And what I'll talk about is the last so
so what

we've seen a lot nowadays is algorithms
for high dimensional problems.

And then you try to prove theoretical
properties of prisms this is about

inference and a more or less classical
inference in the sense of trying to find

confidence intervals in this TZ and
I'll just consider a very simple case.

So in the end will be to look
at the as it will take normality

of esteem a thirst in high damage and

want to fess up to the dramatic to you can
build your confidence into false and just.

OK so.

Let's start on the whiteboard.

We have like on Tuesday but

don't worry I'm not going to use to
research their response variable X.

Y.
and dimensions and design matrix X..

Damages and Steins.

And the linear model so why because X..

Does However this is the linear model
Right Sector bita is an unknown.

And I'm going to assure for

simplicity that the noise that's
to noise is standard Goetia.

OK now I'm going to construct
a confidence interval for

a parameter of interest to one dimensional
parameter you can extend that to higher

dimensions and if you have higher very
high dimensions you might use a multiple

distinct technique but that's consider
a one dimensional parameter of interest.

Let's say that the premise of interest is
to first component of the effect or Peter.

Could be also and the other components
are without loss of generality or

it could be multiple components but
for the moment or for today.

We issue that we're interested in
is to mating the first component.

B.

one.

OK.

Now.

To paste in the further context
let's say that the design is random.

And that the rows of the matrix X.

are Goetia.

Lets.

The rules of X.

be ID copies.

Of some factor X.

maybe a little bit different no dish and.

In this X..

Is say means zero.

And some call Ferentz matrix Sigma.

OK now just go back back
back to the basics so

both you can just do least squares.

Then we know what to do right case one of.

The number of parameters is B.

it's both speed is less than and are less
than equal maybe lets say peace fixed.

And the rank.

Of X.

is B.

Yeah.

You can do the square stats like this.

X.
transpose X.

inverse X.

trends why the squares estimator right.

And then you have it's.

Distribution SO B.

to have.

SARAH So given X.

say he's done normal zero.

X.
transpose X.

inverse OK.

And OK we're interested just
in the first component.

So.

If this is the first component.

It's just a first.

It's just to the entry of
this matrix in the top.

Left corner.

So we have to add square root and

the least squares estimator of
the first component minus the true one.

Over in distribution.

To Normal zero.

One one.

Where that's just in first of
that matrix zero zero where.

The third.

He's sick.

In first and this is just a point of
course assuming that there exists.

Now so the rank of Texas be so
that it should exist.

I think up and she went on I want every.

Young.

Little boy school.

Boy maybe maybe we'll know nobody saw it
if you write and see what you see or.

So this is the it's made of it so
this is already normalized Yeah yeah.

I was so sick in my head Culver
just a sick mind it's in first.

Should go virtue I think
if he's fixed now.

So anyway now so we want to mimic
that in hide them ancients.

Be much Archer and.

Now my main idea of the stalk will
be what is the S A bull tick so

this is the best until tick fair and said
You call this to some dull tick fairy and.

What is the efficient as adult
experience of the estimates of the first

component in the linear model Well we
have to come of our lower bound and

maybe you know it maybe do not know
that it's in the low dimensions.

And he put a tear thick SIKMA Ron Ron.

East.

Calmer lower bunk.

If what.

If you have no assumptions
on the parameters of course.

So I was saying you know that
the parameters are in some space

if you have no assumptions.

Of.

The parameters to hold space and
then this is to comment hour about.

Them or
you just know to mark off what is called.

For best linear unbiased estimators you
can all better than the ghost Mark.

It's far away from.

So.

You might want to mimic this and just
showed it also hired him mentions you get

some estimate you construction estimator
which has this simple tick Ferentz and

what I want to show in this stock is to
sort of maybe also start a discussion and

show that no in the height I'm
ancients in high dimension

sparsity so we don't have to we
should something about the parameter.

And so in high dimensions it's not
the whole parameter space it's

an issue mean that some.

Sparsity like that some of the cofee
chanst of the parameter bitter are zero.

So I'm going to show you that
in higher dimensions indeed

this is not the comet our lower bound.

It's actually a little bit larger it's
going to be larger just by examples and

that means that it's good news so it's.

You don't have to be so pessimistic but
you also have to be careful

if you construct an estimator which
has this as until tick variance

that you're not supposed to cry out and
say you it's it's official and.

