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Question: Statistics:-Mode does not work properly

Question:Statistics:-Mode does not work properly

mmcdara 6149 Maple
statistics
November 12 2016
0

reference :

Question:Quantile function
Posted: Mikhail Drugov 88 

 

In the reference above, Mikhail has raised a problem concerning the function Statistics:-Quantile.
A problem of the same kind exists for the function Mode.

In fact  Mode returns the value of the mode only for unimodal distributions ; but for "bimodal" distributions it does not work properly.
Theoritically the mode is the value where the PDF reaches its maximum maximorum. Except in very particular cases this maximum is unique, even if common language speaks of "bimodal distributions" instead of "two bumped distributions".

Here is an example of a two bumped distribution (Z) obtained by mixing two gaussians distributions.
It has two bumps (z=-1, z=2) but only one mode (z=-2).
It could be hopefully acceptable that Mode returns the {-2, 2} (even if only -2 is the true mode), but Mode returns also the value of z that minimizes PDF(Z, z), which is not correct at all.


 

> 

restart:
with(Statistics):
> 

X := RandomVariable(Normal(-2,1)):
Y := RandomVariable(Normal(2,1)):
> 

r    := 0.4:
f__Z := unapply((1-r)*PDF(X,t)+r*PDF(Y,t), t);
Z    := Distribution(PDF=f__Z):

proc (t) options operator, arrow; .1692568750*2^(1/2)*exp(-(1/2)*(t+2)^2)+.1128379167*2^(1/2)*exp(-(1/2)*(t-2)^2) end proc

 (1)
> 

plot(PDF(Z,t), t=-4..4);

 
> 

Mode(Z);

{-1.999102417, .1352239093, 1.997971857}

 (2)
> 

 


 

Download ProblemWithMode.mw

 
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