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DEWEY
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©
Estimating The Variance Parameter From
Noisy High Frequency Financial Data
Bin Zhou
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
02142
MA
#3739
November, 1994
iv'iASSACHUSETTS INSTITUTE
OF TECHNOLOGY
DEC 01
1994
LIBRARIES
Estimating The Variance Parameter From
Noisy High Frequency Financial Data
Bin Zhou
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
MA
02139
Latest Revision: November, 1994
Abstract
Many
financial data are
such as tick-by-tick.
now
collected at an ultra-high frequency,
However, increasing the observation frequency
while keeping the time span of the observation fixed does not always
help in estimating parameters.
A
different type of consistency, the
consistency of a estimator as the observation frequency goes to infinity,
becomes important
in
studying high frequency data. In addi-
tion to the consistency, the deviation of a financial time series from
a continuous process
frequency increases.
is
also increasingly significant as the observation
This deviation
is
not negligible and causes an-
other difficulty in estimating parameters. This paper concentrates on
constructing estimators of variance parameter using contaminated observations;
i.e.,
observations from a continuous process with deviation
at time of observation.
The
consistencies of these estimators, as the
observation frequency goes to
Key Words:
infinity, are
analyzed.
f-consistency; observation noise; quadratic estimator.
Introduction
1
an example of estimating the mean parameter /j in a simple
process dB(t) = fidt + adW(t). For a fixed span of observation interval and
a = 1, does increasing the observation frequency help the estimation? This
type of question has come up recently in studying high frequency financial
time series. We now accumulate more and more financial data not because
time goes fast, but because data are recorded more frequently. We have
gone from quarterly data to monthly data to weekly, daily and now tick-bytick data. How does more data help us in estimating parameters? It may
I
start with
surprise
many
people to
know
that increasing observation frequency while
keeping the span of observation fixed does not always help in estimating
the parameter.
In the case of estimating the
mean parameter,
observation frequency does not help the estimation at
the
minimum
The
the two end observation points.
When a
all.
mean parameter
sufficient statistic for the
difference of these
increasing
is
is
given,
the difference of
two end observation
points does not change as the observation frequency increases.
However,
when the variance parameter is interested, increasing observation frequency
does help the estimation. The quadratic variation is a consistent estimator
of the variance parameter as the observation frequency in the limit.
raises a
new
consistency problem, f-consistency, the consistency
This
when the
observation frequency goes to infinity while time span keeping fixed.
There
is
another issue associated with using high frequency data.
When
the observation frequency increases, the difference of financial data from a
continuous process becomes increasingly significant. For example, as observation frequency increases, the variance of price increment does not approach
zero.
We
The
first
order autocorrelation of the increments
is
strongly negative.
often neglect such a difference in using low frequency data such as daily
or monthly prices.
This difference is not negligible in high frequency data.
However, this should not keep us from using a continuous process for high
frequency data. I suggested (Zhou 1991) that the high frequency financial
data can be viewed as observations from a diffusion process with observation
noises:
S(t)
where P(t)
is
=
P(t)
+t
t
,
te[a,b),
(1)
a diffusion process
dP(t)
=
fi(t)
+ a(t)dW
t
.
(2)
I
the diffusion process the signal process and the
call
The observation
and
noise
is
assumed to be independent from the
is
et
non-binding quoting error
is
diffusion process.
part of the noise.
things
In other markets, bid
Many
offer difference also contributes to the observation noise.
all
Many
In the currency market, for example,
contribute to this observation noise.
structural behaviors are
observation noise.
the deviation of data from the continuous process
and
other micro-
included in this so-called observation noise. For
low frequency observations, the observation noise is overwhelmed by the signal change. When observation frequency increases, the signal change becomes
smaller and smaller while the size of the noise remains the same.
totally
dominates the price change
in ultra-high frequency data.
high frequency data as observation with noise certainly captures
characteristics of high frequency financial time series
In this paper,
I
The
noise
Viewing
many
basic
mentioned above.
concentrate on constructing the estimators of the vari-
ance parameter using noisy high frequency observations. The f-consistency
is
investigated for each estimator.
the time span considered here
parameter to be estimated
a
2
=
Without loss of generality, I assume that
which can be an hour or a month. The
is [0,1],
is
f a 2 {t)dt
=
Var(P(l)
-
P(0)).
