GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 6, 1999, 501-516

LIMIT DISTRIBUTION OF THE MEAN SQUARE DEVIATION OF THE GASSER–MÜLLER

NONPARAMETRIC ESTIMATE OF THE REGRESSION FUNCTION

R. ABSAVA AND E. NADARAYA

Abstract. Asymptotic distribution of the mean square deviation of the Gasser–Müller estimate of the regression curve is investigated. The testing of hypotheses on regression function is considered. The asymptotic behaviour of the power of the proposed criteria under contiguous alternatives is studed.

Let {Y_{nk}, k = 1, …, n}_{n≥1} be a sequence of arrays of random variables defined as follows:

Y_{nk} = g(x_{nk}) + Z_{nk}, x_{nk} = \frac{k}{n}, k = 1, …, n, n ≥ 1,

where g(x), x ∈ [0, 1], is the unknown real-valued function to be estimated by given observations Y_{nk}; Z_{nk}, k = 1, …, n, is a sequence of arrays of independent random variables identically distributed in each array such that EZ_{nk} = 0, EZ_{nk}^{2} = σ^{2}, k = 1, …, n, n ≥ 1.

Let us consider the estimate of the function g(x) from [1]:

g_{n}(x) = ∑_{i=1}^{n} W_{ni}(x)Y_{ni}, x ∈ [0, 1],

where

W_{ni} = b_{n}^{−1} ∫_{Δ_{i}} K(\frac{x − t}{b_{n}})dt, Δ_{i} = [\frac{i − 1}{n}, \frac{i}{n}].

Here K(x) is some kernel, b_{n} is a sequence of positive numbers tending to 0.

Assume that the kernel K(x) has a compact support and satisfies the conditions:

501 1^{∘}. supp(K) ⊂ [−τ, τ], 0 < τ < ∞, sup |K| < ∞,

1991 Mathematics Subject Classification. 62G07.

Key words and phrases. Limit distribution, consistency, power of criteria, contiguous alternatives.

1072-947X/99/1100-0501$16.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

∫ K(u)du = 1, K(−x) = K(x),

2^{∘}. ∫ K(u)u^{j}du = 0, 0 < j < s, ∫ K(u)u^{s}du ≠ 0, 3^{∘}. K(x) has a bounded derivative in R = (−∞, ∞).

Denote by F_{s} the family of regression functions g(x), x ∈ [0, 1], having derivatives of order up to s (s ≥ 2), g^{(s)}(x) is continuous. Note that ∑_{i=1}^{n} W_{ni}(x) = 1 for x ∈ Ω_{n}(τ) = [τb_{n}, 1−τb_{n}] and Eg_{n}(x) = g(x)+O(b_{n}^{2}) if the kernel K(x) satisfies condition 1^{∘} and g(x) ∈ F_{2}. On the other hand, ∑_{i=1}^{n} W_{ni}(x) ≠ 1 for x ∈ [0, τb_{n}) ∪ (1 − τb_{n}, 1] and it may happen that Eg_{n}(x) ↛ g(x), for example, Eg_{n}(0) → \frac{g(0)}{2} (or Eg_{n}(1) → \frac{g(1)}{2}). If the estimate g_{n}(x) is divided by ∑_{i=1}^{n} W_{ni}(x), then the proposed estimate g̃_{n}(x) becomes asymptotically unbiased and, moreover, Eg̃_{n}(x) = g(x) + O(b_{n}) for x ∈ [0, τb_{n}) ∪ (1 − τb_{n}, 1]. Hence the asymptotic behaviour of the estimate g_{n}(x) near the boundary of the interval [0, 1] is worse than within the interval Ω_{n}(τ). A phenomenon of such kind is known in the literature as the boundary effect of the estimator g_{n}(x) (see, for examle, [2]). It would be interesting to investigate the limit behaviour of the distribution of the mean square deviation of g_{n}(x) from g(x) on the interval Ω_{n}(τ) and this is the aim of the present article.

The method of proving the statements given below is based on the functional limit theorems for semimartingales from [3].

We will use the notation:

U_{n} = nb_{n} ∫_{Ω_{n}(τ)}(g_{n}(x) − Eg_{n}(x))^{2}dx, σ_{n}^{2} = 4σ^{4}(nb_{n})^{2} ∑_{k=2}^{n} ∑_{i=1}^{k−1} Q_{ik}^{2}(n),

Q_{ij} ≡ Q_{ij}(n) = ∫_{Ω_{n}(τ)} W_{ni}(x)W_{nj}(x)dx,
η_{ik} ≡ η_{ik}(n) = 2nb_{n}Q_{ik}Z_{ni}Z_{nk}σ_{n}^{−1},

ξ_{nk} = ∑_{i=1}^{k−1} η_{ik}, k = 2, …, n, ξ_{n1} = 0, ξ_{nk} = 0, k > n,

F_{k}^{(n)} = σ(ω : Z_{n1}, Z_{n2}, …, Z_{nk}),

where F_{k}^{(n)} is the σ-algebra generated by the random variables Z_{n1}, Z_{n2}, …, Z_{nk}, F_{0}^{(n)} = (∅, Ω).

Lemma 1 ([4], p. 179). The stochastic sequence (ξ_{nk}, F_{k}^{(n)})_{k≥1} is a martingale-difference.

