Some asymptotic properties of a kernel spectrum estimate with different multitapers

Multitapering method maintains the good bias properties that tapering provides and at the same time produces an estimate with less variability (see [7, 11, 12, 14]). Some asymptotic statistical properties of spectral estimates were studied by several authors (see [1, 3, 4]) using a tapered data. The authors of this paper argued in [9, 10] the asymptotic expressions of the first and second-order moments of some spectral estimates, on non-overlapped and overlapped segments via different tapers and different weight functions (kernels) for both continuous time and discrete time stationary processes.

In this paper we study the problem of estimating a spectral density function (spectrum) on non-overlapped and overlapped segments using different multitapers and different kernels with a bandwidth parameter, for a discrete parameter stationary time series. In section 2 we introduce an estimate of the spectral density function using different multitapers and different kernels. Moreover, we give asymptotic expressions of the mean and variance of the average of the constructed different multitapered periodograms. In section 3 we obtain the asymptotic expressions of the mean and variance for the suggested estimate, assuming that direct spectral estimates are uncorrelated. Also, we obtain an optimal choice of the bandwidth. Furthermore, we formulate an asymptotic expression of the integrated relative mean squared error of the estimate.

• © 2007 A. A. M. Teamah, H. S. Bakouch

• 2.    The model

Suppose that X(1), X(2),..., X(N) is a realization of N observations from a real-valued stationary and discrete parameter process X(t) , t = 0, ± 1,..., with a zero mean. The spectral density function of X (t) is

∞ fxx(A) = — X Cxx(t)e-iXT, -n 6 A 6 n; i = \ I, т = — ^

where C xx ( т ) is the autocovariance function of X (t) and given by

π

C xx ( t ) = I f xx ( A ) e^ dA,

-π

∞ provided that ^2 |Cxx (т)| < ro. т= — ^

If the process X(t) is invertible, then the inverse spectral density function is defined by

∞

fXX(A) = 2; E dxx (тKiX',                      (3)

т = — ^

where d xx (т) is the inverse autocovariance of X (t) (see [2]) and given by

π dxx(т) = j fxX(A)eiAT dA,

-π

∞ such that ^2 |dxx(т)| < ro. т= — ^

We construct L segments by dividing the given observations:

X ^{( j )} (t)= X [(j - 1)M + t], j = 1, 2,...,L; t = 1, 2,...,M + q; 0 6 q < M, (4) where X ^{( j )} (t) is the set of observations in the j th segment. If N = LM + q, 0 < q < M, then the number of overlapped segments L = (N — q)/M and each segment contains M + q observations. Also, if q = 0 , then the number of non-overlapped segments L = N/M.

Now, we define the average of different multitapered periodograms as an estimate of ^f XX ^{(A) :}

L

®(A) = - XI^j(A),                      (5)

XX         XX

j=i where IXmj (A) is the multitapered periodogram of X(j)(t) and given by

K

i^j (a) = - X i(j (a)

XX          XX k=i

with

IXX (A) = 2n

M+q

i

M+q      I 1 M+q

X j(t)    E^ (t)X(j)(t)

t=i        J t=i

M+q

e-iλt

.

h j,k (t) is called the data taper for the (j, k) th direct spectral estimate I XdX ^,k (A) and equals zero outside the interval [1, M + q] and K is the number of components of multitaper in each segment.

Let h j,k (t) be a set of orthonormal tapers, that is

MM+q hTt)>i fc(t) =    ,            k

= ^m ’ ^j,kV ’   (0, otherwise ,

(see [12]). From the orthonormality we deduce that direct spectral estimates УУ(A), j = 1, 2,..., L; k = 1, 2,..., K, are uncorrelated (see [13]), and then iXX (A) = ^

M + q

E h j,k ( t ) X j ' ft ,e — t=1

In fact, the direct spectral estimate I^Xdx'^k (A) has the asymptotic properties (see [8, 13])

E [i x j (A)] * f xx (A), Var [l ^d j (A)] « f Xx (A).                (6)

Equations (5), (6) and the uncorrelation of direct spectral estimate Т ^ У (A) imply ^E [ f xX ^{) (A)} ] ~ f xx ^{(A) and Var} [ f xx ^{) (A)} ] ~ ^f xx ^(A)/KL . Obviously, ^Var [ / xx ^{) (A)} ] becomes less variability as at least K or L increases. The case when L = 1 was investigated in [13].