Jo
This paper
of the
is
maximum
organized as follows. In Section
2,
I
(3)
study the f-consistency
likelihood estimator under the assumption of Gaussian noise
and a constant variance parameter. In Section 3, I explore the estimator
by the method of moment under more relaxed assumptions. In Section 4,
I construct an optimal quadratic estimator.
In section 5, I investigate the
sensitivity of each estimator to its assumptions and give an overall evaluation
of each estimator.
2
F-consistency of The
Maximum Likelihood
Estimator
In this section,
I
assume that the process (1) has independent Gaussian
and the signal process (2) has ^(t) — //£, a(t) =
noise with constant variance
a.
Under these assumptions,
I
can obtain a
maximum
likelihood estimator
(MLE). Solving the
different equation (2),
S(t)
where W{t)
random
is
mean
spaced observations from
X-in
of
is
+ aW(t) +
nt
have
te[0,l],
tu
a standard Winner process and
variables with
where Zj
=
1
—
zero and variance
[0,1],
"Jt.n
—
et
I
are independent Gaussian
2
rj
is
.
Taking n
+
1
equally
have {So,n,Si,n,~-,Sntn } such that
<Jt-l,Ti
——+
n
r=^i
y/n
~^~
a standard Gaussian random variable.
{Xi n,...,Xn ,n}
(4)
C»
—
"
e *-l
The
(5)
joint distribution
a multivariate normal distribution with
mean
zero and
:
variance matrix
n
where
/„
is
an identity matrix and
(
An =
2
In
+
2
V
K
(6)
The
derivatives of the log-likelihood function with respect to each
param-
eter are
dl(n,a
2
,r}
2
;X)
=
/
^^^
=
a-_jWnt e-i£
n
n
-itr( S „-M„)
+
z
077^
A
Rewrite matrix
is
n)
I(X-^
Sn-M„ Sn-'(X-^)(13)
z
n
n
as
^n
where e
/
n
dp
a vector with
= Vn A n V?,
and Vn = (vi,...,vn ) is the matrix
and A n =diag(Aj) is a diagonal matrix
elements
all
(14)
1
consisting of eigenvectors defined in (8)
with diagnal elements being the eigenvalues defined in
The
(9).
inverse of
the covariance matrix
1
^n
=
V^diag(
)V?
\
t
Let
(Y
l
Let
v.i
=
J2j Vij
,...,Yn )
and notice that £,
T
=
Vij
= V?X
The
1.
MLE
of p,
a 2 and
2
the
is
T]
solution of equations:
ft
(<r
2
/n
+
2
n tt
v K)
V 2n{a /n +
t
2
2
tt 2(a /n
t
(
2
/(/i.a ,^
2
)
=
V
)
of (p,a
,tt
2(a 2 /n
2
2
,r)
)
+
'
^A,) 2
^
is
Eij[Z-%/n 2
^trfE^
1
)
;
rttt
K
2
2
V K)
t
*
Ai)
,n)2
{Yi
1
+ r/2A
The Fisher information matrix
2
77
~^
V
/n
h M° + rfKY
l
£{
+
(a 2 /n
^(E-M^-
1
)
£tr(5£M n £^) ^tr(S-M n S;M n
)
'
(T.
i
n2(<72/n+T7 2 Ai)
._2\12
£— 9«2^2/„
2n 2 ( -2/n+T,2A
i )2
^ 2n(cr2/n+r)2A.)
\
The
MLE of the
mean
—
Z-,
2n(<r2/„ + T,2A0 2
A?
^ 2(7PJW+^iF
2
J
is
Vfo
Am = n(£
tt
(<r
t
2
Ai
+
i
»W ^ (*7* +
)/(£
v
2
(18)
:)
V K)
Using the scoring method, the variance parameter and the variance of noise
can be solved by the iteration of
^
2,(fc)
(
2,(fc)
k ~ l)
l
+ i(°r- \vT-
Vm
1)
r
1
(19)
where
^
dt
^H^-V/nW^K)
[
l)
2-A
and I(a 2 ,r) 2 )
is
2,(fc-l)
„/
A.
;
"
2,(Jfc~l)
.
+ 2n2(^ Vn-H7^ - >A
+ 2.(fc-l)/„
J_„2,(k-l) A 2
Vn+r,
1
fc
2J
1
i)
x
)
'
the lower-right corner sub-matrix of the information matrix
(18).