Lemma 2. Suppose the kernel K(x) satisfies conditions 1^{∘} and 3^{∘}. If nb_{n}^{2} → ∞, then

b_{n}^{−1}σ_{n}^{2} → 2σ^{4} ∫_{−2τ}^{2τ} K_{0}^{2}(u) du,  (1)

and

EU_{n} = σ^{2} ∫ K^{2}(u)du + O(b_{n}) + O(\frac{1}{nb_{n}} ),  (2)

where K_{0} = K ∗ K (the symbol ∗ denotes the convolution).

Proof. It is not difficult to see that

σ_{n}^{2} = 2σ^{4}(nb_{n})^{2}[ ∑_{k=1}^{n} ∑_{i=1}^{n} Q_{ki}^{2} − ∑_{i=1}^{n} Q_{ii}^{2}] = d_{1}(n) + d_{2}(n).  (3)

Here by the definition of Q_{ki} we have

d_{1}(n) = 2σ^{4}n^{2}b_{n}^{−2} ∑_{i,j} ( ∫_{Δ_{i}} ∫_{Δ_{j}} Ψ_{n}(t_{1}, t_{2})dt_{1}dt_{2})^{2},

where

Ψ_{n}(t_{1}, t_{2}) = ∫_{Ω_{n}(τ)} K(\frac{x − t_{1}}{b_{n}} ) K(\frac{x − t_{2}}{b_{n}} ) dx.

Using the mean value theorem, we can rewrite d_{1}(n) as

d_{1}(n) = 2σ^{4}(nb_{n})^{−2} ∑_{i,j} Ψ_{n}^{2}(θ_{i}^{(1)}, θ_{j}^{(2)}),

with θ_{i}^{(1)} ∈ Δ_{i}, θ_{i}^{(2)} ∈ Δ_{j}.

Let P (x) be the uniform distribution function on [0, 1] and P_{n}^{(i)}(x) the empirical distribution function of the “sample” θ_{1}^{(i)}, …, θ_{n}^{(i)}, i = 1, 2, i.e., P_{n}^{(i)}(x) = n^{−1} ∑_{k=1}^{n} I_{(−∞,x)}(θ_{k}^{(i)}), where I_{A}(·) is the indicator of the set A. Then d_{1}(n) can be written as the integral

d_{1}(n) = 2σ^{4}b_{n}^{−2} ∫_{0}^{1} ∫_{0}^{1} Ψ_{n}^{2}(x, y)dP_{n}^{(1)}(x)dP_{n}^{(2)}(y).  (4)

It is easy to check that

| ∫_{0}^{1} ∫_{0}^{1} Ψ_{n}^{2}(x, y)dP_{n}^{(1)}(x)dP_{n}^{(2)}(y) − ∫_{0}^{1} ∫_{0}^{1} Ψ_{n}^{2}(x, y)dP (x)dP (y)|≤I_{1}+I_{2},

where

I_{1} = | ∫_{0}^{1} ∫_{0}^{1} Ψ_{n}^{2}(x, y)dP_{n}^{(2)}(y)[dP_{n}^{(1)}(x) − dP (x)]|,

I_{2} = | ∫_{0}^{1} ∫_{0}^{1} Ψ_{n}^{2}(x, y)dP (x)[dP_{n}^{(2)}(y) − dP (y)]|.

Furthermore, the integration by parts of the internal integral in I_{1} yields

I_{1} ≤ 2 ∫_{0}^{1} dP_{n}^{(2)}(y) ∫_{0}^{1} (|P_{n}^{(1)}(x) − P (x)| |Ψ_{n}(x, y)|b_{n}^{−1} ×
× ∫_{Ω_{n}(τ)} |K^{(1)}(\frac{t − x}{b_{n}})| |K(\frac{t − y}{b_{n}} )|dt)dx.  (5)

Since sup_{0≤x≤1} |P_{n}^{(i)} − P (x)| ≤ \frac{2}{n} and |Ψ_{n}(x, y)| ≤ cb_{n}, we have

I_{1} = O(b_{n}/n).

Here and in what follows c is the positive constant varying from one formula to another.

In the same manner we may show that

I_{2} = O(b_{n}/n).

Therefore

d_{1}(n) = 2σ^{4}b_{n}^{−2} ∫_{0}^{1} ∫_{0}^{1} Ψ_{n}^{2}(t_{1}, t_{2})dt_{1}dt_{2} + O(\frac{1}{nb_{n}} ).  (6)

It is easy to show also that

d_{1}(n)=2σ^{4}∫_{Ω_{n}(τ)}∫_{Ω_{n}(τ)}( ∫_{(x−1)b_{n}^{−1}}^{xb_{n}^{−1}}K(u)K(\frac{x − y}{b_{n}} −u)du)^{2}dxdy+O(\frac{1}{nb_{n}} ).

Since [−τ, τ] ⊂ [\frac{x−1}{b_{n}}, \frac{x}{b_{n}}] for all x ∈ Ω_{n}(τ), we have

d_{1}(n) = 2σ^{4} ∫_{Ω_{n}(τ)} ∫_{Ω_{n}(τ)} K_{0}^{2}(\frac{x − y}{b_{n}} )dx dy + O(\frac{1}{nb_{n}} ).

