Georgian Mathematical Journal 1(1994), No. 2, 197-212

LIMIT DISTRIBUTION OF THE INTEGRATED SQUARED

ERROR OF TRIGONOMETRIC SERIES REGRESSION ESTIMATOR

E. NADARAYA

Abstract. Limit distribution is studied for the integrated squared error of the projection regression estimator (2) constructed on the basis of independent observations (1). By means of the obtained limit theorems, a test is given for verifying the hypothesis on the regression, and the power of this test is calculated in the case of Pitman alternatives.

Let observations Y_{1}, Y_{2}, …, Y_{n} be represented as

Y_{i} = μ(x_{i}) + ε_{i}, i = \overline{1, n},  (1)

where μ(x), x ∈ [−π, π], is the unknown regression function to be estimated by observations Y_{i}; x_{i}, i = \overline{1, n}, are the known numbers, and −π = x_{0} < x_{1} < ⋯ < x_{n} ≤ π, ε_{i}, i = \overline{1, n}, are independent equidistributed random variables; Eε_{1} = 0, Eε_{1}^{2} = σ^{2}, and Eε_{1}^{4} < ∞.

The problem of nonparametric estimation of the regression function μ(x) for the model (1) has a recent history and has been treated only in few papers. In particular, a kernel estimator of the Rosenblatt–Parzen type for μ(x) was proposed for the first time in [1].

Assume that μ(x) is representable as a converging series in L_{2}(−π, π) with respect to the orthonormal trigonometric system

{(2π)^{−1/2}, π^{−1/2} cos ix, π^{−1/2} sin ix}_{i=1}^{∞} .

Consider the estimator of the function μ(x) constructed by the projection method of N.N. Chentsov [2]

μ_{nN}(x) = \frac{a_{0n}}{2} + ∑_{i=1}^{N} a_{in} cos ix + b_{in} sin ix,  (2)

1991 Mathematics Subject Classification. 62G07.

where N = N(n) → ∞ for n → ∞ and

a_{in} = \frac{1}{π} ∑_{j=1}^{n} Y_{j}Δ_{j} cos ix_{j}, b_{in} = \frac{1}{π} ∑_{j=1}^{n} Y_{j}Δ_{j} sin ix_{j},

Δ_{j} = x_{j} − x_{j−1}, j = \overline{1, n}, i = \overline{0, N}.

The estimator (2) can be rewritten in a more compact way as

μ_{nN} = ∑_{j=1}^{n} Y_{j}Δ_{j}K_{N}(x − x_{j}),

where K_{N}(u) = \frac{1}{2π} ∑_{|r|≤N} e^{iru} is the Dirichlet kernel.

In [3], p.347, N.V. Smirnov considered estimators of the type (2) for a specially chosen class of functions μ(x) in the case of equidistant points x_{j} ∈ [−π, π] and of independent and normally distributed observation errors ε_{i}. In [4] an estimator of the type (2) is obtained, which is asymptotically equivalent to projection estimators which are optimal in the sense of some accuracy criterion. The asymptotics of the mean value of the integrated squared error of the estimator (2) is considered in [5].

It is of interest to investigate the limit distribution of the integrated squared error

∫_{−π}^{π}[μ_{nN}(x) − μ(x)]^{2}dx,

which is the goal pursued in this paper. The method used to prove the theorems below is based on the functional limit theorem for a sequence of semimartingales [6].

Denote

U_{nN} = \frac{n}{2π(2N + 1)} ∫_{−π}^{π} [μ_{nN}(x) − Eμ_{nN}(x)]^{2} dx,

Q_{ir} = Δ_{i}Δ_{r}K_{N}(x_{i} − x_{r}), σ_{nN}^{2} = \frac{n^{2}σ^{4}}{π^{2}(2N + 1)^{2}} ∑_{r=2}^{n} ∑_{j=1}^{r−1} Q_{jr}^{2},

η_{ik} = \frac{n}{π(2N + 1)σ_{nN}} ε_{i}ε_{k}Q_{ik},

ξ_{1} = 0, ξ_{k} = ∑_{i=1}^{k−1} η_{ik}, k = \overline{2, n}, ξ_{k} = 0, k > n,

and assume that F_{k} is σ-algebra generated by random variables ε_{1}, ε_{2}, …, ε_{k}, F_{0} = (ϕ, Ω).

Lemma 1 ([7], p.179). The stochastic sequence (ξ_{k}, F_{k})_{k≥1} is a martingale-difference.

Lemma 2. Let p(x) be the known positive continuously differentiable distribution density on [−π, π], and points x_{i} be chosen from the relation ∫_{−π}^{x_{i}} p(u) du = \frac{i}{n}, i = \overline{1, n}.

If \frac{N ln N}{n} → 0 for n → ∞, then

EU_{nN} = θ_{1} + O (\frac{N ln N}{n} ) , θ_{1} = \frac{σ^{2}}{(2π)^{2}} ∫_{−π}^{π} p^{−1}(u) du,  (3)
(2N + 1)σ_{nN}^{2} → θ_{2} = \frac{σ^{4}}{4π^{3}} ∫_{−π}^{π} p^{−2}(u) du.  (4)

Proof. From the definition of x_{i} we easily obtain

Δ_{i} = \frac{1}{np(x_{i})} [1 + O(\frac{1}{n})] ,

where O(\frac{1}{n}) is uniform with respect to i = \overline{1, n}.

