On automatic kernel density estimate-based tests for goodness-of-fit

Abstract

Although estimation and testing are different statistical problems, if we want to use a test statistic based on the Parzen–Rosenblatt estimator to test the hypothesis that the underlying density function f is a member of a location-scale family of probability density functions, it may be found reasonable to choose the smoothing parameter in such a way that the kernel density estimator is an effective estimator of f irrespective of which of the null or the alternative hypothesis is true. In this paper we address this question by considering the well-known Bickel–Rosenblatt test statistics which are based on the quadratic distance between the nonparametric kernel estimator and two parametric estimators of f under the null hypothesis. For each one of these test statistics we describe their asymptotic behaviours for a general data-dependent smoothing parameter, and we state their limiting Gaussian null distribution and the consistency of the associated goodness-of-fit test procedures for location-scale families. In order to compare the finite sample power performance of the Bickel–Rosenblatt tests based on a null hypothesis-based bandwidth selector with other bandwidth selector methods existing in the literature, a simulation study for the normal, logistic and Gumbel null location-scale models is included in this work.

Proofs

1.1 Proof of Theorem 1

Consider the expansion

$$\begin{aligned} (n\hat{h})^{-1} I_n(\hat{h})&= \int \{ f_{\hat{h}}(x) - K_{\hat{h}}*f(x) \}^2 \mathrm{d}x \nonumber \\&\quad + \int \{ K_{\hat{h}}*f(x) - K_{\hat{h}}*g(x; \hat{\theta }_1, \hat{\theta }_2 ) \}^2 \mathrm{d}x \nonumber \\&\quad + 2 \int \{ f_{\hat{h}}(x) - K_{\hat{h}}*f(x) \} \{ K_{\hat{h}}*f(x) - K_{\hat{h}}*g(x; \hat{\theta }_1, \hat{\theta }_2 ) \} \mathrm{d}x \nonumber \\&=: I_{n,1} + I_{n,2} + 2 I_{n,3}. \end{aligned}$$

(21)

In order to establish the asymptotic behaviour of each one of the previous terms, we use the approach of Tenreiro (2001), which is based on the Taylor expansion

$$\begin{aligned} K_h(u) := W(u,h) = \sum _{\ell =0}^{\omega -1} (h-1)^\ell K^{\partial (\ell )}(u) + (h-1)^\omega K^{\partial (\omega )}(u,h), \end{aligned}$$

where $u\in \mathbb {R}$, $h>0$,

$$\begin{aligned} K^{\partial (\ell )}(u) := \frac{1}{\ell !}\frac{\partial ^\ell W}{\partial h^\ell }(u,1), \; \ell =0,\ldots ,\omega -1, \end{aligned}$$

and

$$\begin{aligned} K^{\partial (\omega )}(u,h) := \frac{1}{(\omega -1)!} \int _0^1 (1-t)^{\omega -1} \frac{\partial ^\omega W}{\partial h^\omega }(u,1+t(h-1)) \mathrm{d}t. \end{aligned}$$

Note that, from assumption (K) the functions $K^{\partial (\ell )}$ are bounded and integrable on $\mathbb {R}$, for $\ell =1,\ldots ,\omega -1$, and there exists $\eta \in \,]0,1[$ such that the function $K^{\partial (\omega ),\eta }(u) := \sup _{|h-1|\le \eta } |K^{\partial (\omega )}(u,h)|,$ is bounded and integrable on $\mathbb {R}$. From the previous Taylor expansion we deduce the following expansions for $f_{\hat{h}}$, $K_{\hat{h}}*f$ and $K_{\hat{h}}*g(\cdot ; \hat{\theta }_1, \hat{\theta }_2 )$, that play a crucial role in what follows. For $x\in \mathbb {R}$ and denoting by h the deterministic bandwidth h(f) given in assumption (B), we have

$$\begin{aligned} f_{\hat{h}}(x)= & {} \sum _{\ell =0}^{\omega -1} \xi _n^\ell \frac{1}{n} \sum _{i=1}^n K^{\partial (\ell )}_h(x-X_i) + \xi _n^\omega \frac{1}{n} \sum _{i=1}^n K^{\partial (\omega )}_h(x-X_i,\hat{h}), \end{aligned}$$