But it is easy to see that

b_{n}^{−1}d_{1}(n) = 2σ^{4} ∫_{0}^{1} ( ∫_{τ−y/b_{n}}^{(1−y)/b_{n}−τ} K_{0}^{2}(u) du)dy + O(b_{n}) + O(\frac{1}{nb_{n}} ).

Therefore

b_{n}^{−1}d_{1}(n) → 2σ^{4} ∫ K_{0}^{2}(u) du.  (7)

We can directly verify that

b_{n}^{−1}|d_{2}(n)|≤n^{2}b_{n}^{−3} ∑_{i=1}^{n} ∫_{Δ_{i}} ∫_{Δ_{i}}∫_{Δ_{i}}∫_{Δ_{i}}( ∫_{Ω_{n}(τ)}|K(\frac{x − t_{1}}{b_{n}} )K(\frac{x − t_{2}}{b_{n}} )|dx×

× ∫_{Ω_{n}(τ)} |K(\frac{y − t_{3}}{b_{n}} )K(\frac{y − t_{4}}{b_{n}} )|dy)dt_{1} dt_{2} dt_{3} dt_{4} ≤ c\frac{1}{nb_{n}} .  (8)

Statement (1) immediately follows directly from (3), (7), and (8). Let us now show that (2) holds. We have

Dg_{n}(x) = \frac{σ^{2}}{nb_{n}^{2}} ∫_{0}^{1} K^{2}(\frac{x − t}{b_{n}}) dP_{n}(x),

where

P_{n}(x) = n^{−1} ∑_{k=1}^{n} I_{(−∞,x)}(θ_{k}), θ_{k} ∈ Δ_{k}, k = 1, …, n.

Applying the same argument as in deriving (6), we find

Dg_{n}(x) = \frac{σ^{2}}{nb_{n}^{2}} ∫_{0}^{1} K^{2}(\frac{x − t}{b_{n}}) dt + O(\frac{1}{(nb_{n})^{2}} ).  (9)

Furthermore, taking into account that [−τ, τ] ⊂ [\frac{x−1}{b_{n}}, \frac{x}{b_{n}}] for all x ∈ Ω_{n}(τ) by (9) we can write

Dg_{n}(x) = \frac{σ^{2}}{nb_{n}} ∫ K^{2}(u) du + O((nb_{n})^{−2}).

Thus

EU_{n} = σ^{2} ∫ K^{2}(u) du + O(b_{n}) + O(\frac{1}{nb_{n}} ).

Theorem 1. Suppose the kernel K(x) satisfies conditions 1^{∘} and 3^{∘}. If

sup_{n≥1} EZ_{n1}^{4} < ∞ and nb_{n}^{2} → ∞, then

b_{n}^{−1/2}(U_{n} − σ^{2}θ_{1})σ^{−2}θ_{2}^{−1}⟶^{d}ξ,

where

θ_{1} = ∫ K^{2}(u) du, θ_{2}^{2} = 2 ∫ K_{0}^{2}(u) du.

By the symbol ⟶^{d} we denote the convergence in distribution, ξ is a random variable distributed according to the standard normal distribution. Proof. Note that

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

where

H_{n}^{(1)} = ∑_{k=1}^{n} ξ_{nk}, H_{n}^{(2)} = \frac{nb_{n}}{σ_{n}} ∑_{i=1}^{n}(Z_{ni}^{2} − EZ_{ni}^{2})Q_{ii}.

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

DH_{n}^{(2)} = \frac{(nb_{n})^{2}}{σ_{n}^{2}} ∑_{i=1}^{n} EZ_{ni}^{4}Q_{ii}^{2} ≤sup_{n≥1} EZ_{n1}^{4} \frac{(nb_{n})^{2}}{σ_{n}^{2}} ∑_{i=1}^{n} Q_{ii}^{2} = sup_{n≥1} EZ_{n1}^{4} \frac{|d_{2}(n)|}{σ_{n}^{2}}.

This and (8) yield DH_{n}^{(2)} = O(1/(nσ_{n}^{2})) = O(1/(nb_{n})). Hence H_{n}^{(2)}⟶^{P}0. (By the symbol ⟶^{P} we denote the convergence in probability).

Now let us prove the convergence H_{n}^{(1)}⟶^{d}ξ. For this it is sufficient verify the validity of Corollaries 2 and 6 of Theorem 2 from [3]. We will show that the asymptotic normality conditions from the corresponding statement are fulfilled by the sequence {ξ_{nk}, F_{k}^{(n)}}_{k≥1} which is a square-integrable martingale-difference according to Lemma 1.

A direct calculation shows that ∑_{k=1}^{n} Eξ_{nk}^{2} = 1. Now the asymptotic normality will take place if for n → ∞ we have

∑_{k=1}^{n} E[ξ_{nk}^{2}I(|ξ_{nk}| ≥ ϵ) F_{k−1}^{(n)}]⟶^{P}0  (10)

and

∑_{k=1}^{n} ξ_{nk}^{2}⟶^{P}1.  (11)

But, as shown in [3], the validity of (11) together with the condition sup_{1≤k≤n} |ξ_{nk}|⟶^{P}0 implies the statement (10) as well.

Since for ϵ > 0

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

to prove H_{n}^{(1)}⟶^{d}ξ we have to verify only (11), since relation (12) given below is fulfiled.