Theorem
1
77ie asymptotic behavior of the information
(
2^ + °(VM
/(/^W) =
o
is
o
8
*+
o
8<r«
V
matrix
o(n)
/
(20)
2
where y=T] /a
7
The proof can be found
It is
in the
Appendix.
easy to prove that
Var(/t
M =
)
E ^(aVn +
1
1
^Ai)
J"
= O(v^)
The
variance diverges as the observation frequency increase.
than the estimator
= £„„ — 5
ft
MLE
Instead of using the
observation frequency.
used
,n,
in this section as the
asymptotic results about
estimator of
MLE do not
It
is
which has constant variance
of
fi,
ft
mean parameter
=
fi.
£„,„
worse
for
any
— S0>n
The
classical
apply here because we are considering
To
the observation frequency, rather than the time span, goes to infinity.
investigate the f-consistency of the estimator,
a2
for using different
frequency n,
I
,
signal-to- noise ratio
Then I calculate
The results are given
in
Table
1.
and the convergence
f-consistent
I
conduct a
=
7
if /a
2
.
series of simulations
For each observation
simulate 100 series of noisy observations as in process
100 MLE's and
their
rate of the variance of the
to the inverse of the information matrix (20).
^
Var(^) =
(4).
sample mean and sample variance.
Empirical results indicate that the
Var(^) =
is
That
MLE is
MLE
is
simmilar
is
+ (-L)
(21)
^-+0^)
n
n
(22)
no observation noise, the MLE of the variance parameter is the
quadratic variation a 2 = £ X? n — (£ Xhn ) 2 /n. The variance of the quadratic
4
variation estimator is cr /n. For both mean and variance parameter, the
When
there
is
variances of
MLE's converge
y/n slower
Because the eigenvalues of matrix
An
when
there are observation noise.
are known,
it is
not too expensive to
computing the MLE. However, there are several setbacks
First, the variances of high frequency financial
distributed
among
all
observations;
i.e.,
the
<r t
for this estimator.
data are extremely unevenly
=
a
is
often violated. Second,
the noise in (4) is often not normally distributed and may be dependent.
Third, the iteration (19) needs a reasonable initial guess. In the next section,
I
look for the estimator by the
assumptions.
method
of
moment under more
relaxed
Table
1:
Empirical
Mean and Variance
of
MLE
Estimating the Variance Parameter by the
3
Method
Moments
of
Assuming that the observation noises are independent with a finite fourth
moment, I can construct a very simple estimator by the method of moment:
? = £(*?« + 2Xi,nXi-i,n) - (E *>,n)
The second term converges
simplicity,
assume that
I
\i
to zero
=
0.
and therefore
The estimator
is
2
/n.
(23)
negligible as
—
n
oo.
For
(23) simplifies to
&MU = £(*?n + 2Xi,„X _ lin )
(24)
i
This estimator does not require any distribution assumption on the noises.
The noise can be non-stationary. The estimator is nearly unbiased. The
mean
of this estimator
is
E{g 2mm )
—
= Y,Wn + ri! + vLi-2vU)
= ° 2 + V$-vl
(25)
^
a 2 dt and rj2 is the variance of observation noise e*. UnforJt
n
tunately, this estimator is not f-consistent if the majority of noises is non-zero.
where a 2 n
The variance
Var(a^ M )
of the estimator
is
= £( 2 <n + 4a 2 nV 2 + Aa 2_ l<n a 2 n + Aa 2_ hn
2
-q
_x
+
A)
+±nhU + toLijti + toilmU) + vn + vi
{
where
n\
is
the fourth
non-zero value of
2
rj
,
moment
m
let
Suppose that
be the number of rj2 > rj2 and
of noise
Cj.
2
rj
^i-i,nVi
4
is
\
(26)
the
minimum
m be proportional
to n, then
Var(^ M ) >
If all
with
a2n
=
mean
a 2 In and
J>ftk)
e*
>
4
2
5>.
are independently
zero variance
2
7y
,
E^-
and
2
/«
= 4m 2 /n V \
identically distributed
(i.i.d.)
then
Var(<7^ M )
=
4
6a /n
+
I6a 2 n 2
+
Sriv
4
.