You have to be careful there so this is
in the literature you see that often that

people say it's efficient but
then it's actually not OK.

Just to get an idea about what
can we do in high dimensions.

We say OK.

We first estimate.

The parameter.

And that's to be dimensional factor and
I take for

instance the law so and
I estimate it's like that.

And then I take one step
Newton rests an estimator.

Which he's like this.

I have to I forget to
write something sure.

So let's just for to get it into it
that you should that the matrix is no

you know we take the estimator like this.

And Ickes.

And what's this.

This is to first.

Column or row or do you ever.

See fish that for station matrix
is to inverse to go fish matrix.

And then this one is specified
will tickle the normal

under some conditions to me put them down.

So it's really easy.

You see work not.

To put the lemma.

Up.

Suppose you have some rate of convergence
for your initial estimate or.

In L.
one norms mall order one over screwed.

Then.

Square root and.

Five by.

OK I normally lies I should really
normalize because if you think

that's important I keep on
forgetting it myself but

everything depends on the stamp of size so
everything changes with sample size if not

I have to say that explicitly so
I can't really write this because

the limit the pencil still in the and
so I have to dorm a lice OK.

OK.

Cost distribution normal.

My.

Question.

So let's proof this.

Proof.

OK so you have B..

One runs one zero across the definition.

There be to one.

Plus.

See one transpose X.

transposed and this Y.

is equal to to noice plus something us so
it's epsilon minus.

X.

B.

it is zero.

Then.

So I put to beat the heads together.

You can write it as well this is
the units vector of the first units

vector transposed times told factor so
I can write this.

First unit vector and here you get minus.

One transpose X.

transpose X.

that sick my hat if you divide by and.

And then you gets.

Lines zero and so this is.

Transposed transposed and
suppose here this transpose that

is the first element and
then this is just that instead.

And then there's a conditionally
all next linear term.

OK Now let's look at this park this
party specimen told the girl you can

forget about it.

So.

You can write it as.

Sick my mind is sick my head's.

On.

Transpose.

Here so Sigma time stick to one.

Wealthy toast in first so sick my thanks
to one is to first units Victor now.

And then.

See if I can do it in absolute foul
you just do dual norm inequality.

He won.

The super normal and here to L.

one norm.

One.

OK And then maybe I should write it but

there's no much base you know
that this is small enough for.

Me to write it here.

That's order that's by assumption order
one over square a little luck here.

And here you have.

The maximum of B..

Well at least some exploration going
ferals in that they can get efforts and

then you are you know that maximum of
those things is always a foreigner all.

Over and.

So the whole thing is so
small order square root of N.

so you can use until Tex You
can forget about it so

disturb you that you can forget about
it as until to call the foundations.

And then this term.

Just a final.

Argument.

You go up here.

What about that there.

So it's distributed given X.

so condition you are conditionally on X.

to destruction of that thing.

In this.

Of the.

That term Get Next is just normal.

Zero and then to go Francis Zito
one transpose SIKMA had.

One.

I think I have to divide by and.

Right.

Not so sure.

And then Sigma had converted to sick mind
you know that sick man want to transpose

Sigma.

Three to one transpose Sigma three
to one you see the wood transpose.

The first units vector which
use the two one one so

this this term goes to T two one one.

So that's the idea.

OK.

OK now this is a very simple result if
you look in the literature of friends it

in particular papers by your
mom to ninety he proved.

A result like this.

With a not this condition but
the usual condition so

the usual sparse deconditioned
that the sparsity.

Is of small order and divided by
a lot of people in the right appear.

Maybe you've still remember or
maybe you were there on on juice day.

Well so far as to completion.

Yes Yes I'm all order and
if I know people where S.

is to X..

S.
just sometimes written like this.

Just a number of nuns.

Zero elements of fact be.

OK so in the literature you can
find results which say OK if I

have only disk on dish and

some technical conditions I don't use
you in this then I can prove this.

So the point is now OK so

it's nice that you can prove that you
don't need any conditions on the.

Sparsity of the over the precision
matrix It's all very nice but

the estimator can be inefficient.

OK.

So there is some.

Discussion about that.

OK this inefficiency is what I
want to show just a final remark.

Remark.

What is this first.

Column off to receive to make
tricks you can write it is.

One here mine is gone my two.

Lines go be.

Defied it by.

The residual sum of squares if
you project the first factor.

On all the others.

Whole where.

Going to zero is to.