Hence it follows that

Q_{ir} = \frac{1}{n^{2}p(x_{i})p(x_{r})}K_{N}(x_{i} − x_{r}) [1 + O(\frac{1}{n})] .  (5)

Taking into account the relation

max_{−π≤u≤π} |K_{N}(u)| = O(N)  (6)

and (5), we find

σ_{nN}^{2} = \frac{σ^{4}}{2π^{2}(2N +1)^{2}n^{2}} ∑_{i=1}^{n} ∑_{j=1}^{n} K_{N}^{2}(x_{i}−x_{j})\frac{1}{[p(x_{i})p(x_{j})]^{2}} +O(\frac{1}{n}). (7)

Let F (x) be a distribution function with density p(x) and F_{n}(x) be an empirical distribution function of the “sample” x_{1}, x_{2}, …, x_{n}, i.e., F_{n}(x) = n^{−1} ∑_{k=1}^{n} I_{(−∞,x)}(x_{k}), where I_{A}(·) is the indicator of the set A. Then the right side of (7) can be written as the integral

σ_{nN}^{2} = \frac{σ^{4}}{2π^{2}(2N + 1)^{2}} ∫_{−π}^{π} ∫_{−π}^{π} K_{N}^{2}(t − s)\frac{dF_{n}(t) dF_{n}(s)}{[p(t)p(s)]^{2}} + O(\frac{1}{n}).

Further we have

|∫_{−π}^{π} ∫_{−π}^{π} K_{N}^{2}(t − s)\frac{dF_{n}(t) dF_{n}(s)}{[p(t)p(s)]^{2}} − ∫_{−π}^{π} ∫_{−π}^{π} K_{N}^{2}(t − s)\frac{dF (t) dF (s)}{[p(t)p(s)]^{2}} | ≤

≤ I_{1} + I_{2},

I_{1} = |∫_{−π}^{π} ∫_{−π}^{π} K_{N}^{2}(t − s)\frac{dF_{n}(s)}{[p(t)p(s)]^{2}} [dF_{n}(t) − dF (t)]| ,

I_{2} = |∫_{−π}^{π} ∫_{−π}^{π} K_{N}^{2}(t − s)\frac{dF (t)}{[p(t)p(s)]^{2}} [dF_{n}(s) − dF (s)]| .

By integration by parts in the internal integral in I_{1} we readily obtain

I_{1} ≤ 2 ∫_{−π}^{π} \frac{dF_{n}(s)}{p^{2}(s)} ∫_{−π}^{π} |dF_{n}(t) − dF (t)| |(K′_{N}(t − s)p(t) −

−K_{N}(t − s)p′(t))K_{N}(t − s)/p^{3}(t)| dt.  (8)

Since sup_{−π≤x≤π} |F_{n}(x) − F (x)| = O(\frac{1}{n}) and the relations [8]^{1}

\begin{matrix}max_{−π≤u≤π} |K′_{N}(u)| = O(N^{2}), ∫_{−π}^{π} K_{N}^{2}(u) du = 2N + 1, \\ ∫_{−π}^{π} |K_{N}(u)| du = O(ln N) \end{matrix}  (9)

are fulfilled, from (8) we have the estimate

I_{1} = O (\frac{N^{2} ln N}{n} ) .

In the same manner we show that

I_{2} = O (\frac{N^{2} ln N}{n} ) .

Therefore

(2N +1)σ_{nN}^{2} = \frac{σ^{4}}{4π^{3}} ∫_{−π}^{π} ∫_{−π}^{π} Φ_{N}(s−t)\frac{dt ds}{p(s)p(t)} +O (\frac{N ln N}{n} ) , (10)

where Φ_{N}(u) = \frac{2π}{2N+1}K_{N}^{2}(u) is the Fejér kernel.

We shall complete the definition of the function p^{−1} outside [−π, π] as regards its periodicity and also note that K_{N}(u) and Φ_{N}(u) are periodic functions with the period 2π. The continued function will be denoted by g(x). Then

∫_{−π}^{π} ∫_{−π}^{π} Φ_{N}(s − t)\frac{dt ds}{p(s)p(t)} = ∫_{−π}^{π} p^{−2}(x) dx + χ_{n},

1See p. 115 in the Russian version of [8]: “Mir”, Moscow, 1965.

where

|χ_{n}| ≤ ∫_{−π}^{π} |σ̄_{N}(x) − g(x)|dx,
σ̄_{N}(x) = ∫_{−π}^{π} Φ_{N}(u)g(x − u)du.

Hence, on account of the theorem on convergence of the Fejér integral σ̄_{N}(x) to g(x) in the norm of the space L_{1}(−π, π) (see [9], p.481), we have χ_{n} → 0 for n → ∞.

Therefore

(2N + 1)σ_{nN}^{2} → \frac{σ^{4}}{4π^{3}} ∫_{−π}^{π} p^{−2}(x) dx.