(27)
The optimal observation frequency
of the estimator
=
n
is
JZ/A/^. The minimum variance
is
Var(^ M ) =
29.867/V.
This estimator, after a certain point,
getting worse as the observa-
is
However, because of its simplicity, I want
an f-consistent estimator with this simplicity. In
construct an optimal quadratic estimator of the variance
tion frequency goes to infinity.
to investigate
there
if
the next section,
I
is
parameter.
4
Quadratic Estimator
In this section,
study the estimator
I
X=
section,
I
(AT 1>n
assume
,
...,
/i
Xntn
=
)
T and
Q
is
Eo% =
Var(aJ)
= a 2 /n
Assume that of n
and
2
rj
=
(28)
any n x n matrix. Similar to the
Oq has mean and
0.
Vn
=
V? Vn
where
.
and
Aj
last
variance:
tr(QE)
(29)
= tr(gEQS)
(30)
rf.
2
En =
quadratic form
XTQX
=
a%
where
in following
2
—In + r?A n = Kn diag(— + W)K
n
n
and
are defined in (8)
Vn
(9).
is
symmetric, therefore
Let
<2„
= Vn Q n Vn
(31)
,
then
E&1 =
£&(—n + W)
(32)
t
Var(aJ)
=
^ -J(^n + A
9
2 2
l
r7
a
10
)
=
^
4
2
+ A 7)
E9j( 1
n
i
ij
)
(33)
where 7
only
=
if /a
2
,
the signal-to-noise ratio. Obviously, a Q
unbiased
is
if
and
if
^2qi,=n
Yl^^ = °
and
i
Theorem
34 )
(
i
2 For any given
r, the solution of
minQ^4(- + A r) 2
i
subject to
^
=n
g.i
^
and
i
is
g'jj
=
for
i
and
^j
a+
*"
w/ierie o:
and
/?
= a
a
is
pXi
+
2(l/n
(35)
A,r) 2
satisfy
2n
The proof
=
g-jjAj
i
^(l/n +
A t r)2
Ai
^ (l/n +
A,r)
2
+/?
+
^(l/n +
^
A
(l/n
A
l
r)2
2
+
A,r) 2
given in the Appendix.
For any given
r,
the best quadratic estimator
*5
is
= £&!?,
(
36 )
i
where g«
is
Theorem
and
defined in (35).
3 For the optimal quadratic estimator aq, with
(35), the
Q
defined in (31)
asymptotic convergence rate of
y/r
11
2^ V™
V"
The proof is again given in the Appendix.
From Theorem 3, for any given r, the quadratic estimator &q, with Q
defined in (31) and (35) is f-consistent. The prefered value of r is 7, which
unknown, r should be chosen
order of 7, but should not be too small.
When r is close to 7, the performance of the quadratic estimator is similar to
and it does not need any distribution assumption about noises. Other-
is
in
MLE
wise, the variance does not decrease as the signal-to-noise ratio
Without assumption of constant variances,
it is
very
7
dereases.
difficult to find
an
I empirically examine the sensitivboth MLE and the quadratic estimator to the assumptions of constant
variance and Gaussian noises.
f-consistent estimator. In the next section,
ity of
5
Non-normal Noises and Unequal Variances
In applications to financial market, both assumptions of constant variance
and Gaussian noises are not
over time, especially
Gaussian.
among
In this section,
valid.
The variance
of the prices
high frequency observations.
I
examine the
sensitivity of
The
is
changing
noise
both the
is
2
MLE a M
and the quadratic estimator Oq to their assumptions. The following
series are simulated and used in estimating the overall variance:
Series
I:
a2n
=
a 2 /n and the noise
a degree of freedom
Series
II:
Series
III:
such that
e* is i.i.d.
r/t(5),
a
t
random
=
a 2 /n and the noise a
a2 n
is
sampled from uniform distribution U(0,
Series IV: of n is
=
2
(T
six
time
variable with
5.
a2n
J2<Ti >n
hardly
,
the noise
is i.i.d.
e» is i.i.d.
^Bernoulli (p) with p
1)
scaled such that Yjv"in
°"
2
>
the noise
.5.
and then re-scaled
ryt(5).
sampled from lognormal distribution LN(0,
=
=
e^ is i.i.d.