Project sions filter
coefficients just projections.

Are made in.

One minus six lines.

It's where.

They are so this is a theoretical
projection you project a first victory

on all the others these are its goldfish
and this is to residual sum of squares

set foot square here and
he's go fish in state occur.

As the first row.

After normalization that's one one.

TEACHER one one is here so it's next
to in first of this research yes and

maybe you remember distance is not
a new result but just to remember.

There's a comma.

You know.

Squared and
then the inverse of stepped on over.

Yes I'll do it if you do projection is.

If the projection this very small you
have your left over with a lot and

then you're happy because stand
some tilt experience is small.

This variable has nothing to
do with all the others and

this will be a very large Sure.

And the little.

You solve.

This is not true with
the others this will be.

A small number.

OK And also let me just
before going to the slides.

Just to normalize it and Hugh.

From now on there on the diagonal.

There are ones of these matrix.

May be less or equal to one but let's
say just equal to one just an army OK.

OK now we can go to the slides.

OK So dish the linear model.

A no go fish is in the noise and the high
dimensional case in front of him just OK.

Right.

As you look these so
the noise is Gaussian the go variables X.

are Gaussian in so everything is Gazans
are used to fifty plus one dimensional

Gulshan Fichter conditional
good mean of why given X.

is this linear function in
the Goshen case of course.

And then the covariance matrix of.

The axis you shoot as an infers.

We normalise that the elements in
this matrix are bounded I'm doing

a some thought to do some normalization
say the diagonal is just all once

an important assumption
which I should throughout

is that the minute more I can value
of this matrix stays away from zero.

Eight.

Here's the last so.

That's could be a good candidate for
the initial estimator here.

Right.

And this is the device last so which
I've written on the board it's now gone.

So I just do a one step new term vests and

that's only one step and
this is that the river thief.

Of the of the last function there if
the function you're trying to minimize and

this is its in first to his feeling you
take the inverse of the heavy and so

it's just really that.

Big now you know in the high
dimension see here.

That this sick man Matrix was known in
higher dimensions it's well if it's

not known you have to estimate it and then
if you estimate it's it's not so clear

you can just used the empirical covariance
matrix because it's going to be inverted.

So you have to do something so
that's why I put here till the here so

I'm going to use two choices or several
choices for this estimator Well you

can think of that as an estimate of the
first column of the perception matrix but

actually what I want to show is that
you should not really do that always

you should maybe do something
else to oped in efficiency but so

far it's been that in literature
groups that's wrong theatre one was.

Always considered as some estimator.

Of the first column of
the precision matrix.

In the years to literature.

OK.

OK this what we just showed.

This condition then you have
asymptotic normality with this S.

until tic Ferentz T.

to one one.

That's a proof we did that.

Now the question is what is the best
possible as adult experience

what is to come of our lower
bound then you just go back.

To your lecture notes or.

You look at what is done in the finite
dimensional case and you try to extend it

to the infinite dimensional case and
that involves a little bit of work because

we are to talking about triangle arrays So
everything depends on the sample size.

In Ya.

Have to be lead with careful there.

But this is how it goes so
you have a model for

your parameter in our case it will
be some kind of sparsity assumption.

Then we're going to look
in a certain direction.

Normalized by square root of N..

That's all square of
the event comes from the S.

until the commodity normalization is
what we want this direction to stay

inside the model class we're not
allowed to go outside the model class.

And then you call an estimator T.

It's actually a sequence
of estimators regular.

If well if your parameter is not be
to one so zero zero say fixed but

if the trooper enters not beat
the serial but something close to it.

So B.
to zero plus something the small per to

Bishan then this estimate there is
still less until TACL the normal.

Will.
The same variance.

Case.

So death means if your model is so
you have some B.

to zero to fix some some parameter
if the if this is not exactly true

from the read you're in the little small
neighborhood that you're still OK And so

this is saying that it goes beyond
pointwise a simple ticks it's something

like uniformly in a small neighborhood
type of a simple tick you want

some unity uniformity in a neighborhood
and if you don't have that you're

killed also in practice because then you
will see in the simulation that your X.

until to.

Use distributions do are not
valid in finite sample sizes.

So this is something to realize it's
really a very necessary assumption

that your one trick estimates to be
regular It's something like unbiased or

something like that that for some topics.

So you're restricting I'm going to
restrict myself now to regular estimators

because these are the ones which
work well also in practice.