Now we will establish ∑_{k=1}^{n} ξ_{nk}^{2}⟶^{P}1. It is sufficient to verify that

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

as n → ∞, i.e., due to ∑_{k=1}^{n} Eξ_{nk}^{2} = 1

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

LIMIT DISTRIBUTION OF THE MEAN SQUARE DEVIATION 507

First show that ∑_{k=1}^{n} Eξ_{nk}^{4} → 0 as n → ∞. By virtue of the definitions of ξ_{nk} and η_{ij} we can write

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

where

I_{n}^{(1)} = \frac{16(nb_{n})^{4}}{σ_{n}^{4}} ∑_{k=2}^{n} EZ_{nk}^{4} ∑_{j=1}^{k−1} EZ_{nj}^{4}Q_{jk},
I_{n}^{(2)} = \frac{48(nb_{n})^{4}}{σ_{n}^{4}} ∑_{k=2}^{n} ∑_{i≠j}^{k−1} EZ_{ni}^{2}EZ_{nj}^{2}Q_{ik}^{2}Q_{jk}^{2}.

On the other hand, since sup_{n≥1} EZ_{n1}^{4} < ∞, |Q_{ij}| ≤ cb_{n}^{−1}n^{−2} and b_{n}^{−1}σ_{n}^{2} →

σ^{4}θ_{2}^{2}, we have I_{n}^{(1)} = O((nb_{n})^{−2}), I_{n}^{(2)} = O(1/(nσ_{n}^{4})) = O(1/(nb_{n}^{2})). Hence

∑_{k=1}^{n} Eξ_{nk}^{4} → 0, n → ∞.  (12)

Now let us establish that

2 ∑_{1≤k_{1}<k_{2}≤n} Eξ_{nk_{1}}^{2}ξ_{nk_{2}}^{2} → 1

as n → ∞. The definition of ξ_{ni} yields

ξ_{nk_{1}}^{2} · ξ_{nk_{2}}^{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)},

where B_{k_{1}k_{2}}^{(1)} = σ_{2}(k_{1})σ_{2}(k_{2}), B_{k_{1}k_{2}}^{(2)} = σ_{2}(k_{1})σ_{1}(k_{2}), B_{k_{1}k_{2}}^{(3)} = σ_{1}(k_{1})σ_{2}(k_{2}), B_{k_{1}k_{2}}^{(4)} = σ_{1}(k_{1})σ_{1}(k_{2}),

σ_{1}(k) = ∑_{i≠j} η_{ik}η_{jk}, σ_{2}(k) = ∑_{i=1}^{k−1} η_{ik}^{2}.

Therefore

2 ∑_{1≤k_{1}<k_{2}≤n} Eξ_{nk_{1}}^{2}ξ_{nk_{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 = 1, 2, 3, 4.

508 R. ABSAVA AND E. NADARAYA

Let us consider the term A_{n}^{(3)}. According to the definition of η_{ij} we have

Eη_{ik_{2}}^{2}η_{sk_{1}}η_{tk_{1}} = 0, s ≠ t, k_{1} < k_{2}.

Thus

A_{n}^{(4)} = 0.  (13)

To estimate the term A_{n}^{(2)}, note that sup_{n≥1} EZ_{n1}^{4} < ∞ and |Q_{ij}| ≤ cb_{n}^{−1}n^{−2}. So we obtain

|A_{n}^{(2)}| ≤ c\frac{nb_{n}^{2}}{σ_{n}^{4}} ∑_{k_{1}=2}^{n} ∑_{i=1}^{k_{1}−1} Q_{ik_{1}}^{2} ≤ c\frac{1}{nσ_{n}^{2}} = O(\frac{1}{nb_{n}} ).  (14)

Now we will verify that A_{n}^{(1)} → 1 as n → ∞. For this let us represent A_{n}^{(1)} in the form

A_{n}^{(1)} = D_{n}^{(1)} + D_{n}^{(2)},

where

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

D_{n}^{(2)} = 2( ∑_{k_{1}<k_{2}} B_{k_{1}k_{2}}^{(1)} − ∑_{k_{1}<k_{2}} ( ∑_{i=1}^{k_{1}−1} Eη_{ik_{1}}^{2})( ∑_{j=1}^{k_{2}−1} Eη_{jk_{2}}^{2})).

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

D_{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{nb_{n}}{σ_{n}} )^{4} ∑_{k=2}^{n} ( ∑_{i=1}^{k−1} Q_{ik}^{2})^{2} ≤ c\frac{1}{nσ_{n}^{4}} = O(\frac{1}{nb_{n}^{2}} ).

Therefore

D_{n}^{(1)} → 1 as n → ∞.  (15)

Next we will show that D_{n}^{(2)} → 0 as n → ∞. It is easy to verify that

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

But

Eη_{ik_{1}}^{2}η_{ik_{2}}^{2} ≤ c(\frac{nb_{n}}{σ_{n}} )^{4}Q_{ik_{1}}^{2}Q_{ik_{2}}^{2} ≤ c \frac{1}{n^{4}σ_{n}^{4}} .

Similarly,

Eη_{ij}^{2} = O(n^{−2}σ_{n}^{−2}).