Smoothing the multitapered periodograms УУ (A), j = 1, 2,...,L, in equation (5) by the different kernels (weight functions) Wej)(a), — n < a 6 n, and taking their average, we get π fAmt)spma 1 X w® (A - ^^ Imtj fxx (A) = eL / v J W l в ) Ixx W d^ ' ' j=1-n X 7

which is a smoothed estimate (kernel estimate) of f xx (A) with W j (a) = 1 W ^{( j )} ( ^a ) , such that J — _n W ^{j )} (a)da = 1 ; W ^{( j )} ( — a) = W ^{( j )} (a) . в is called the bandwidth. Also, we can deduce that

L

-mt.         1     7(mt)jMi fxx (A) = LZ—jfxx (A),                            (8)

j=i where

π

l^(mt)j/\ \ _ ¹ [ Ty^{(j )} f ^A - ^ Imj (lAdu fxx ^(A) = в J W \ в ) Ixx ^W d^

-π is the jth smoothed multitapered periodogram of X(j)(t).

• 3.    Statistical properties of f ^mx)³¹¹ (A)

• 3.1.    Expected value. Taking expectation of equation (7), we get

In this section we obtain the asymptotic expressions of expectation, variance and integrated relative mean squared error of the smoothed (kernel) spectrum estimate f x^ sp (A) :

Lπ

E [. f xmx )" (A) ] « в_ XI W ^{( j )} A-^f f xx O) d^.             (9)

¹    j=¹ - n         ^{X 7}

Making use of the transformation ^ = A + ва, with small в; A G ( — n,n] , then equation (9) becomes

. L ~

E [fK ^{) s}p (A) ] « L X    W (³) ( a ) f _xx (A + ва ) da,

j - \ from a Taylor expansion for fxx (A + ва) about A, equation (10) has the form:

∞ г ,                      1 L Г                 Г                                          <12^2                             "I

E[fXX)sp(A)] = LE    W(j)(a) fxx(A) + вafXx(A) + 2.'x(A) + О(в2) da = j —^

L ^

= f xx (AH^f xx (A) X [ a ² W ^{( j )} (a) da + О(в ² ),     (11)

7=1 ^j —^

where f x_x (A) is the second derivative of the spectrum f _xx (A) . Therefore

∞

2L

Bias [fXX)sp(A)] « 2Lfxx(A) X   a2W^(а) da.

7 = 1

^j —^

It is clear that the bias of f X-Xt s (A) is of the order в ² L 1 .

• 3.2.    Variance. Since the tapered periodogram ordinates lXX’^k (^ ' ) , 1 6 ' < (M + q)/2, are asymptotically independent (see [6, 8]), then equation (8) can be put in the form:

®^{) J} (A) =    X X Z W « ( ) I(j to ■ .. «

^{XX v} ' вк J кв ) '              ■

• -____2П____X XX nz(j) ( A - ^' ^ Id)ji-ku,>

вК ( M + q) ^k=1     V в r^{XX (}^ ' )‘

Using equations (6) and (8), we get

Var fX"(A)] « вТЕ^К^Р X X [W(j) (fXX<«) ^

• " ^²УК ( M + q ) X / [ W ^{( j )} ( в )] f ^{XX (p)} d^'