You have to shoot something because
otherwise you can always find get super

efficiency and say things like that.

OK And then comes to Camarillo
about supposed to estimate or

it's as simple to call linear that means.

It's approximately
approximately an effort.

With a term of smaller order there so

that one over a square root of and
from the central limit theorem.

This is called the influence function.

Is as I have means zero and
finite variance.

And I'm showing this trains
is also remaining finite so

it's order one for us and
Ghost infinity and

the Linda birth condition to
have the central limit theorem.

And regularity of the estimator
then you have to scam our own.

Some tilt experience is at least.

This maximal over an Important
to note it's the maximum over

directions that stay within the model.

Of and it's it's the squint.

OK And this is just if you look more
carefully it's something like this.

What's written there.

Just.

You're trying to do to recommission of
the first variable on all the others.

But the you have to go fissions have to be
such that you stay within the model class.

OK so if you have an estimator.

Which has.

It's See what I write here.

Which has an asymptotic variants
which is exactly of the form for

some direction within the model cost and
you know it's efficient OK so that's.

A tool to prove its efficiency.

Now let's look at the next sample.

Suppose the model is that my underlying
parameter is sparse meaning that

the number of non-zero coefficients
is smallest and some value S..

Let's S.

Ciro be the extra set of the Go Fish and

we size Ciro So
this serial is smaller than S..

Now if the number of not so if you
have sparsity of the precision matrix.

That's what this says.

Then Well let's look here here it is.

Here are these Go Fish and I suppose
it's sparse in fact in fact most of them

are zero then this direction
Stace was in the model class and

so you have an efficiency.

And the state and some toxic effort the
efficient variance is the one one OK So

under sparsity assumptions this procedure
gives should as a fish and estimate.

If you have that the first variable that
we're trying to estimate is act if and

some bit I'm in condition you
have to think a little bit.

And if effect where is it.

You're at the boundary so

to speak of parameter space then the as
simple tick efficient as until it

experiences to one the same this to one
where you know which variables are active.

OK.

So that is in sense bad news because
it means that you will not be able to

reach the karma of our lower bound because
you don't know which variables are X.

If you cannot mimic that at least that's
my conjecture of course you can mimic

it on the additional assumptions but
if you have Dish no assumptions you

change your parameter space in
the coming hour Bolt also changes so

that you ended up in the loop and
you're cheating yes so this seems.

Not faring much to something
that can be achieved.

Although people do that what people do
is they do the last so throw away to

variables which are ethical fissions
equal to zero and then refit and then do.

Behave as if as if they know
back deck to ferry bills and

then just say OK I have for
this until to variance.

From from this type of efficiency
point of view if you're honest

then that's not a good idea
it's not a good procedure.

And we're generally if you so this is
just to point out a little bit that

with this kind of sparsity assumptions
maybe this kind of our lower bound

is not the chief as soon
as you have known sparsity.

OK Here it's the chief maybe
that if this one is not sparse.

And it's probably not cheap.

Now let's assume we have a weaker
form of sparsity that the L one norm

over to Go fish and see is bounded
by something like square this.

And that the trooper understates
away from the boundary that means

the true parameter as is not
exactly square root of S.

but the little bit smaller.

Then the common are lower
Bonet is just minimising

this residuals on the square so
you try to project X.

one on all the others but the GO fissions

have to be bounded by square of the face
and then there's a square root of N.

because of the normal I say
shouldn't any school year.

Here so that the erections
are normalized by square root of N and

that square root of and
appears here as well.

OK so you minimize this under L one
restriction and this is more like a lost

soul type Wolf estimated and really
projecting this is really projecting.

And here you are not doing the exact
projection because you have an L.

one Norm restriction.

And so
if you want to improve over this factor.

Then you must have to.

The true production this gum effect or
rather Or maybe this one does matter.

Is very non sparse so its L.

one norm should be a larger ordered square
root of and because if it would be smaller

order you can just do to
standard projection and it's OK.

So that's my idea I want to show
that if this thing is not sparse.

Theda one one is not the common
ground Yeah exactly.

And then this is in between it's gets a
little bit technical but it's got I think

interesting So does this week sparsity
that you sure are should write that there

are is here go fish and just something
fixed between zero and one say one half.

So this is.

Generalizing those two cases
here are a sequel to one

here are as equal to zero and
here are as between Ciro and one.