Now we shall prove (3). We have

Dμ_{nN}(x) = σ^{2} ∑_{j=1}^{n} \frac{1}{np^{2}(x_{j})}K_{N}^{2}(x − x_{j}) [1 + O(\frac{1}{n})] .

Applying the same reasoning as in deriving (10), we find

Dμ_{nN}(x) = \frac{σ^{2}}{n} ∫_{−π}^{π} K_{N}^{2}(x − s)\frac{ds}{p(s)} + O(\frac{N^{2} ln N}{n^{2}} ).  (11)

Therefore

EU_{nN} = \frac{σ^{2}}{(2π)^{2}} ∫_{−π}^{π} ∫_{−π}^{π} Φ_{N}(t − s)\frac{ds dt}{p(s)} + O(\frac{N ln N}{n} ) =

= \frac{σ^{2}}{(2π)^{2}} ∫_{−π}^{π} p^{−1}(s) ds + O(\frac{N ln N}{n} ).

Denote by the symbol →^{d} the convergence in distribution, and let ξ be a random variable having normal distribution with zero mean and variance 1.

Theorem 1. Let x_{i} i = \overline{1, n} be the same as in Lemma 2 and \frac{N^{2} ln N}{n} → 0

for n → ∞. Then, as n increases, \sqrt{2N + 1}(U_{nN} − θ_{1})θ_{2}^{−1/2} →^{d} ξ.

Proof. We have

\frac{U_{nN} − EU_{nN}}{σ_{nN}} = H_{n}^{(1)} + H_{n}^{(2)},

where

H_{n}^{(1)} = ∑_{j=1}^{n} ξ_{j}, H_{n}^{(2)} = \frac{n}{2π(2N + 1)σ_{nN}} ∑_{i=1}^{n}(ε_{i}^{2} − Eε_{i}^{2})Q_{ii}.

H_{n}^{(2)} converges to zero in probability. Indeed,

DH_{n}^{(2)} ≤ \frac{n^{2}Eε_{1}^{4}}{(2π)^{2}(2N + 1)^{2}σ_{nN}^{2}} ∑_{i=1}^{n} Q_{ii}^{2} =

= \frac{Eε_{1}^{4}}{(2π)^{2}(2N + 1)^{2}σ_{nN}^{2} · n^{2}} ∑_{i=1}^{n} \frac{1}{(p(x_{i}))^{4}} K_{N}^{2}(0) (1 + O(\frac{1}{n})) ≤

≤ C \frac{1}{nσ_{nN}^{2}} = O(\frac{N}{n} ),

whence H_{n}^{(2)} →^{P} 0. Here and in what follows C is the positive constant varying from one formula to another and the letter P above the arrow denotes convergence in probability.

We will now prove that H_{n}^{(1)} →^{d} ξ. To this end we will verify the validity of Corollaries 2 and 6 of Theorem 2 from [6]. We have to show whether the conditions contained in these statements are fulfilled for asymptotic normality of the square-integrable martingale-difference, which, by Lemma 1, is our sequence {ξ_{k}, F_{k}}_{k≥1}.

A direct calculation shows that ∑_{k=1}^{n} Eξ_{k}^{2} = 1. Asymptotic normality will take place if for n → ∞

∑_{k=1}^{n} E[ξ_{k}^{2} · I(|ξ_{k}| ≥ ε) | F_{k−1}] → 0  (12)

and

∑_{k=1}^{n} ξ_{k}^{2} →^{P} 1.  (13)

It is shown in [6] that the fulfillment of (13) and the condition sup_{1≤k≤n} |ξ_{k}| →^{P} 0

implies the validy of (12) as well.

Since for ε > 0

P { sup_{1≤k≤n} |ξ_{k}| ≥ ε} ≤ ε^{−4} ∑_{k=1}^{n} Eξ_{k}^{4},

to prove H_{n}^{(1)} →^{d} ξ we have to verify only (13) by the relation (15) to be given below.

We will establish ∑_{k=1}^{n} ξ_{k}^{2} →^{P} 1. For this it suffices to make sure that E(∑_{k=1}^{n} ξ_{k}^{2} − 1))^{2} → 0 for n → ∞, i.e., due to ∑_{i=1}^{n} Eξ_{i}^{2} = 1

E( ∑_{k=1}^{n} ξ_{k}^{2})^{2} = ∑_{k=1}^{n} Eξ_{k}^{4} + 2 ∑_{1≤k_{1}<k_{2}≤n} Eξ_{k_{1}}^{2}ξ_{k_{2}}^{2} → 1.  (14)

LIMIT DISTRIBUTION OF THE SQUARED ERROR 203

In the first place we find that ∑_{k=1}^{n} Eξ_{k}^{4} → 0 for n → ∞. By virtue of the definitions of ξ_{k} and η_{ij} we write

∑_{k=1}^{n} Eξ_{k}^{4} = L_{n}^{(1)} + L_{n}^{(2)},

where

L_{n}^{(1)} = \frac{n^{4}}{π^{4}(2N + 1)^{4}σ_{nN}^{4}} Eε_{1}^{4}(Eε_{1}^{4} − 3σ^{4}) ∑_{k=2}^{n} ∑_{j=1}^{k−1} Q_{jk}^{4},

L_{n}^{(2)} = \frac{3n^{4}σ^{4}Eε_{1}^{4}}{(2N + 1)^{4}σ_{nN}^{4}π^{4}} ∑_{k=2}^{n} ( ∑_{j=1}^{k−1} Q_{jk}^{2})^{2}.