(22)

$$\begin{aligned} K_{\hat{h}}*f(x)= & {} \sum _{\ell =0}^{\omega -1} \xi _n^\ell K^{\partial (\ell )}_h\!*\!f(x) + \xi _n^\omega K^{\partial (\omega )}_h(\cdot ,\hat{h})\!*\!f(x), \end{aligned}$$

(23)

and

$$\begin{aligned} K_{\hat{h}}*g(x; \hat{\theta }_1, \hat{\theta }_2 ) = \sum _{\ell =0}^{\omega -1} \xi _n^\ell K^{\partial (\ell )}_h\!*\!g(x; \hat{\theta }_1, \hat{\theta }_2 ) + \xi _n^\omega K^{\partial (\omega )}_h(\cdot ,\hat{h})\!*\!g(x; \hat{\theta }_1, \hat{\theta }_2 ), \end{aligned}$$

(24)

where $K^{\partial (\ell )}_h(u)=K^{\partial (\ell )}_h(u/h)/h$ and $K^{\partial (\omega )}_h(u,\hat{h})=K^{\partial (\omega )}_h(u/h,\hat{h}/h)/h$. Moreover, for $|\hat{h}/h-1| \le \eta$ we have $| K^{\partial (\omega )}_h(u,\hat{h})| \le K^{\partial (\omega ),\eta }_h(u)$, for $u\in \mathbb {R}$.

Each one of the terms in (21) is studied in the following propositions. We denote by h the deterministic sequence h(f) given in assumption (B).

Proposition 1

We have

$$\begin{aligned} \begin{aligned} I_{n,1}&= (1-\xi _n) \frac{1}{nh} R(K) + \frac{1}{nh^{1/2}} U_n (1+ o_p(1)) \\&\quad + O_p\Big ( n^{-1} h^{-1/2} \xi _n + (nh)^{-1} \xi _n^2 + \xi _n^\omega \Big ), \end{aligned} \end{aligned}$$

where $U_n$ given by (25) is asymptotically normal with zero mean and variance $2R(K\!*\!K)R(f)$.

Proof

Using equalities (22) and (23), and assumptions (D), (K) and (B), from Proposition 2 of Tenreiro (2001, p. 290) we have

$$\begin{aligned} \begin{aligned}I_{n,1}&= \int \{ f_h(x)-K_h*f(x) \}^2 dx \mathrm {d}x - \xi _n \frac{1}{nh} R(K)\\&\quad + O_p\Big ( n^{-1} h^{-1/2} \xi _n + (nh)^{-1} \xi _n^2 + \xi _n^\omega \Big ). \end{aligned} \end{aligned}$$

Moreover, using degenerated U-statistics techniques (see Hall 1984; Tenreiro 1997) we have

$$\begin{aligned} \int \{ f_h(x)-K_h*f(x) \}^2 \mathrm{d}x = \frac{1}{nh} R(K) + \frac{1}{nh^{1/2}} U_n (1+ o_p(1)), \end{aligned}$$

with

$$\begin{aligned} U_n= & {} \frac{2}{n} \sum _{1\le i < j \le n} q_n(X_i,X_j),\\ q_n(u,v)= & {} h^{1/2} \int \big \{ K_h(x-u) - K_h\!*\!f(x) \big \} \big \{ K_h(x-v) - K_h\!*\!f(x) \big \} \mathrm{d}x,\nonumber \end{aligned}$$

(25)

and $U_{n}$ is asymptotically normal with zero mean and variance equal to $2R(K*K) R(f)$.

$\square$

Proposition 2

We have

$$\begin{aligned} I_{n,2} = R\big ( f - g(\cdot ; \theta _1(f),\theta _2(f) ) \big ) + o_p(1). \end{aligned}$$

Moreover, under the null hypothesis we have

$$\begin{aligned} I_{n,2} = O_p\big ( n^{-1} \big ). \end{aligned}$$

Proof

From (23) and (24) we have

$$\begin{aligned} I_{n,2}&= \sum _{\ell ,\ell '=0}^{\omega -1} \xi _n^{\ell +\ell '} \int K^{\partial (\ell )}_h \!* \hat{\delta }_n(x) K^{\partial (\ell ')}_h \!* \hat{\delta }_n(x) \mathrm{d}x \\&\quad + 2 \sum _{\ell =0}^{\omega -1} \xi _n^{\omega +\ell } \int K^{\partial (\ell )}_h \!* \hat{\delta }_n(x) K^{\partial (\omega )}_h (\cdot ,\hat{h}) \!* \hat{\delta }_n(x) \mathrm{d}x \\&\quad + \xi _n^{2\omega } \int \big ( K^{\partial (\omega )}_h (\cdot ,\hat{h}) \!* \hat{\delta }_n(x) \big )^2 \mathrm{d}x, \end{aligned}$$