Therefore

cov(η_{ik_{1}}^{2}, η_{ik_{2}}^{2}) = O((nσ_{n})^{−4}).  (16)

Furthermore, since ∑_{1≤k_{1}<k_{2}≤n}(k_{1} − 1) = O(n^{3}), equation (16) implies

D_{n}^{(2)} = O(\frac{1}{nσ_{n}^{4}} ) = O(\frac{1}{nb_{n}^{2}} ).  (17)

Thus, according to (15) and (17),

A_{n}^{(1)} = 1 + O(\frac{1}{nb_{n}^{2}} ).  (18)

Finally, we will show that A_{n}^{(4)} → 0 as n → ∞. Again, by the definition of η_{ij} we can write

|A_{n}^{(4)}| = 4| ∑_{k_{1}<k_{2}} ∑_{t<s}^{k_{1}−1} Eη_{sk_{1}}η_{tk_{1}}η_{sk_{2}}η_{tk_{2}} | ≤

≤ c(\frac{nb_{n}}{σ_{n}} )^{4}[| ∑_{s,t,k_{1},k_{2}} Q_{sk_{1}}Q_{sk_{2}}Q_{tk_{1}}Q_{tk_{2}} | +

+ | ∑_{k,s,t} Q_{ks}^{2}Q_{kt}^{2}| + | ∑_{k,s,t} Q_{kt}Q_{st}Q_{ks}Q_{ss}|] =

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

By the same argument as in (4), it can be shown that

E_{n}^{(1)} = n^{−7}b_{n}^{−8} ∑_{s,t,k} Ψ_{n}(θ_{s}^{(1)}, θ_{k}^{(2)})Ψ_{n}(θ_{t}^{(1)}, θ_{k}^{(2)}) ×

× ∫_{0}^{1} Ψ_{n}(θ_{s}^{(1)}, u)Ψ_{n}(θ_{t}^{(1)}, u)dP_{n}^{(1)}(u).

Hence, integrating by parts and taking into account that sup |Ψ_{n}(t_{1}, t_{2})| ≤ cb_{n} and sup |K^{(1)}(u)| < ∞, we obtain

E_{n}^{(1)} = n^{−7}b_{n}^{−8} ∫_{0}^{1} ∑_{s,t,k} Ψ_{n}(θ_{s}^{(1)}, θ_{k}^{(2)})Ψ_{n}(θ_{t}^{(1)}, θ_{k}^{(2)}) ×

510 R. ABSAVA AND E. NADARAYA

×Ψ_{n}(θ_{s}^{(1)}, u)Ψ_{n}(θ_{t}^{(1)}, u))du = O(\frac{1}{n^{5}b_{n}^{4}} ).  (20)

In the same manner, we can rewrite (20) as

E_{n}^{(1)} = n^{−4}b_{n}^{−8} ∫_{0}^{1} ∫_{0}^{1} ∫_{0}^{1} ∫_{0}^{1} Ψ_{n}(z, u)Ψ_{n}(z, t)Ψ_{n}(y, u) ×

×Ψ_{n}(y, t) du dt dy dz + O(n^{−5}b_{n}^{−4}).

It is easy to verify that

n^{−4}b_{n}^{−8} ∫_{0}^{1} ∫_{0}^{1} ∫_{0}^{1} ∫_{0}^{1} |Ψ_{n}(z, u)Ψ_{n}(z, t)Ψ_{n}(y, u)Ψ_{n}(y, t)| du dt dy dz ≤

≤ cn^{−4}b_{n}^{−2} ∫_{Ω_{n}(τ)} ∫_{Ω_{n}(τ)} |K_{0}(\frac{x − v}{b_{n}} )| dx dv ≤ cn^{−4}b_{n}^{−1}.

Thus

(\frac{nb_{n}}{σ_{n}} )^{4}E_{n}^{(1)} = O(b_{n}) + O(\frac{1}{nb_{n}^{2}} ).  (21)

Furthermore, it is not difficillt to show

(nb_{n})^{4}σ_{n}^{−4}E_{n}^{(2)} = O(\frac{1}{nb_{n}^{2}} )

and

(nb_{n})^{4}σ_{n}^{−4}E_{n}^{(3)} = O(\frac{1}{nb_{n}^{2}} ).  (22)

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

A_{n}^{(4)} → 0 as n → ∞.  (23)

Combining relations (12), (13), (14), (18), and (23), we obtain

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

Therefore

\frac{U_{n} − EU_{n}}{σ_{n}} ⟶^{d} ξ.

But, due to Lemma 2, we have EU_{n} = σ^{2}θ_{1} + O(b_{n}) + O((nb_{n})^{−1}) and b_{n}^{−1}σ_{n}^{2} → σ^{4}θ_{2}^{2}, and hence we obtain

b_{b}^{−1/2} (\frac{U_{n} − σ^{2}θ_{1}}{σ^{2}θ_{2}} ) ⟶^{d} ξ.

Let us denote by

T_{n} = nb_{n} ∫_{Ω_{n}(τ)}[g_{n}(x) − g(x)]^{2}dx.

Theorem 2. Suppose g(x) ∈ F_{s}, s ≥ 2 and K(x) satisfies conditions 1^{∘} − 3^{∘}. If, in addition, sup_{n≥1} EZ_{n1}^{4} < ∞, nb_{n}^{2} → ∞ and nb_{n}^{2s} → 0, then

b_{n}^{−1/2}(T_{n} − σ^{2}θ_{1})σ^{−2}θ_{2}^{−1} ⟶^{d} ξ.