From (5) and (6) we obtain

|L_{n}^{(1)}| = C \frac{1}{n^{4}N^{4}σ_{nN}^{4}} ∑_{k=2}^{n} ∑_{j=1}^{k−1} \frac{K_{N}^{4}(x_{j} − x_{k})}{[p(x_{j})p(x_{k})]^{4}}[1 + O(\frac{1}{n})] ≤

≤ Cn^{−2}σ_{nN}^{−4} = O((\frac{N}{n} )^{2})

and also

|L_{n}^{(2)}| = C \frac{1}{n^{2}N^{4}σ_{nN}^{4}} ∑_{k=2}^{n} (\frac{1}{n} ∑_{j=1}^{k−1} \frac{K_{N}^{2}(x_{j} − x_{k})}{p(x_{j})p(x_{k})}[1 + O(\frac{1}{n})])^{2} ≤

≤ C \frac{1}{n^{2}N^{4}σ_{nN}^{4}} ∑_{k=1}^{n} (\frac{1}{n} ∑_{j=1}^{k−1} K_{N}^{2}(x_{j} − x_{k}))^{2} =

= C \frac{1}{n^{2}N^{4}σ_{nN}^{4}} ∑_{k=2}^{n} (∫_{−π}^{π} \frac{K_{N}^{2}(x_{k} − u)}{p^{2}(u)}dF_{n}(u))^{2} ≤

≤ C \frac{1}{n^{2}N^{4}σ_{nN}^{4}} ∑_{k=2}^{n} \{[K_{N}^{2}(x_{k} − u)p^{−1}(u) du]^{2}+

+[ ∫_{−π}^{π} K_{N}^{2}(x_{k} − u)p^{−2}(u) d(F_{n}(u) − F (u))]^{2}}^{2} .

Hence, taking into account the relation (9) and the formula of integration by parts, we have |L_{n}^{(2)}| = O(\frac{1}{n}). Therefore

∑_{k=1}^{n} Eξ_{k}^{4} → 0 for n → ∞.  (15)

Let us now establish that 2 ∑_{1≤k_{1}<k_{2}≤n} Eξ_{k_{1}}^{2}ξ_{k_{2}}^{2} → 1 for n → ∞. The definition of ξ_{i} implies

ξ_{k_{1}}^{2}ξ_{k_{2}}^{2} = ( ∑_{i=1}^{k_{1}−1} η_{ik_{1}}^{2})( ∑_{i=1}^{k_{2}−1} η_{ik_{2}}^{2}) + ( ∑_{i=1}^{k_{1}−1} η_{ik_{1}}^{2})( ∑_{i≠s=1}^{k_{2}−1} η_{ik_{2}}η_{sk_{2}}) +

+( ∑_{i=1}^{k_{2}−1} η_{ik_{2}}^{2})( ∑_{s≠t=1}^{k_{1}−1} η_{sk_{1}}η_{tk_{1}}) + ( ∑_{s≠t=1}^{k_{1}−1} η_{sk_{1}}η_{tk_{1}})( ∑_{k≠r=1}^{k_{2}−1} η_{kk_{1}}η_{rk_{2}}) =

= B_{k_{1}k_{2}}^{(1)} + B_{k_{1}k_{2}}^{(2)} + B_{k_{1}k_{2}}^{(3)} + B_{k_{1}k_{2}}^{(4)}.

Therefore

2 ∑_{1≤k_{1}<k_{2}≤n} Eξ_{k_{1}}^{2}ξ_{k_{2}}^{2} = ∑_{i=1}^{4} A_{n}^{(i)},

where

A_{n}^{(i)} = 2 ∑_{1≤k_{1}<k_{2}≤n} EB_{k_{1}k_{2}}^{(i)}, i = \overline{1, 4}.

In the first place we consider A_{n}^{(3)}. By the definition of η_{ij} we obtain Eη_{ik_{2}}^{2}η_{sk_{1}}η_{tk_{1}} = 0, s ≠ t, k_{1} < k_{2}. Thus

A_{n}^{(3)} = 0.  (16)

Let us derive an estimate of A_{n}^{(2)}. Divide the sum EB_{k_{1}k_{2}}^{(2)} into two parts:

EB_{k_{1}k_{2}}^{(2)} = ∑_{i=1}^{k_{1}−1} ∑_{r≠s=1}^{k_{1}} Eη_{ik_{1}}^{2}η_{rk_{2}}η_{sk_{2}} + ∑_{i=1}^{k_{1}−1} ∑_{r≠s=k_{1}+1}^{k_{2}−1} Eη_{ik_{1}}^{2}η_{rk_{2}}η_{sk_{2}}.

The second term is equal to zero, since i cannot coincide with r or with s and r ≠ s; in this case Eη_{ik_{1}}^{2}η_{rk_{2}}η_{sk_{2}} = 0, and Eη_{ik_{1}}^{2}η_{rk_{2}}η_{sk_{2}} = 0 also in the first term each time except for the case s = k_{1} or r = k_{1}.