where $\hat{\delta }_n(x)=f(x) - g(x; \hat{\theta }_1, \hat{\theta }_2 )$. Moreover,

$$\begin{aligned} \bigg | \int K^{\partial (\ell )}_h \!* \hat{\delta }_n(x) K^{\partial (\ell ')}_h \!* \hat{\delta }_n(x) \mathrm{d}x \bigg | \le ||K^{\partial (\ell )}||_1 ||K^{\partial (\ell ')}||_1 || \hat{\delta }_n ||_2^2, \end{aligned}$$

and for all $\epsilon \in \,]0,\eta [$ and for $|\hat{h}/h-1| \le \epsilon$ we have

$$\begin{aligned} \bigg | \int K^{\partial (\ell )}_h \!* \hat{\delta }_n(x) K^{\partial (\omega )}_h (\cdot ,\hat{h}) \!* \hat{\delta }_n(x) \mathrm{d}x \bigg | \le ||K^{\partial (\ell )}||_1 ||K^{\partial (\omega ),\eta }||_1 || \hat{\delta }_n ||_2^2 \end{aligned}$$

and

$$\begin{aligned} \bigg | \int \big ( K^{\partial (\omega )}_h (\cdot ,\hat{h}) \!* \hat{\delta }_n(x) \big )^2 \mathrm{d}x \bigg | \le ||K^{\partial (\omega ),\eta }||_1^2 || \hat{\delta }_n ||_2^2. \end{aligned}$$

Therefore, from assumption (B) we can write

$$\begin{aligned} I_{n,2} = R\big ( K_h*\hat{\delta }_n \big ) + O_p\big ( || \hat{\delta }_n ||_2^2 \xi _n \big ). \end{aligned}$$

(26)

On the other hand, from assumption (F) the function $(\theta _1,\theta _2) \mapsto g(x; \theta _1,\theta _2)$ has continuous first-order partial derivatives, and the functions $(\theta _1,\theta _2) \mapsto \big |\big | \frac{\partial g}{\partial \theta _k} (\cdot ; \theta _1,\theta _2)\big |\big |_2$ are locally bounded on $\mathbb {R} \times ]0,+\infty [$ for $k=1,2$. Therefore, for each $x\in \mathbb {R}$, a Taylor expansion of $g(x; \hat{\theta }_1, \hat{\theta }_2 )$ at the point $(\theta _1(f), \theta _2(f))$ leads to

$$\begin{aligned} \hat{\delta }_n(x) = f(x) - g(x; \hat{\theta }_1, \hat{\theta }_2 ) = f(x) - g(x; \theta _1(f), \theta _2(f) ) + u_n(x), \end{aligned}$$

(27)

where

$$\begin{aligned} || u_n ||_2 = O_p\big ( |\hat{\theta }_1-\theta _1(f) | + |\hat{\theta }_2-\theta _2(f) | \big ). \end{aligned}$$

(28)

The first part of the stated result follows now from (26) and the following convergence that can be established from standard arguments as h tends to zero, when n tends to infinity:

$$\begin{aligned} R\big ( K_h \!*\! (f - g(\cdot ; \theta _1(f),\theta _2(f) )) \big ) = R\big ( f - g(\cdot ; \theta _1(f),\theta _2(f)) \big ) + o(1). \end{aligned}$$

Finally, taking into account that $\hat{\delta }_n = u_n$ under the null hypothesis, where $|| u_n ||_2 = O_p(n^{-1/2})$ from assumption (P), we deduce that $I_{n,2} = O_p(n^{-1})$ under the null hypothesis. $\square$

To establish the order of convergence of $I_{n,3}$ we need the following lemma. Note that we are always assuming that $\hat{h}$ satisfies assumption (B).

Lemma 1

Let $\varphi$ be a real-valued function defined on $\mathbb {R} \times ]0,+\infty [$, and assume that there exists $\eta \in \,]0,1[$ such that the function $\varphi ^\eta (u)= \sup _{|h-1|\le \eta } |\varphi (u,h)|$ is bounded and integrable.