But if R.

is not zero you get this phone
to gain you have to issue.

To restrict yourself when you try to
project that you have to restrict

yourself to.

As you put here are probably I
think there's a mistake here

vectors with this are Norm I
should put here are three our.

Spector said this find this condition.

OK.

Finally let's go to some another notation.

If you have a sequence of random
variables I want to show and

which distribution depends on parameters
I want to show that something is true

uniformly in the parameter because if you
want to construct confidence intervals.

You need that right it's not just for
one parameter but for all of them so

there's a soup outside the probability.

OK.

So government zero it's
approaching action of X.

one on the other side.

And that's just this is the formula for
projection.

Standard formula Suppose X.

inverse extra Suppose Y.

but now in a disguised form
because it's different notation.

So this is extra Suppose X.

inverse but now it's just all
the other variables transpose and so

on it's always a minus when you.

Don't take the first variable here.

OK.

Now let's start with the sick man no now.

So I'm going to shoot that we have.

Eligible bear that is so to speak gum a
zero is the thing which is say not sparse.

The true projection and

I'm going to approximated by something
which is sparse in this sense.

One sense.

Love the sharp is here something
small depends on the situation and

gum up sharply something on
spark something sparse and

if this is true she will dangerously
be called a spare and she will pair.

It for such pair we can sort
of mimic the first call

column of depreciation matrix like
here you take one and mine this.

But now sparse approximation and
divide by two receipts will sum of

squares but now something like
a sparse version of that So

here's to residual sum of squares Yes So
this is just.

X.
one minus this one X.

one squared expectation of that
is equal to one that this one.

And minus these other quadratic.

OK it's written in there so
it's similar as stare here

OK now I'm going to do simple splitting
that's just to avoid a necessary condition

said to make the proof simple you can do
without sample splitting even then it's

like in the paper by not means now do you
need additional technical assumptions so

let's just as existence proof do it with
a simple splitting split the simple into.

On the first half you you take your
last so as to make the for instance.

And then C.

So here it is the first half you do the
last so and on the second tough you use.

This X.

and Y.

fairy boss and then you reverse the roles
you do on the first half the X.

and Y.
variables and

on the second tof the last so
estimate or so then you have to D.

biased is to make this like I've written
was written here and then I average them

and then you have the following
result supposed these last so

as you makers are consistent in
prediction error so to speak.

In a quick enough rate for.

L.
one norm.

Then you have as I'm told
the Clean Air three uniformly

in the underlying parameter.

And a simple to come manatee and
now I forgot the square root here but

some tilt experience is now this sharp.

And not seek up one one and
sort a synthetic variance

it's not broke don't usually
smaller than three to one one.

That's if you look at this condition
there's nothing that prevents you from

taking lump those sharp equal to zero
where if you look at the dish and then you

take love to sharp equal to zero you just
take the musher are equal to God must zero

and then let's see then this is
automatically true just as just zero.

And then the sharp is C.

to one one so that's a special case you
see there are no conditions then any more

on that zero one Norm you don't need
this condition result in those you'll.

Go.

Now.

Looked as sharp and
just take a tooth for a.

Look the sharp equals to zero I can take
it you're allowed to take it equal to

zero and then you get black back to
the usual by a month or not but now

I'm of course interested where it's not
going to zero with this maybe very small.

And then you may improve over the result

tonight by getting the better
of a simple tick Ferentz.

So.

And so on the condition of this
error you see here this is

a simple to clean the error think you.

Have this is just linear in the.

In these variables so it's as simple
to call the linear and regular.

And so

it means that the comma of our lower bound
applies this is false within the class for

which the cameras are lower bound to hold
So if I can show for this is to mater.

That this variance is smaller
OK if this is did I want

to show them that this is just simple to
fairly efficient a simple tick fairies.

So let's look at the next sample
suppose you have L's hero sparsity So

the number of non-zero
cofee census bounded by S.

and S.
Is this usual condition that S.

is a small order and over a long P.

that's to usual sparse decomposition.

Then you get this rate of call virtue and
for the prediction error so

disco's to zero by assumption.

And this rate of core virtue is for
a one hour.

NET standard literature theory.

And then you want this
time square root and

lump the sharp to go to zero so you want.

So there's a square root and
here it's cancels and

this is the condition on loved a sharp.

K.

if you have L.

one sparsity then again
dispersed should be the standard

form that should not be too large if
you have only one sparsity you get.