Thus

EB_{k_{1}k_{2}}^{(2)} = 2 ∑_{i=1}^{k_{1}−1} E(η_{ik_{1}}^{2}η_{ik_{2}}η_{k_{1}k_{2}}).

Hence, using the definition of η_{ij} and the inequality |Q_{ij}| ≤ C \frac{N}{n^{2}} obtained from (5) and (6), we find

|EB_{k_{1}k_{2}}^{(2)} | ≤ C \frac{1}{(2N + 1)^{2}σ_{nN}^{4}} ∑_{i=1}^{k_{1}−1} Q_{ik_{1}}^{2}.  (17)

LIMIT DISTRIBUTION OF THE SQUARED ERROR 205

Next, taking into account statement (4) of Lemma 2 and the definition of σ_{nN}^{2}, from (17) we have

|A_{n}^{(2)}| ≤ C \frac{n}{N^{2}σ_{nN}^{4}} ∑_{k_{1}=2}^{n} ∑_{i=1}^{k_{1}−1} Q_{ik_{1}}^{2} ≤ C \frac{1}{nσ_{nN}^{2}} = O(\frac{N}{n} ).  (18)

Consider now A_{n}^{(4)}. By the definition of η_{ij} we obtain

A_{n}^{(4)} = \frac{8n^{4}}{π^{4}(2N + 1)^{4}σ_{nN}^{4}} ∑_{s<t<k_{1}<k_{2}} Q_{sk_{1}}Q_{sk_{2}}Q_{tk_{1}}Q_{tk_{2}} ≤

≤ C \frac{n^{4}}{N^{4}σ_{nN}^{4}} [| ∑_{s,t,k_{1},k_{2}} Q_{sk_{1}}Q_{sk_{2}}Q_{tk_{1}}Q_{tk_{2}} | +

+| ∑_{k_{1},s,t} Q_{k_{1}s}^{2}Q_{k_{1}t}^{2}| + | ∑_{k_{1},s,t} Q_{k_{1}t}Q_{st}Q_{k_{1}s}Q_{ss}|] =

= C \frac{n^{4}}{N^{4}σ_{nN}^{4}} [|E_{1}| + |E_{2}| + |E_{3}|].  (19)

According to (5) and (6) we write

E_{1} = n^{−7} ∑_{s,t,k_{1}} K_{N}(x_{s} − x_{k_{1}})K_{N}(x_{t} − x_{k_{1}}) ×

× ∫_{−π}^{π} K_{N}(x_{s} − u)K_{N}(x_{t} − u) dF_{n}(u) + O(\frac{N^{2}}{n} ).

Hence, integrating by parts and taking into account (9), we obtain

E_{1} = n^{−7} ∫_{−π}^{π} ∑_{s,t,k_{1}} K_{N}(x_{s} − x_{k_{1}})K_{N}(x_{t} − x_{k_{1}}) ×

K_{N}(x_{s} − u)K_{N}(x_{t} − u)p(u) du + O(\frac{N^{4} ln N}{n^{5}} ).  (20)

Applying the same operations three times, we represent (20) in the form E_{1} = n^{−4} ∫_{−π}^{π} ∫_{−π}^{π} ∫_{−π}^{π} ∫_{−π}^{π} K_{N}(z − u)K_{N}(z − t)K_{N}(y − u)K_{N}(y − t) ×

×p(y)p(u)p(z)p(t) du dt dz dy + O(\frac{N^{4} ln N}{n^{5}} ) =

= O(\frac{N ln^{3} N}{n^{4}} ) + O(\frac{N^{4} ln N}{n^{5}} ).

206 E. NADARAYA

Thus

\frac{n^{4}}{N^{4}σ_{nN}^{4}} |E_{1}| = O(\frac{ln^{3} N}{N} ) + O(\frac{N^{2} ln N}{n} ).  (21)

Further, it is not difficult to show

\begin{matrix}\frac{n^{4}}{N^{4}σ_{nN}^{4}} |E_{2}| = O(\frac{N^{2}}{n} ), \\ \frac{n^{4}}{N^{4}σ_{nN}^{4}} |E_{3}| = O(\frac{N^{2}}{n} ). \end{matrix}  (22)

Therefore (19), (21), and (22) imply

A_{n}^{(4)} = O(\frac{N^{2} ln N}{n} ) + O(\frac{ln^{3} N}{N} ).  (23)

Finally, we will show that A_{n}^{(1)} → 1 for n → ∞. For this represent A_{n}^{(1)} in the form A_{n}^{(1)} = Q_{n}^{(1)} + Q_{n}^{(2)}, where

Q_{n}^{(1)} = 2 ∑_{k_{1}<k_{2}} ( ∑_{i=1}^{k_{1}−1} Eη_{ik_{1}}^{2})( ∑_{j=1}^{k_{2}−1} Eη_{ik_{2}}^{2}),

Q_{n}^{(2)} = 2( ∑_{k_{1}<k_{2}} EB_{k_{1}k_{2}}^{(1)} − ∑_{k_{1}<k_{2}} ( ∑_{i=1}^{k_{1}−1} Eη_{ik_{1}}^{2})( ∑_{j=1}^{k_{2}−1} Eη_{ik_{2}}^{2})).