j 1 —П

Putting ^ = A + ва ,

L ^

Var [fX).,(Л)] = вк(Mn q)L2 X J [W(«)] fXX(A + ва) do j=l ^

L ^

• ” вк(M*+ q)L2 fXX (A) X j [W

j 1—ro which is of the order £_PKL2(M + q)] 1

From equations (12) and (13) the mean squared error (MSE) of f XXt s (A) is

h f(mt)sp \1 Лп;яч h I(mt)sp \1A 2 h f(mt)sp 1

^MSE f xx ^(A)j — ^Bias f xx ^(A)_|/ ^{+ Var} f xx ^(A)_| ‘

The MSE of an estimate can be small only if both bias term and variance term are small. We show that the two terms are of the orders в ² L ^{— 1} and [ вKL ² ( M + q)] ^{- 1} . Then it follows that the variance and the squared bias terms of f XX) ^s (A) are balanced for ^KL ² ( M +q)] ^{- 1} ~ в 4 L ^{- 2} . This implies an optimal choice of bandwidth equals to в ~ [K(M + q)] ^{- 1 / 5} . Hence, в ^ 0 as M ^ ro . Using equations (12), (13) and the optimal choice of в , we get

Bias fX^ (A)] , Var [fXX^ (A)] ^ 0, M ^ ro ,

that is f Xx s (A) is a consistent estimate of f xx (A) as M ^ ro . Also, Var f XX sp (A )] becomes less variability as at least K or L increases.

• 3.3.    Integrated relative mean squared error. We take IMSE as a measure for the goodness of fit of a spectral estimate. IMSE of f Xx s (A) is defined by (see [5]):

  IMSE h f XmX^ sp (A) 1 — / E " f ^{XX ( A )} - f ^{XX ( A )} 1 dA —

  ^{L J} 7             f xx ^(A)

  - π

  
  

  π

  ^—

  - π

  
  

  e [/Xmt ^{) sp} (A)]

  
  

  f X X (A)

  
  
  
  

  1 dA +

  
  

  π

  Var

  
  

  X ( mt ) sp ^f XX

  
  
  
  

  (A)

  
  

  - π

  
  

  f X X (A)

  
  

  dλ.

  
  

Hence,

π

( mt )                                 2        ( mt )

^IMSE [/Xx ^(A)]^{—2n +} / f XX ^(A)] (Ef Xx ^(A)J) ^dA

-πππ

9 f f^-1(\ 'I F h f^(mt) p( 1 _rl\ I f £ f ^{— 1} /\^1²Vяг h f^(mt) p( X^l /7Х

• ² / f xx ^{( A ) E} f xx ^(A)] ^{dA +} / f xx ^(A)J ^Var f xx ^(A)] ^dA*

-π                          -π

From equations (3), (11) and (13), then formula (15) has the form:

∞∞

IMSE f f mX p ^p (A)] » 2п + (2п) ^{- 2} X X d xx ( t ) d xx (r) x T = — ^ r = -^

π

+

π

L

∞

x

j f xx (A) + 2L/ Xx (A) ^X j a ² W ^{( j )} ( a ) da n                     j ^-1 -^

-

π

∞

e

-

^{iA ( T + r )} dA

^                      22L

52 d xx (T)     f xx (A) + 2Lf Xx (A) £    a ² w ^{( j )} ( a ) da

= -oo        27=1

—nj

∞

T=

(2n) ^{— 1}

вKL ² ( M + q)

π

π

L

∞

∞

∞∞

X X d xx ^{( t )d} xx ^{(r )} / ^f fX x ^(A) X [^{W ( j ) (a)}]

T = — m r= — m               ____ 7=1

-∞ -π

∞

π

L

e

e

-

-

iλτ

dA+

^{iA ( T + r )} dAda.

If we consider the optimal choice of в, в ~ [K(M + q)] ^{1 / 5} , then equation (16) can be put in the form:

∞∞

IMSE [f XX ^{) sp} (A)] и 2n + (2n) ^{- 2} 52 E d XX ( t№x ( r ) T = — ^ r = -^

π

/ fXX (A)e ^{-iA ( T + r )} dA

- π

-

∞

- п ^{- 1} X d XX ( t )

T = — ^

π

У f XX ( A ) e^^dA

-π

∞ и 2п - п-1 X dXX(t)Cxx(-t) +

T = — ^

∞∞

+ (2п) ^{- 2} 52 E d XX ( t ) d XX ( r )

T = — ^ r = -^

π

У f XX (A)e ^{-iA ( T + r )} dA

-π

M → ∞ .