• (a) If $\gamma _n : \mathbb {R} \mapsto \mathbb {R}$ is such $||\gamma _n||_2 = O(1)$ then

  $$\begin{aligned} \frac{1}{n} \sum _{i=1}^n \int \big \{ \varphi _h(x-X_i) - \varphi _h*f(x) \big \} \gamma _n(x) \mathrm{d}x = O_p\big ( n^{-1/2} \big ). \end{aligned}$$

• (b) If $\gamma _n : \mathbb {R} \mapsto \mathbb {R}$ is such $||\gamma _n||_r = O(1)$, for some $r\in [1,\infty ]$, then

  $$\begin{aligned} \frac{1}{n} \sum _{i=1}^n \int | \varphi _h(x-X_i,\hat{h}) - \varphi _h(\cdot ,\hat{h})*f(x) | \gamma _n(x) \mathrm{d}x = O_p( 1 ). \end{aligned}$$

• (c) If $\tilde{\gamma }_n = \tilde{\gamma }_n(\cdot ;X_1,\ldots ,X_n): \mathbb {R} \mapsto \mathbb {R}$ is such that $||\tilde{\gamma }_n||_r = O_p(1)$, for some $r\in [1,\infty ]$, then

  $$\begin{aligned} \frac{1}{n} \sum _{i=1}^n \int | \varphi _h(x-X_i,\hat{h}) - \varphi _h(\cdot ,\hat{h})*f(x) | \tilde{\gamma }_n(x) \mathrm{d}x = O_p\big ( h^{-1/r} \big ). \end{aligned}$$

Proof

Write $S_{n,a}$, $S_{n,b}$ and $S_{n,c}$ for the sums considered in each one of the parts a), b) and c). The order of convergence stated in part a) follows from the inequalities

$$\begin{aligned} {E}(S_{n,a}^2)&\le \frac{1}{n} {E}\bigg ( \int \varphi _h(x-X_i) \gamma _n(x) \mathrm{d}x \bigg )^2 \\&\le \frac{1}{n} \iint \varphi (u)^2 \gamma _n(z+uh)^2 f(z) \mathrm{d}u \mathrm{d}z \\&\le \frac{1}{n} ||f||_\infty ||\varphi ||_2^2 ||\gamma _n||_2^2. \end{aligned}$$

In order to establish parts b) and c), it is enough to note that for all $\epsilon \in \,]0,\eta [$ and for $|\hat{h}/h-1| \le \epsilon$ we have

$$\begin{aligned} |S_{n,b}| \le \frac{1}{n} \sum _{i=1}^n \bigg \{ \int \varphi _h^\epsilon (x-X_i) |\gamma _n|(x) \mathrm{d}x + \int \varphi _h^\epsilon *f(x) |\gamma _n|(x) \mathrm{d}x \bigg \}=:S_{n,b}^\epsilon , \end{aligned}$$

and

$$\begin{aligned} |S_{n,c}| \le \frac{1}{n} \sum _{i=1}^n \bigg \{ \int \varphi _h^\epsilon (x-X_i) |\tilde{\gamma }_n|(x) \mathrm{d}x + \int \varphi _h^\epsilon *f(x) |\tilde{\gamma }_n|(x) \mathrm{d}x \bigg \}=:S_{n,c}^\epsilon , \end{aligned}$$

where

$$\begin{aligned} {E}(S_{n,b}^\epsilon ) \le 2 \int \varphi _h^\epsilon *f(x) |\gamma _n|(x) \mathrm{d}x \le 2 ||f||_\infty ||\varphi ^\epsilon ||_s ||\gamma _n||_r, \end{aligned}$$

and

$$\begin{aligned} S_{n,c}^\epsilon \le 2 h^{-1/r} ||\tilde{\gamma }_n||_r ||\varphi ^\epsilon ||_s, \end{aligned}$$

with $1/r+1/s=1$. Therefore, $S_{n,b}^\epsilon = O_p(1)$ and $S_{n,c}^\epsilon = O_p\big ( h^{-1/r} \big )$ which implies the stated results as $\hat{h}/h -1 = o_p(1)$. $\square$

Proposition 3

We have

$$\begin{aligned} I_{n,3} = O_p\big ( (nh)^{-1/2} \big ). \end{aligned}$$

Moreover, under the null hypothesis we have

$$\begin{aligned} I_{n,3} = O_p\big ( n^{-1} h^{-1/r} + n^{-1/2} \xi _n^\omega \big ), \end{aligned}$$

where $r \in \,]2,\infty ]$ is given in assumption (F).

