$3.$ An engineer is choosing between systems to predict a signal x $2$ Rd $.$ For a prediction

y $2$ Rd of the signal$$s values at times i D $1$; $2$; : : : ; d $,$ the engineer considers three loss

measures. The first takes a threshold t $>0$ and evaluates

L$0.$x; y$/$ D t $1$

d

dX

i D$1$

$1$fjyi xi j t g

where $1$fg D $1$ if its argument is true and zero otherwise. The second and the third are

the mean absolute error $($MAE$)$

L$1.$x; y$/$ D $1$

d

dX

iD$1$

jyi xi j D $1$

d ky xk$1$

and the root mean square $($RMS$)$ error

L$2.$x; y$/$ D rms$.$y x$/$ D

v

u

u

t $1$

d

dX

i D$1$

$.$yi xi $/2$ D

r $1$

d ky xk$2$

$2$

respectively. The engineer claims that

L$0.$x; y$/($a$)$

L$1.$x; y$/($b$)$

L$2.$x; y$/$:

$($a$)[3($bonus$)]$ Justify inequality $($a$).$

$($b$)[2($bonus$)]$ Justify inequality $($b$).$

$($c$)[5($bonus$)]$ There are m different possible prediction systems which produce prediction vectors

y$\mathrm{.}1/$; y$\mathrm{.}2/$; : : : ; y$.$m$/2$ Rd $.$ The engineer will choose one of the losses Lk $,$ and for the

choice k $2$ f$0$; $1$; $2$g$,$ will use the system j satisfying

j $?$ D arg min

j $2$f$1$;:::;mg

Lk $.$x; y$.$j $//$:

The engineer only cares that the predictions satisfy jyi xi j $<0$:$5$ the majority of

the time. Which loss most reflects the engineer$$s preferences? Why