From the definition of σ_{nN}^{2} it follows that

Q_{n}^{(1)} = 1 − ∑_{k=2}^{n} ( ∑_{i=1}^{k−1} Eη_{ik}^{2})^{2},

where

∑_{k=2}^{n} ( ∑_{i=1}^{k−1} Eη_{ik}^{2})^{2} ≤ C \frac{n^{4}}{N^{4}σ_{nN}^{4}} ∑_{k=2}^{n} ( ∑_{i=1}^{k−1} Q_{ik}^{2})^{2} ≤

≤ C \frac{1}{nσ_{nN}^{4}} = O(\frac{N^{2}}{n} ).

Therefore

Q_{n}^{(1)} = 1 + O(N^{2}/n).  (24)

Let us now show that Q_{n}^{(2)} → 0. Q_{n}^{(2)} can be written as

Q_{n}^{(2)} = 2 ∑_{k_{1}<k_{2}} [ ∑_{i=1}^{k_{1}−1} (cov(η_{ik_{1}}^{2}, η_{ik_{2}}^{2}) + cov(η_{ik_{1}}^{2}, η_{k_{1}k_{2}}^{2}))].

But

Eη_{ik_{1}}^{2}η_{ik_{2}}^{2} ≤ C \frac{n^{4}}{N^{4}σ_{nN}^{4}} Q_{ik_{1}}^{2} · Q_{ik_{2}}^{2} ≤

≤ C \frac{1}{n^{4}N^{4}σ_{nN}^{4}} ( max_{−π≤u≤π} |K_{N}(u)|)^{4} = O(\frac{1}{n^{4}σ_{nN}^{4}} ).

Similarly, Eη_{ij}^{2} = O(n^{−2}σ_{nN}^{−2}). Therefore

cov(η_{ik_{1}}^{2}, η_{ik_{2}}^{2}) = O(\frac{1}{n^{4}σ_{nN}^{4}} ).  (25)

Further, since ∑_{1≤k_{1}<k_{2}≤n}(k_{1} − 1) = O(n^{3}), (25) implies

Q_{n}^{(2)} = O(\frac{N^{2}}{n} ).  (26)

Thus, according to (24) and (26)

A_{n}^{(1)} = 1 + O(N^{2}/n).  (27)

Combining the relations (16), (18), (23) and (27), we finally obtain

E( ∑_{k=1}^{n} ξ_{k}^{2} − 1)^{2} → 0 for n → ∞.

Therefore

\frac{U_{nN} − EU_{nN}}{σ_{nN}} →^{d} ξ.

Further, due to Lemma 2, EU_{nN} = θ_{1} + O(\frac{N ln N}{n}) and (2N + 1)σ_{nN}^{2} → θ_{2}, and hence we obtain

(2N + 1)^{1/2}(U_{nN} − θ_{1})θ_{2}^{−1/2} →^{d} ξ.

Denote

T_{nN} = \frac{n}{2π(2N + 1)} ∫_{−π}^{π} [μ_{nN}(x) − μ(x)]^{2} dx.

Theorem 2. Let x_{i}, i = \overline{1, n}, be the same as in Lemma 2 and the function μ(x) with period 2π have bounded derivatives up to the second order. Moreover, if N^{2} ln N/n → 0 and n ln^{2} N/N^{9/2} → 0 for n → ∞, then

\sqrt{2N + 1}(T_{nN} − θ_{1})θ_{2}^{−1/2} →^{d} ξ.

208 E. NADARAYA

Before we proceed to proving the theorem, we have to show

∫_{−π}^{π} |K′_{N}(u)|du = O(N ln N).  (28)

Denote D̃_{ν}(u) = ∑_{k=1}^{ν} sin ku. Then by virtue of the Abel transformation we have

K′_{N}(u) = − ∑_{k=1}^{N} k sin ku = ∑_{ν=1}^{N−1} D̃_{ν}(u) + ND̃_{N}.

It is well known [8] that η_{ν} = (ln ν)^{−1} ∫_{−π}^{π} |D̃_{ν}(u)|du → 1 for ν → ∞. Denote b_{N} = ∑_{ν=1}^{N−1} ln ν. Then by the Toeplitz lemma

R_{N} = \frac{1}{b_{N}} ∑_{ν=1}^{N−1} ln ν · η_{ν} → 1.

Therefore

∫_{−π}^{π} |K′_{N}(u)|du ≤ ∑_{ν=1}^{N−1} ∫_{−π}^{π} |D̃_{ν}(u)|du + N ∫_{−π}^{π} |D̃_{N}(u)|du =

= b_{N} · R_{N} + N ∫_{−π}^{π} |D̃_{N}(u)|du = O(N ln N).

Let us return to the proof of the theorem. We have

T_{nN} = U_{nN} + A_{1n} + A_{2n},

A_{1n} = \frac{n}{π(2N + 1)} ∫_{−π}^{π} [μ_{nN}(x) − Eμ_{nN}(x)][Eμ_{nN}(x) − μ(x)]dx,

A_{2n} = \frac{n}{2π(2N + 1)} ∫_{−π}^{π} [Eμ_{nN}(x) − μ(x)]^{2}dx.