Proof. Note that

T_{n} = U_{n} + d_{n}^{(1)} + d_{n}^{(1)},

where

d_{n}^{(1)} = 2nb_{n} ∫_{Ω_{n}(τ)}[g_{n}(x) − Eg_{n}(x)] [Eg_{n}(x) − g(x)]dx,
d_{n}^{(2)} = nb_{n} ∫_{Ω_{n}(τ)}[Eg_{n}(x) − g(x)]^{2}dx.

It is easy to verify that

Eg_{n}(x) − g(x) = b_{n}^{−1} ∑_{i=1}^{n} ∫_{Δ_{i}} K(\frac{x − t}{b_{n}})dx g(\frac{i}{n}) − g(x) =

= b_{n}^{−1} ∫_{0}^{1} K(\frac{x − t}{b_{n}}) [g̃_{n}(t) − g(t)]dt +

+b_{n}^{−1} ∫_{0}^{1} K(\frac{x − t}{b_{n}}) [g(t) − g(x)]dt, x ∈ Ω_{n}(τ),  (24)

with

g̃_{n}(x) = ∑_{i=1}^{n} g(\frac{i}{n}) I_{Δ_{i}}(x), Δ_{i} = [\frac{i − 1}{n}, \frac{i}{n}].

But

sup_{0≤x≤1} |g̃_{n}(x) − g(x)| ≤ max_{1≤i≤n} sup_{x∈Δ_{i}} |g(\frac{i}{n}) − g(x)| =

= max_{1≤i≤n} sup_{x∈Δ_{i}} |\frac{i}{n} − x| |g′(τ_{i})| = O(\frac{1}{n}), τ_{i} ∈ Δ_{i}.

Therefore the second term in (24) is O(\frac{1}{n}). Since g(x) ∈ F_{s} and K(x) satisfies conditions 1^{∘}–3^{∘}, we have

sup_{x∈Ω_{n}(τ)} | ∫_{0}^{1} K(\frac{x − t}{b_{n}}) [g(t) − g(x)]dt| =

= \frac{b_{n}^{s}}{(s − 1)!} sup_{x∈Ω_{n}(τ)} | ∫_{−τ}^{τ} ∫_{0}^{1}(1 − t)^{s−1}g^{(s)}(x + tub_{n})u^{s}K(u)du| ≤ cb_{n}^{s}. (25)

Thus from (24) and (25) it follows that

sup_{x∈Ω_{n}(τ)} |Eg_{n}(x) − g(x)| ≤ c(b_{n}^{s} + \frac{1}{n}).  (26)

Therefore

b_{n}^{−1/2}d_{n}^{(2)} ≤ c(nb_{n}^{2s}\sqrt{b_{n}} + b_{n}^{s+1/2} + n^{−1}\sqrt{b_{n}}).  (27)

On the other hand, (9) and (26) yield

b_{n}^{−1/2}E|d_{n}^{(1)}| ≤ nb_{n}^{1/2}{ ∫_{Ω_{n}(τ)} E(g_{n}(x) − Eg_{n}(x))^{2}dx ×

× ∫_{Ω_{n}(τ)}(Eg_{n}(x) − g(x))^{2}dx\}^{1/2} ≤ c\sqrt{n}b^{s}.  (28)

Finally, the statement of Theorem 2 follows directly from Theorem 1, (27) and (28).

Using Theorem 2, it is easy to solve the problem of testing the hypothesis about g(x). Given σ^{2}, it is required to verify the hypothesis

H_{0} : g(x) = g_{0}(x), x ∈ Ω_{n}(τ).

The critical region is defined approximately by the inequality

T_{n}^{0} = nb_{n} ∫ (g_{n}(x) − g_{0}(x))^{2}dx ≥ q_{n}(α),  (29)

where

q_{n}(α) = σ^{2}(θ_{1} + λ_{α}\sqrt{b_{n}}θ_{2}), θ_{1} = ∫ K^{2}(u)du, θ_{2}^{2} = 2 ∫ K_{0}^{2}(u)du,

and λ_{α} is the quantile of level 1 − α of the standard normal distribution, i.e.,

Φ(λ_{α}) = 1 − α, Φ(u) = (2π)^{−1/2} ∫_{−∞}^{u} exp ( − \frac{x^{2}}{2} )dx.

Suppose now that σ^{2} is unknown. We will use the b_{n}^{−1/2}-consistent estimate of variance σ^{2} (see, for example, [5]):

S_{n}^{2} = \frac{1}{2(n − 1)} ∑_{i=1}^{n−1}(Y_{i+1} − Y_{i})^{2}, Y_{i} ≡ Y_{ni}.

Indeed,

ES_{n}^{2} = \frac{1}{2(n − 1)} ∑_{i=1}^{n−1} Eϵ_{i}^{2} + \frac{1}{2(n − 1)} ∑_{i=1}^{n−1} [g(\frac{i − 1}{n}) − g(\frac{i}{n})]^{2},

where ϵ_{i} = Z_{i+1} − Z_{i}, Z_{i} ≡ Z_{ni}. Moreover, since Eϵ_{i}^{2} = 2σ^{2}, we can write

ES_{n}^{2} = σ^{2} + \frac{1}{2(n − 1)} \frac{1}{n^{2}} ∑_{i=1}^{n−1}(g^{(1)}(τ_{i}))^{2} = σ^{2} + O(n^{−2}).