Consistency but
no rates so-called slow rates.

And here maybe not so you need a really
strong assumptions on luck to sharpe.

And.

If it does go my sharp so
dispersed approximation of gamma zero

the true projection is sufficiently
sparse we see that the cameras are lower

bound to achieve and it's better than the
one they're better than three to one one

if the true projection go
fissions are very known sparse So

in normal arches and square and in this.

Oak and you can extend it's two.

Zero one and this is are between zero and
one three years a little table.

So for known Sigma.

So this is zero.

Sparsity the conditions in S.

are the usual conditions.

The conditions on the slope
the sharp are like this.

And this is always by assumption true for
an eligible bear

you always have if this is true you
have some tool to normality and

if this is true you have
some tilted efficiency.

OK where in this case a simple to
deficiency is with some quote Well not

really because this is something that
depends on Colson's of the model calls and

also on the year to year to some just
actually depend on the true because zero

and here I only need to show that
the Truby to zero states away from the by.

Andris So this is a very next situation
this situation is not so nice.

If you look more carefully you see that
it's sort of fits had the condition.

Sparsity of the sparse
approximation of the projection.

Is of order S.

here and it's this S.

is this S.

There's only a long term here and
here there's no look term gap here is

she square root and this square root and
this there's no local Gap So

they they come back and if you look at
where is it are between zero and S.

It's to say meso what you see here
is conditions on lump the sharp and

they are to same here in these conditions

there's no look there's
a look at between S.

until that normality and
simple to get a fish and see so for

a simple to get fish is unique need a
little bit more but what you need more is

only local arithmetic it's only
this locked term which is not here.

It's.

OK.

Now I don't think I have very much time so
just quickly

what does this sparse approximation
of these projections so

it's a projection itself is not
sparse we do sparse approximation and

the case we consider is where
you have a sparse vector.

And the true protection is like a.

Least Squares version of the sparse one so

if you do if you have a sparse regression
model and you do least squares

without any assumptions then
the estimator will not be sparse.

And this is sort of
what I'm having here so

they've got my Sharpie sparse I do
approach action more or less I can get.

As to details and then I get to restrict
the square stop of projections and

then this is to make almost zero.

And this one will then not be sparse so
if this is true and this error term.

Some fire fairy and
she doesn't go to zero then the.

Feet of one one will be much larger or

at least the difference between the two
will still stay away from zero So yes and

it's a politically known finishing
improvement over three to one one.

So just this is saying I have examples.

Namely these type of pairs
which I call eligible for

these examples I have an improvement
over three to one one.

No I don't have a general result
that you what you can do.

So here are some some work this is too
technical for this for the late hour so

if you do some calculations you get
an improvement over three to one one but

then this true comes here to two
projection has to be known sparse.

OK if the matrix is unknown.

We do something which everybody
already does that doing the law so.

I so the government zero is the project.

Fissions of the projection of X.

one and all the others in theoretical
sense if you don't know if you can

compute it because you don't know Sigma
you do you replace Sigma by stick my head.

So you just do the projection.

Of the first.

Column of X.

on all the other columns and
then you use the last so because if you

just straightforward projections because
of the high dimensionality one variable is

always a linear combination of all dealers
because and then you get just projection

doesn't bring anything so then there's no
residual So you have to do regularization.

So for instance the last so it's listed.

So there you get a gun my head and then.

The biased estimate and
it's maybe go back to where is it.

Here So here you have your freedom and so

we know this is the general notation
we know with the snow in sick

we place despite it up
a shark with unknown

six might we be placed his by Sigma
pets and sick my head is the loss of.

Until we get this that's
what people do in practice

because in practice to go variance
make use of course not know.

OK that's what we do here.

OK.

So you have.

One like there one minus come
ahead the last so divided by

the residual sum of squares and plus this
is quite natural if you look at the.

At the formulas I thought I would
explain it here but this is more or

less the residual sum of squares
like there but now for the last.

OK So this is the most estimated
with here to last so no.

Issue this condition which is
also here one overlock P..

Consider an eligible pair.

Some technical conditions.

Then you get a simple tick well not
be there Spears a little point here

you get this expansion with a remainder
term which is a small order

this expansion is not does not
give us a little bit linearity

because this is a random thing so
this is not a linear function

this is not an effort this is
something with a random part.

But you still by slip ski steering
you get some topic normality and

you get this until to the one one shot.