1)
and then
re-
r/t(5).
sampled from Bernoulli(p) with p = 0.1 and then re-scaled
such that Z)of n = a 2 and the noise et is i.i.d. 7/Bernoulli(p) with p = .5.
Series V: of n is
12
Series VI:
noise
a2n
=
a 2 /n and the noise
t{
MA(1) with
MA
coefficient 0.5
and
77t(5).
In the simulation, following values are used for various parameters:
=
2
I, T]
Table
0.01
and n
=
100,500 and 1000. The empirical
a2
=
results are listed in
2.
first two series have non-Gaussian noises. For t and Bernoulli random
both a 2M and Oq show the variance convergence rate of l/y/n. The
MLE takes advantages of smaller sigmal-to-noise ratio in series II and has
smaller sample variance of the estimates.
The next three series have unequal variances over time. Again, both a 2M
The
noises,
and Oq show the variance convergence rate of l/-y/n. The performances of
two estimators are somewhat similar. The MLE is slightly better. For small
n, both estimators slightly under estimate the variance. The bias disappears
2
as the observation frequency increases. More variation in a n causes more
bias in both estimators. However, asymptotically, both estimators are not
sensitive to this deviation from the assumption of equal variance. Many other
simulations have confirmed above findings.
Series
cant bias
VI has correlated observation noises. The MLE clearly has signifithat does not go away as observation frequency increases. However,
much better. The bias, if any, is negliThe variance of o 2M converges at rate of 1/y/n. For this set of data,
mean squared error of the quadratic estimator is about the same as ones
the quadratic estimator performed
gible.
the
using series I-V. Therefore, the quadratic estimator has advantages of being
not sensitive to correlation
cases,
can be used as an
a summary
initial
noises.
guess of the
The quadratic
MLE.
I
end
estimator, in other
this section
by giving
table (Table 3).
Discussion
6
A
among
misleading perception
is
that the
more data there
is,
the better. Increas-
ing observation frequency while keeping time span constant does not always
help parameter estimation.
may
An
estimator developed for low frequency data
not be usable for high frequency data.
The observation
does not decrease as the observation frequency increases,
cle.
The name
of observation noise
is
is
sometimes misleading
13
noise,
which
the key obstain the financial
Table
2:
Sensitivity of
a 2M and Oq to Their Various Assumptions
Table
3:
Summary and Comparason
of
2
and Oq
aM
Gaussian
equal
sensitive to:
noises?
variances?
noises?
MLE
No
Yes—
Significant
O(fijn)
O(yffjn)
Not Converge
No.
Yes^
Small
Q(l/M
Q(UM
the estimator
Is
Bias
Var(<7^)
Q.E.
Bias
Var(^)
0(1 /y/n)
The summary of this table
is
spaced
correlated
based on empirical simulations
incuding ones not listed in this paper.
community. Currency spot quotes have widely recognized noises. However, a
may still have so-called observation
noise. The noise is simply the deviation of the price from an assumed underlying continuous process and may prefer to be called a different name in such
stock transaction price recorded precisely
application.
that
is
The observation
noise can include micro-activies of the market
not interested in applications.
If
high frequency data
is
used in study-
ing the macro-activity of the market such as daily volatility, the variance of
important not to be overwhelmed by micro-activities.
The advantage of using high frequency data to estimate parameter such as
daily price change,
it is
is that we can estimate the volatility day-by-day rather than
an average. The MLE estimator developed here has been applied to many
financial data such as the currency exchange rates and prices from futures
daily volatility
The results are not listed here because that the true volatilities are
unknown and no comparisons can be made. The estimators developed in
market.
paper can be generalized to estimate the multi-dimensional covariance
matrix. Treating micro-activities as observation noise, one can construct a
this
consistent estimator of covariance matrix (Zhou 1994).
The
f-consistency
plications.
A
data series in
lot of
and observation noise
issues also exist in
many other ap-
manufacturing processes have generating high frequency
last decade.
To study such type data and
to perform parameter
estimation, one has to be aware of the observation noises and examine the
consistency of high frequency.
15
Appendix:
i)
Proof of Theorem
matrix
the asymptotic behavior of the Fisher information
1,
(20):
From
(18),
r
^
f
\
_ _fa _
n2(<r'/n+»? 2 Ai)
/(jx,*W) =
^ 2n
"
V
First,
I
5-
2
(<7
2 /n+r, 2
2n(cr 2 /n+r, 2 A i ) 2
Recall that A,
=
4sin
+
2
(a /n
2
*
(
2
,
).
