001
+----------------------------
 
 Table: $\chi^2$ results compatibility.
 
-It can be concluded that only the third parameter, $\gamma$ is not compatible
-with the expected one in both cases. An in-depth analysis of the algebraic
-arrangement of $F$ would be required in order to justify this outcome.
+It can be concluded that, with both methods, the parameters $\alpha$ and $\beta$
+were recovered succefully, while $\gamma$ is incompatible. However, the
+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
 
-What if the probability distribution function was isotropic? Could it be
-compatible with the found results?  
-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
-$p$-value approximately zero for all the three parameters, meaning that there is
-no compatibility at all with this hypothesis.
+What if the probability distribution function were isotropic?
+Is this hypothesys compatible with the observation?
+
+If $F$ is isotropic, $\alpha_I$, $\beta_I$ and $\gamma_I$ would be $1/3$ , 0,
+and 0 respectively, since this gives $F_I = 1/{4 \pi}$. The t-test gives a
+$p$-value approximately zero for all the three parameters, meaning that there
+is no compatibility at all with this hypothesis.