OK So this is something.

Depending on what you want I mean
our common are lower Bonet issuing

that the estimator you're considering
is essential to clearly linear so

if we want to say something about this
simple to give fissions here we have to.

Have strengthened the assumptions so
that we can have a simple to clean your.

House so that's what this says.

If you are sure that your low so
estimator of grandma.

Is.

Converging fast enough
one over squared lock B.

then you have a simple to
clean the air three and

then you can see OK the camera
our lower bound applies.

Not so if I wrote this kind of assumption
in the paper and then the referee says

something similar the referee says you
don't need that assumption because you

have a low Slutsky Sam is a yacht but I'm
trying to prove a simple to clean your.

And there I need this assumption and
so that's a little discussion of

why would you want a simple to
seniority Well that's for that

kind of situation I have my camera lower
bound and I don't otherwise I don't know.

What to do OK.

So I'm running out of time so
let me just give you the table.

So this is for unknown Sigma and for

unknown sick might you need to do this
sparsity assumption for instance if you

have else zero sparsity Square would
look a square root and overlook.

So again this is a sumption.

My referee said you don't need
that assumption look at the papers

to not only but.

This is some Sion is necessary
sumption almost there is a big O.

I think here if you want
to proof a simple to clean

unbiased miss if your estimates are so
there's no way this is.

Very truly a necessary assumption for
a simple ticket I'm biased in this

as soon as you start thinking
statistically not point wise but

uniformly you need to search
kind of conditions for

the case unknown if it's no no but if it's
unknown Yes you do need this condition.

OK then the other rates and you see again.

Something for a simple tick you're
mad at the essence of the Clean Air

routine now you need to issue
an additional things and

simple to efficiency you
need to stay within the.

Parameter space but
that's this kind of assumptions.

Sigmas unknown I have to be sure that my

sparsity description the week sparsity
is really for are less than one for

are equal to one you don't get anything
any more you can see it happening here.

Where is it or is it or is it so if R.

is equal to one you can't HEAR THAT S.

should be off order one overlock be so
do that makes no sense OK.

It's just not good enough so it.

L.
one sparsity is of course very weak form

of sparsity you would like only that but
it doesn't work in this case.

So if you are less than one and then.

Hasn't thought of normality.

In the fish and

see if they have these condition if you
compare these two they're not even though

one is not a subset of the others it gets
a little bit messy they're OK Let me see.

Yeah just very quickly some stimulations
this done by Francesco or tele.

Just so that you really
shows up in the simulations.

So here the C.

to one one is five and
the see the one one sharp is one.

And the first parameters equal to one and

the number of unknown parameters and the
number of active parameters is fair you.

Pay So you see that it's
least if you have more X.

the ferryboats.

Difference doesn't show
in the beginning it for

a larger sample size indeed see that
a simple to variance just smaller than

the five that you would get if you
would use that procedure there.

And if the group underlying parameters
is not active Well you do the last so

as an initial estimate or so
you're sort of profiting for

that and then you see that there is no

worse starting years older as some thought
experience is always already quite good.

According to the theory so
it's one and you see

a simple experience she's already
quite clear close to that OK.

Thank you very much you.

Were.

You know we're.

All.

Like.

This.

It is a standard thing in this kind of
theory because you see you have some.

Maybe it's written here somewhere.

You know if you look at this here.

So you just have some something
something in there or

I don't know yeah you know
I should maybe write some.

Got to look at this kind of expression.

If this would be fixed fixed or
not the random factor.

You just have a linear combination
of as you have you just haven't

ever chain you know how being swung over
squared of an average grades like one over

square of two then but
now you have here random coefficients and

then the trick is of course OK I
take one from another set so

that they're random but
independent of what's going on here.

And then I can just condition on it.

And so as if it's not random and
then you just have ever just.

Because of the sample size splitting you
split it in half so you would say OK I

lose half of the sample that's why I do
it so put in efforts again not to lose.

Efficiency they're like they're perfect
the one half suddenly in the variant

would not be a good idea so you know.

What we should.

Be.

It is and we're just hopeful that
this will still make all of us

just won't go all the way to say
racism but we should go for it was.

Really awful.

You know for.

You but you depends on the girl fissions
of the linear form if that sparser is

you're fine with otherwise.

Always you're in trouble
you're not finding.

What we should.

All look so
far awful thank you thank you very much.