+1)
(l/n
r)
2
\i)
We
8a 4 ^y
2
/
i
(l/n + 47sin
4 7 si
^r(l/n
r
2
[0, 1]
<
-dx
(7ra:/2))
+
o(y/n).
can easily prove that
7 >
+ 47sin 2 (7rx/2)) 2:
a decreasing function of x G
r
/
A?
2(<r 2 /n+rj 2 A i ) 2
i
2
1
•
2n(<r 2 /n+tj 2 A i ) 2
prove that the (2,2)-th element of the matrix
E 2n
is
^
^
A0 2
and
1
E
n+l
(l/n
„
+
2
(7ra:/2)) 2
dx.
(37)
47sin
"n+l''"'n+l
<
A n+l
(l/n
Jo
The maximum
value of 1/(1 jn
^
E
1
n+
l
+ 47X 2
1
2
)
over
+
47sin 2 (7rx/2)) 2
[0,1] is
n 2 Therefore
.
1
,
+ 47sin 2 (ra/2)) 2
(l/n
-
'n+l' ''n+l
Jo
(i
[t
+ 4477 sin 2 (fx))^
(?x)r
(5
+ 47)
Arctan(
L(i + 4>v ^+ 4 ^
/
16
(
^^
)tan(
^)
+
(£+ 4
X^ +
4 7-47Cos(7rx))
(^+47)
47
1
7T
+
+
0(n)
x=0
0(n)
0(n 3 / 2 )
4-
7;
n=i
47 sin (7r a:)
3/2
+ o(n 3 ' 2
)
4^/7
Therefore
E 2n
n+
1
2
(a 2 /n
I
n 3/2
^+
.
,
/2 ..
o(V")-
prove the (3,3)-th element of the matrix
A2
E 2(a /n +
2
It is
.
+ ^Ai) 2
8a 4
Next,
1
2
easy to see that x /(l/n + 7x)
n
T)
2
2
\i)
is
2
+
o(n)
Irf
an increasing function of x. Therefore,
4
sin (7nr/2))
+
(l/n
2
47sin (7rx/2)) 2
,
7
>0
an increasing function of x. The maximum value of above function
2
Using the similar technique as used above,
l/(47)
is
is
.
4
n
^E
-\
B
2
(4 +
167 2 (j+
27ri __
sin4
sin (7rx/2))
(1/n
+ 47 sin
2
=
(7rx/2))
2
(f")
L a + 47sin
(I
2d:C
dx
2
+
(fx))
V
1
C
(
-)
12 2 )
47)^+4*
I2yirx
2
167 7rx
2
(167 (^
+
+
+ 4^(7rxcos(7rx) —
- 47 + 47cos(7rx))
2
167 7rxcos(7rx)
47)7r(-^
17
1=1
sin(7rx))
+0
J
1=0
&
(£ + 12*)
2 I
1 67
+ 4 7)7r
JL
yT^I 2
2
(16 7 (1
(
(12*)
-32 7 2
1
1
+
16 7 2 (+4 7 )(-8 7 )
16 7 2(4 7 )y4^2
+ 4 7 )7r(-£
a
7r + ?(-7r)
- 47 - 47)
-S-^-167^-167
,
1
V
o(l)
+ o(l).
16 7 2
Therefor
16 2 (n
A?
2(<r
2
/n
+
r/
2
A,)
2
+
2a
l).
1
+ °(
(t^
16 7
v
A
1
2
))
=
n
^+
°(")-
2rf
tricky to prove the asymptotic result of (2,3)-th element
It is slightly
y/n
A,
£ 2n(a /n +
2
r)
2
\x )
2
"
8a 4
v^
+
o(y/n).
Function
2
sin (7ra;/2)
(l/n
is
not monotone. However,
The maximum
it is
+ 4 7 sin 2 (7ra:/2)) 2
and has maximum one turning point.
^-. The similar technique can be used
positive
value of the function
is
here
2
n+
1
.
sin (fx)
L a
(± + 4 7 sin
+ 4 7 sin 2 (7rx/2)) 2
(l/n
n
1
2
i
sin (7rx/2)
1
Jo
"n+1 '"'n+l
—+
1
,(^
Arctan(^
+
(i
47WA +
4
4 7 )tan(fx).