It is not difficult to find

\sqrt{2N + 1}E|A_{1n}| ≤ \frac{nσ^{2}}{2π\sqrt{2N + 1}}( ∑_{j=1}^{n} Δ_{j}^{2}[ ∫_{−π}^{π} K_{n}(y − x_{j}) ×

×(Eμ_{nN}(y) − μ(x))dy]^{2})^{1/2}.

But

Eμ_{nN}(y) = ∫_{−π}^{π} μ(x)\frac{1}{p(x)}K_{N}(y − x) dF_{n}(x)(1 + O(\frac{1}{n}))

and

∫_{−π}^{π} μ(x)p^{−1}(x)K_{N}(y − x) dF_{n}(x) =

= ∫_{−π}^{π} μ(x)K_{N}(y − x) + O(\frac{1}{n} ∫_{−π}^{π} |K′_{N}(u)|du).

It is well known ([10], p.22) that

∫_{−π}^{π} μ(x)K_{N}(y − x) dx = μ(y) + O(\frac{ln N}{N^{2}} )

uniformly in y ∈ [−π, π]. By virtue of (28) this gives us

Eμ_{nN}(x) = μ(x) + O(\frac{ln N}{N^{2}} ) + O(\frac{N ln N}{n} ).  (29)

Therefore

\sqrt{2N + 1}E|A_{1n}| ≤ C[(\frac{n ln^{2} N}{N^{9/2}} )^{1/2} \frac{ln N}{N^{1/4}} +

+(\frac{N^{2} ln N}{n} )^{1/2} \frac{ln^{3/2} N}{\sqrt{N}} ] → 0.  (30)

Further, from (29) we have

\sqrt{2N + 1}A_{2n} ≤ C(\frac{n ln^{2} N}{N^{9/2}} + \frac{N^{2}}{n} \frac{ln^{2} N}{\sqrt{N}} ) → 0.  (31)

Finally, the statement of Theorem 2 directly follows from Theorem 1, (30), and (31).

Using Theorems 1 and 2, it is easy to solve the problem concerning testing of the hypothesis on μ(x). Given σ^{2}, it is required to verify the hypothesis H_{0} : μ(x) = μ_{0}(x). The critical region is defined approximately by the inequality U_{nN} ≥ d_{n}(α) or T_{nN} ≥ d_{n}(α), where

d_{n}(α) = σ^{2}(L_{1} + (2N + 1)^{−1/2}L_{2})λ_{α},

L_{1} = ((2π)^{−2} ∫_{−π}^{π} p^{−1}(x) dx, L_{2} = (\frac{1}{4π^{3}} ∫_{−π}^{π} p^{−2}(x) dx)^{1/2},

and λ_{α} is the quantile of level α of standard normal distribution.

Let now σ^{2} be unknown. We call an \sqrt{N}-consistent estimate of variance σ^{2}, for instance,

S_{n}^{2} = \frac{1}{n} ∑_{i=1}^{n} (Y_{i} − μ_{nλ}(x_{i}))^{2},

where λ = λ(n) → ∞ is a sequence such that \frac{λ}{N} → 0, \frac{N ln^{2} λ}{λ^{4}} → 0 and \frac{Nλ^{4}}{n} → 0 for n → ∞.

Indeed, using the expressions (11) and (29), we easily find

\sqrt{N}(ES_{n}^{2} − σ^{2}) = O((\frac{Nλ}{n})^{1/2}) + O(\frac{N^{1/2} ln λ}{λ^{2}}).  (32)

Denote

Z_{j} = Y_{j} − R_{j},

R_{j} = ∑_{k=1}^{n} Y_{k}Δ_{k}K_{λ}(x_{j} − x_{k}).

Then

n^{2}DS_{n}^{2} = ∑_{j=1}^{n} DZ_{j}^{2} + ∑_{i≠i_{1}} cov(Z_{i}^{2}, Z_{i_{1}}^{2}).

Simple calculations show that cov(Z_{j}^{2}, Z_{j_{1}}^{2}) = O(\frac{λ^{4}}{n} ). Therefore DS_{n}^{2} =

O(\frac{λ^{4}}{n} ). This and (32) imply \sqrt{N}(S_{n}^{2} − σ^{2}) →^{P} 0.

Corollary. Let the conditions of Theorem 2 be fulfilled. Moreover, let \frac{λ}{n} → 0, \frac{Nλ^{4}}{n} → 0 and \frac{N ln^{2} λ}{λ^{4}} → 0. Then

S_{n}^{−2}L_{2}^{−1}\sqrt{2N + 1}(U_{nN} − S_{n}^{2}L_{1}) →^{d} ξ,

S_{n}^{−2}L_{2}^{−1}\sqrt{2N + 1}(T_{nN} − S_{n}^{2}L_{1}) →^{d} ξ.

This corollary enables one to construct a test for verifying H_{0} : μ(x) = μ_{0}(x). The critical region is defined approximately by the inequality U_{nN} ≥ d̃_{n}(α) or T_{nN} ≥ d̃_{n}(α), where d̃_{n}(α) is obtained from d_{n}(α) by using S_{n}^{2} instead of σ^{2}.