Proof

The first statement follows from Propositions 1 and 2 since $|I_{n,3}| \le I_{n,1}^{1/2} I_{n,2}^{1/2}$. On the other hand, from (22), (23) and (24) we have

$$\begin{aligned} I_{n,3}&= \sum _{\ell ,\ell '=0}^{\omega -1} \xi _n^{\ell +\ell '}\frac{1}{n} \sum _{i=1}^n \int \big \{ K^{\partial (\ell )}_h(x-X_i) - K^{\partial (\ell )}_h\!*\!f(x) \big \}K^{\partial (\ell ')}_h \!* \hat{\delta }_n(x) \mathrm{d}x \\&\qquad + \sum _{\ell =0}^{\omega -1} \xi _n^{\omega +\ell }\frac{1}{n} \sum _{i=1}^n \big \{ K^{\partial (\ell )}_h(x-X_i) - K^{\partial (\ell )}_h\!*\!f(x) \big \}K^{\partial (\omega )}_h (\cdot ,\hat{h}) \!* \hat{\delta }_n(x) \mathrm{d}x \\&\qquad + \sum _{\ell =0}^{\omega -1} \xi _n^{\omega +\ell } \frac{1}{n} \sum _{i=1}^n \int \big \{ K^{\partial (\omega )}_h(x\!-\!X_i,\hat{h}) \!-\! K^{\partial (\omega )}_h(\cdot ,\hat{h})\!*\!f(x) \big \} K^{\partial (\ell )}_h \!* \hat{\delta }_n(x) \mathrm{d}x \\&\qquad + \xi _n^{2\omega } \frac{1}{n} \sum _{i=1}^n \int \big \{ K^{\partial (\omega )}_h(x\!-\!X_i,\hat{h}) \!-\! K^{\partial (\omega )}_h(\cdot ,\hat{h})\!*\!f(x) \big \} K^{\partial (\omega )}_h (\cdot ,\hat{h}) \!* \hat{\delta }_n(x) \mathrm{d}x \\&\quad = I_{n,3}^1 + I_{n,3}^2 + I_{n,3}^3 + I_{n,3}^4. \end{aligned}$$

where $\hat{\delta }_n(x)=f(x) - g(x; \hat{\theta }_1, \hat{\theta }_2 )$. From assumption (F), the function $(\theta _1,\theta _2) \mapsto g(x; \theta _1,\theta _2)$ has continuous second-order partial derivatives, and for some $r \in \,]2,\infty ]$ the functions $(\theta _1,\theta _2) \mapsto \big |\big | \frac{\partial ^2 g}{\partial \theta _k \partial \theta _l} (\cdot ; \theta _1,\theta _2)\big |\big |_r$ are locally bounded on $\mathbb {R} \times ]0,+\infty [$, for $k,l=1,2$. Therefore, under the null hypothesis a Taylor expansion of $g(x; \hat{\theta }_1, \hat{\theta }_2 )$ at the point $(\theta _1(f), \theta _2(f))$ leads to

$$\begin{aligned} \hat{\delta }_n(x) = - \sum _k (\hat{\theta }_k-\theta _k(f)) \frac{\partial g}{\partial \theta _k} (x; \theta _1(f), \theta _2(f)) + v_n(x), \end{aligned}$$

(29)

for $x\in \mathbb {R}$, where from assumption (P)

$$\begin{aligned} || v_n ||_r = O_p\big ( (\hat{\theta }_1-\theta _1(f) )^2 + (\hat{\theta }_2-\theta _2(f) )^2 \big ) = O_p\big ( n^{-1} \big ). \end{aligned}$$

Therefore, from Lemma 1 we get $I_{n,3}^1 = O_p \big ( n^{-1} h^{-1/r} \big )$, $I_{n,3}^2 = O_p \big ( (n^{-1/2} + n^{-1} h^{-1/r}) \xi _n^\omega \big )$, $I_{n,3}^3 = O_p \big ( ( n^{-1/2} + n^{-1} h^{-1/r} ) \xi _n^\omega \big )$ and $I_{n,3}^4 = O_p \big ( ( n^{-1/2} + n^{-1} h^{-1/r} ) \xi _n^{2\omega } \big )$, which completes the proof. $\square$