To calculate the variance, it is sufficient to note that

E(S_{n}^{2} − σ^{2})^{2} = \frac{1}{4(n − 1)^{2}} ∑_{i=1}^{n−1} Eϵ_{i}^{4}+O(n^{−2})−σ^{2}+ \frac{1}{4(n − 1)^{2}} ∑_{i≠j} Eϵ_{i}^{2}Eϵ_{j}^{2} =

= \frac{1}{4(n − 1)^{2}} ∑_{i=1}^{n−1} Eϵ_{i}^{4} + \frac{(n − 2)(n − 3)}{4(n − 1)^{2}}(2σ^{2})^{2} − σ^{4} + O(n^{−1}) =

= \frac{1}{4(n − 1)^{2}} ∑_{i=1}^{n−1} Eϵ_{i}^{4} + O(n^{−1}).

Since sup_{n≥1} EZ_{n1}^{4} < ∞, this yields E(S_{n}^{2} − σ^{2})^{2} = O(n^{−1}). Therefore

b_{n}^{−1/2}(S_{n}^{2} − σ^{2}) ⟶^{P} 0.

Corollary. Under the conditions of Theorem 2

b_{n}^{−1/2}(\frac{T_{n}^{0} − S_{n}^{2}θ_{1}}{S_{n}^{2}θ_{2}} ) ⟶^{d} ξ.

This corollary enables us to construct a test for verifying

H_{0} : g(x) = g_{0}(x).

The critical region is defined approximately by the inequality T_{n}^{0} ≥ q̃_{n}(α), where q̃_{n}(α) is obtained from q_{n}(α) by using S_{n}^{2} instead of σ^{2}.

Consider now the asymptotic properties of test (29) (i.e., the asymptotic behaviour of the power function as n → ∞). First, let us study the question whether the corresponding test is consistent.

Theorem 3. Under the conditions of Theorem 2

Π_{n}(g_{1}) = P_{H_{1}}(T_{n}^{0} ≥ q_{n}(α)) → 1, n → ∞,

i.e., the test defined by (29) is consistent under any alternatives

H_{1} : g(x) = g_{1}(x) ≠ g_{0}(x), Δ = ∫_{0}^{1}(g_{1}(x) − g_{0}(x))^{2}dx > 0.

Proof. Denote

Z_{n}(g_{1}) = b_{n}^{−1/2}(nb_{n} ∫_{Ω_{n}(τ)}(g_{n}(x) − g_{1}(x))^{2}dx − σ^{2}θ_{1})σ^{−2}θ_{2}^{−1}.

It is easy to show that

Π_{n}(g_{1}) = P_{H_{1}}{Z_{n}(g_{1}) ≥ −nb_{n}^{1/2}(θ_{2}^{−1}σ^{−2}Δ + o_{p}(1))}.

Since Z_{n}(g_{1}) is asymptoticaly normally distributed with parameters (0, 1) under hypothesis H_{1}, nb_{n}^{1/2} → ∞ and Δ > 0, we have Π_{n}(g_{1}) → 1 as n → ∞.

Thus under any fixed alternative the power of test (29) tends to 1 as n → ∞. Nevertheless, if the alternative hypothesis varies with n and becomes “closer” to the null hypothesis H_{0}, the power of the test may not converge to 1 depending on the rate at which the alternative approaches the null hypothesis. In our case the sequence of “close” alternatives has the form

H_{n} : g̃_{n}(x) = g_{0}(x) + γ_{n}ϕ(x) + o(γ_{n}).  (30)

Theorem 4. Suppose g_{0}(x) and φ(x) are from F_{s}, but K(x) satisfies

coditions 1^{∘}–3^{∘} and sup_{n≥1} EZ_{n1}^{4} < ∞. If b_{n} = n^{−δ}, γ_{n} = n^{−1/2+δ/4}, 1/2s <

δ < 1/2, then under alternatives H_{n} the statistic

b_{n}^{−1/2}(T_{n}^{0} − θ_{1}σ^{2})σ^{−2}θ_{2}^{−1}

has the limiting normal distribution with parameters (\frac{1}{σ^{2}θ_{2}} ∫_{0}^{1} φ^{2}(x)dx, 1).

Proof. Let us represent T_{n}^{0} as the sum

T_{n}^{0} = nb_{n} ∫_{Ω_{n}(τ)}(g_{n}(x) − g̃_{n}(x))^{2}dx + nb_{n} ∫_{Ω_{n}(τ)}(g̃_{n}(x) − g_{0}(x))^{2}dx +

+2nb_{n} ∫_{Ω_{n}(τ)}(g_{n}(x) − g̃_{n}(x))(g̃_{n}(x) − g_{0}(x))dx = T_{n}^{1} + A_{1}(n) + A_{2}(n).

It is easy to check that

b_{n}^{−1/2}A_{1}(n) = ∫_{0}^{1} φ^{2}(u)du + O(n^{−δ}).  (31)

Let us introduce the random variable

d_{n} = ∫_{Ω_{n}(τ)}(g_{n}(x) − E_{1}g_{n}(x))φ(x) dx.

