From d9af3531358c9811f84d53ed2193bac4e14dc820 Mon Sep 17 00:00:00 2001
From: rnhmjoj <rnhmjoj@inventati.org>
Date: Thu, 28 May 2020 20:46:49 +0200
Subject: [PATCH] ex-3: review
---
notes/sections/3.md | 47 ++++++++++++++++++++++++++-------------------
1 file changed, 27 insertions(+), 20 deletions(-)
diff --git a/notes/sections/3.md b/notes/sections/3.md
index 651b60d..d32b974 100644
--- a/notes/sections/3.md
+++ b/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.