Consider now the local behavior of the test power in the case where the critical region is of the form {x ∈ R^{1}, x ≥ d_{n}(α)}. More exactly, find a distribution of the quadratic functional U_{nN} under a sequence of alternatives close to the hypothesis H_{0} : μ(x) = μ_{0}(x). The sequence is written as

H_{1} : μ̄(x) = μ_{0}(x) + γ_{n}φ(x) + o(γ_{n}),  (33)

where γ_{n} → 0 appropriately and o(γ_{n}) is uniform in x ∈ [−π, π].

Theorem 3. Let μ̄_{n}(x) satisfy the conditions of Theorem 2. If 2N + 1 = n^{δ}, γ_{n} = n^{−1/2+δ/4}, \frac{2}{9} < δ < \frac{1}{2}, then under the alternative H_{1} the statistic (2N + 1)^{1/2}(U_{nN} − θ_{1}) is distributed in the limit normally (\frac{1}{2π} ∫_{−π}^{π} φ^{2}(u)du, \sqrt{θ_{2}}).

Proof. Let us represent U_{nN} as the sum

U_{nN} = \frac{n}{2π(2N + 1)} ∫_{−π}^{π} (μ_{nN}(x) − E_{1}μ_{nN}(x))^{2}dx +

+ \frac{n}{π(2N + 1)}γ_{n} ∫_{−π}^{π} [μ_{nN}(x) − E_{1}μ_{nN}(x)]φ̃_{n}(x) dx +

+ \frac{n}{2π(2N + 1)}γ_{n}^{2} ∫_{−π}^{π} φ_{n}^{2}(x) dx = A_{1}(n) + A_{2}(n) + A_{3}(n),

where E_{1}(·) denotes the mathematical expectation under the hypothesis H_{1},

φ̃_{n}(x) = ∑_{j=1}^{n} φ(x_{j})Δ_{j}K_{n}(x − x_{j}).

Due to Theorem 1 one can readily assertain that \sqrt{2N + 1}(A_{1}(n) − θ_{1}) is distributed asymptotically normal (0, \sqrt{θ}_{2}).

By analogy with the proof of Lemma 2 we find

\sqrt{2N + 1}A_{3}(n) = \frac{1}{2π} ∫_{−π}^{π} ( ∫_{−π}^{π} φ(y)K_{N}(x − y)dy)^{2}dx + O(\frac{N^{2} ln N}{n} ).

Hence, by virtue of theorem 2 from [9], p.474, we have

\sqrt{2N + 1}A_{3}(n) → \frac{1}{2π} ∫_{−π}^{π} φ^{2}(u)du.

Further, for our choice of N and γ_{n} we can show by simple calculations that

\sqrt{2N + 1}E|A_{2}(n)| ≤ C(\frac{ln^{2} n}{n^{δ/4}} + \frac{ln n}{n^{1−7δ/4}} ).

Thus the local behaviour of the power P_{H_{1}}(U_{nN} ≥ d_{n}(α)) is

P_{H_{1}}(U_{nN} ≥ d_{n}(α)) → 1 − Φ(λ_{α} − θ_{2}^{−1/2} \frac{1}{2π} ∫_{−π}^{π} φ^{2}(u) du). (34)

Since ∫_{−π}^{π} φ^{2}(u)du > 0 and is equal to zero iff φ(x) = 0, from (34) we conclude that the test for the hypothesis H_{0} : μ(x) = μ_{0}(x) against alternatives of the form (33) is asymptotically strictly unbiased.

Remark. Similar results can be obtained by the same method for the kernel estimator of Priestley and Chao [1].

References

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2. N.N. Chentsov, Estimation of the unknown distribution density by observations. (Russian) Dokl. Akad. Nauk SSSR 147(1962), 45-48.

3. N.B. Smirnov and I.V. Dunin-Barkovskii, A course in probability theory and mathematical statistics for technical applications. (Russian) Nauka, Moscow, 1969.

4. B.T. Polyak and A.V. Tsibakov, Asymptotic optimality of the C_{p}-test in the projection estimation of the regression. (Russian) Teor. veroyatnost. i primenen. 35(1990), No. 2, 305-317.

5. R.L. Eubank, J.D. Hart, and P. Speckman, Trigonometric series regression estimators with application to partially linear models. J. Multivariate Anal. 32(1990), 70-83.

6. R.Sh. Liptser and A.N. Shiryayev, A functional central limit theorem for semimartingales. (Russian) Teor. veroyatnost. i primenen. 25(1980), No.4, 683-703.

7. E.A. Nadaraya, Nonparametric estimation of probability densities and regression curves. Kluwer Academic Publishers, Dordrecht, Holland, 1989.

8. A. Zygmund, Trigonometric series, vol. 1. Cambridge University Press, Cambridge, 1959.

9. A.N. Kolmogorov and S.V. Fomin, Elements of the theory of functions and functional analysis. (Russian) ”Nauka”, Moscow, 1989.

10. D. Jackson, The theory of approximation. American Mathematical Society, New York, 1930.

(Received 1.07.1992)

Author’s address: Faculty of Mechanics and Mathamatics I.Javakhishvili Tbilisi State University 2 University St., 380043 Tbilisi Republic of Georgia