Here E_{1} denotes the mathematical expectation under the hypothesis H_{n}. We can derive the inequality

b_{n}^{−1/2}E|A_{2}(n)| ≤ nb_{n}^{1/2}γ_{n}[E|d_{n}| + ∫_{Ω_{n}(τ)} |E_{1}g_{n}(x) − g̃_{n}(x)|φ(x) dx] =

= nb_{n}^{1/2}γ_{n}E|d_{n}| + O(nγ_{n}b_{n}^{s+1/2}).

But

E|d_{n}| ≤ σ\{ ∑_{i=1}^{n} (b_{n}^{−1} ∫_{Ω_{n}(τ)} ( ∫_{Δ_{i}} K(\frac{x − t}{b_{n}})dt))φ(x) dx)^{2}\}^{1/2} ≤ cn^{−1/2}.

Therefore

b_{n}^{−1/2}E|A_{2}(n)| ≤ c(n^{−δ/4} + n^{\frac{2−(4s+1)δ}{4}} ) → 0.  (32)

Referring to the proof of Theorem 2 it is easy to verify that b_{n}^{−1/2}(T_{n}^{0} − σ^{2}θ_{1})σ^{−2}θ_{2}^{−1} is asymptotically normally distributed with parameters (0,1). Hence, from (31) and (32) we conclude that the theorem is valid.

Remark 1. It follows from Theorem 4 that more closer alternatives of form (30) (i.e., under γ_{n}n^{1/2−δ/4} → 0) are not distinguished from H_{0} by this test (i.e., P_{H_{n}}(T_{n}^{0} ≥ q_{n}(α)) → α), and for more remote alternatives (i.e., under γ_{n}n^{1/2−δ/4} → ∞) the corresponding test preserves the consistency property (i.e., P_{H_{n}}(T_{n}^{0} ≥ q_{n}(α)) → 1).

Thus the local behaviour of the power P_{H_{n}}(T_{n}^{0} ≥ q_{n}(α)) is

P_{H_{n}}(T_{n} ≥ q_{n}(α)) → 1 − Φ(λ_{α} − \frac{1}{σ^{2}θ_{2}} ∫_{0}^{1} φ^{2}(u) du).  (33)

Since ∫_{0}^{1} φ^{2}(u)du > 0 and is equal to zero if and only if φ(u) = 0, it follows from (33) that the test for testing the hypothesis H_{0} : g(x) = g_{0}(x) against the alternative of form (30) is asymptotically strictly unbiased.

Remark 2. Theorem 4 is analogous to Theorem 6.1 from the book by J. D. Hart [6] with the only difference that [6] deals with the statistic based on the Priestley–Chao estimate [7].

Remark 3. If in the interval [0, 1] we choose points x_{nk} in a more general manner, i.e.,

∫_{0}^{x_{nk}} p(u) du = k/n, k = 1, …, n,

where p(x) > 0, x ∈ [0, 1], is the probability density satisfying certain conditions of smoothness, then Theorems 1–4 remain valid provided that the parameters θ_{1} and θ_{2} are changed appropriately.

Remark 4. If instead of K(x) we consider its modification K_{q,r}(x) from [2], then we can give the hypothesis H_{0} on the entire interval [0, 1]. For the corresponding estimate we have g_{n}^{∗}(x) ≡ g_{n}(x), x ∈ Ω_{n}, while the relation

nb_{n}^{1/2}( ∫_{0}^{1}(g_{n}^{∗} − g)^{2}dx − ∫_{Ω_{n}}(g_{n} − g)^{2}dx)⟶^{P}0

will be proved in our forthcoming paper.

Remark 5. If in Theorem 4 we set δ = \frac{2}{2s+1}, then γ_{n} = n^{− \frac{s}{2s+1}} . By Yu. Ingster’s results [8] the test T_{n} is minimax consistent with respect to alternatives of form (30).

Acknowledgement

The authors express their thanks to the referee for his careful reading of the paper and comments that led to adding Remarks 2–5.

References

1. T. Gasser and H. Müller, Kernel estimation of regression functions. Smoothing technique for curve estimation. Eds. T. Gasser and M. Rosenblatt. Lecture Notes in Math. 757, 23–68, Springer, Berlin etc., 1979.

2. J. Hart and T. Wehrly, Kernel regression when the boundary is large. J. Amer. Statist. Assoc. 87(1992), No. 420, 1018–1022.

3. R. S. Lipster and A. N. Shiryaev, A functional limit theorem for semimartingales. (Russian) Teor. Veroyatnost. i Primenen. 25(1980), No. 4, 683–703.

4. E. A. Nadaraya, Nonparametric estimation of probarility densities and regression curves. Kluver Academic Publishers, Dordrecht, 1989.

5. K. A. Brownlee, Statistical theory and methodology in science and engineering. Wiley, New York, 1960.

6. J. D. Hart, Nonparametric smoothing and lack-of-fit tests. Springer, New York, 1997.

7. M. B. Priestley and M. T. Chao, Nonparametric function fitting. J. Roy. Stat. Soc. Ser. B 34(1972), 385–392.

8. Yu. I. Ingster, Asymptotically minimax hypothesis testing for nonparametric alternatives, I. Math. Methods Statist. 2(1993), No. 2, 84–114.

(Received 20.05.1998)

Authors’ address:

I. Javakhishvili Tbilisi State University

2, University St., Tbilisi 380043

Georgia