2 '
"
Vr
^
_
)
n
i=l
sin(7rx)
(i
+ 4 7 )tt(-| -
1
^
+
/
47
+ 4 7 cos(7rx))J x=0
/-\
o(y/n)
4 7 7T^4f 2
+ 0(^/71).
16v9*
18
+ 0(1)
2
;
(fx))
dx
+
0(l)
Therefor
4(n
A,
E 2n{a /n +
2
r?
2
^)
+
y/n
1)
+
Now,
I
come back
.
r
2
o(y/n).
to the (l,l)-th element
^[E-
1
] ii
/n
2
=
-+
o(l).
y
First,
^- = Vn dmg(l/(a 2 /n +
2
x
Therefore
rj
X ))Vn
£[^ = £(t&/(*Vn +
l
.
2
r?
Ai)).
u
«j
Since
XX
XX- W"
1
It is
2
=
^E
=
i
-
i,
-
E
V/(« +
cos(inir/(n
1
n+
=
sin
+
i))
1)) sin(i7r)
sin(z7r/(n
1
n2((72/n
+
1))
+ 7?2Ai y
easy to argue that
^nV/n + A)
n+
"2 ^ 2
-/o
i+4 7 sin 2 (fx) <iE + 0(1/n)
11=1
2
1
712(72
1
=:
W^
2
7T
na 2 n ./4^2
n
/(^
Arctan(-'
a
+4 n
.
1
.
+ o(-/=)
y/n'
V
2a 2 y/yn
y/n
19
+ 47)tan(|x)
4^
,A
yn' + n
+
->
x=0
0(l/n)
ii)
Proof of Theorem
the optimal matrix
2,
Q
for the
quadratic estimator:
For the optimization problem
min $H9y(n
ij
subject to
=n
^<7ii
A« r )
+
2
^2qnK =
and
i
I
0,
i
define Lagrangian function
*(Q;a,/J)
=
E€(-n + A r
«
2
-
)
«(E* - n
ij
Obviously,
(jij
=
for
^
i
For
j.
z
)
- /*(£&*)
i
=
i
j, differentiate 4>
with respect to each
Qu,
Q
=
2qil (-
n
+
\
l
r)
2
-a-p\
.
l
Conditions
J2 On
=
n
]T <?» A
and
i
=
°
lead to follow equations
1
2n
°
" a £(l/n + V)'
= a
A,
£(l/n+\r)»
A2
A
^(l/n +
and
A i r)
2+/?
^(l/n + A r) 2
i
a + /?Ai
9..
iii)
„x-
+/3
Proof of Theorem
From Theorem 1,
3,
'
2(l/n
+ V) 2
the convergence rate of the quadratic estimator
1
S (l/n + V)'
=
20
n 5/2
V?
+
0( "
S/2)
<Jq.
-3/2
A,
^(1/n + V) 2
(1/n
+
=
V) 2
+
4r 3 / 2
~~
H
+
° (n3/2)
° (n)
-
Therefore,
a =
"
(
2 „)(^
+0 („))/(g^ +
^)(-^ + ^
=
3/2
(n^))=^ + o(n-^)
7/2
))/(^^ + °c )) = -|
+ <'(^
Rewrite the variance
Var(aJ)
]Tg 2 (- +
=
a*
=
^E^K 1n +
A l7 )
2
W
+
2(7
-
r)(±
n
i
It is
easy to check that the
first
E*<5+vr
+
A,r)
+
(7
-
r)
2
(^
n
+
A,r)
2
]
term
-
TD^vr
!
+o
i)
(
= -^r + o{-=).
y/n
y/n
The other two terms can be proved by
similar techniques.
It is
a long and
tedious calculus manipulation. Numerically, one can easily verify following
equations:
i
n
y/rn
£*A? =
i
21
^7
2Vr 3 y/n
y/n
+
4)
\/n
Therefore
Var(aJ)
=
^E^ +
.j
.
(4Vf +
n
Ai7) 2
2^ + <l^)^+o(4=).
3 V"
V"
v/r
22
2V7
MIT
3
LIBRARIES
1060 00624050
b
References
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Tim and
Domowitz
Ian
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24
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51
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Date Due
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1
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l
2 y