We can now conclude the proof of Theorem 1. As $\xi _n = o_p(1)$ and $h \rightarrow 0$, as $n\rightarrow \infty$, from Proposition 1 we have

$$\begin{aligned} I_{n,1} = O_p\big ( (nh)^{-1} + \xi _n^\omega \big ). \end{aligned}$$

Therefore, from expansion (21) and Propositions 2 and 3, we get

$$\begin{aligned} (nh)^{-1} \big ( I_n(\hat{h}) - R(K) \big ) = R\big ( f - g(\cdot ; \theta _1(f),\theta _2(f) ) \big ) + O_p\big ( (nh)^{-1/2} + \xi _n^\omega \big ), \end{aligned}$$

which completes the proof of part b) as $nh \rightarrow \infty$, when $n\rightarrow \infty$. Moreover, under the null hypothesis from Propositions 1, 2 and 3 we also have

$$\begin{aligned} h^{-1/2} \big ( I_n(\hat{h}) - R(K) \big ) = U_n + O_p\Big ( h^{-1/2} \xi _n^2 + h^{1/2-1/r} + n h^{1/2} \xi _n^\omega \Big ) + o_p(1). \end{aligned}$$

Taking into account hypothesis (13), this completes the proof of part a) as $r>2$ and $U_n$ are asymptotically normal with zero mean and variance equal to $\nu _f^2 = 2R(K\!*\!K)R(f)$.

$\square$

1.2 Proof of Theorem 2

Let us consider the expansion

$$\begin{aligned} (n\hat{h})^{-1} J_n(\hat{h})&= \int \{ f_{\hat{h}}(x) - f(x) \}^2 \mathrm{d}x \nonumber \\&\quad + \int \{ f(x) - g(x; \hat{\theta }_1, \hat{\theta }_2 ) \}^2 \mathrm{d}x \nonumber \\&\quad + 2 \int \{ f_{\hat{h}}(x) - f(x) \} \{ f(x) - g(x; \hat{\theta }_1, \hat{\theta }_2 ) \} \mathrm{d}x \nonumber \\&=: J_{n,1} + J_{n,2} + 2 J_{n,3}. \end{aligned}$$

(30)

Each one of these terms will be studied in the following propositions. As before, we denote by h the deterministic sequence h(f) which existence is assured by assumption (B).

Proposition 4

We have

$$\begin{aligned} J_{n,1}= & {} \frac{1}{nh} R(K) + R(K_h*f - f) + \frac{1}{nh^{1/2}} U_n (1+o_p(1)) + \frac{1}{\sqrt{n}h^{-2}} V_n\\&+ O_p \Big ( \big ( (nh)^{-1} + h^4 \big ) \xi _n + \xi _n^\omega \Big ), \end{aligned}$$

where $U_n$ is defined in Proposition 1 and $V_n$ given by (31) is asymptotically normal with zero mean and variance $\mu _2(K)^2 \mathrm {Var}_f(f''(X_1))$.

Proof

Taking into account equality (22) and assumptions (D), (D’), (K), (K’) and (B), from Lemma 1 of Tenreiro (2001, p. 286) we have

$$\begin{aligned} J_{n,1} = \int \{ f_{h}(x) - f(x) \}^2 \mathrm{d}x + O_p \Big ( \big ( (nh)^{-1} + h^4 \big ) \xi _n + \xi _n^\omega \Big ). \end{aligned}$$

Using degenerated U-statistics techniques (see Hall 1984) we know that

$$\begin{aligned} \begin{aligned}\int \{ f_{h}(x) - f(x) \}^2 dx \mathrm {d}x&= \frac{1}{nh} R(K) + R(K_h*f - f) \\&\quad + \frac{1}{nh^{1/2}} U_n (1+o_p(1)) + \frac{1}{\sqrt{n}h^{-2}} V_n, \end{aligned} \end{aligned}$$

with $U_n$ given by (25) and

$$\begin{aligned} V_n := \frac{2}{\sqrt{n}} \sum _{i=1}^n \int \{ K_h(x-X_i) - K_h*f(x) \} h^{-2} \{ K_h*f(x) - f(x) \} \mathrm{d}x, \end{aligned}$$

(31)

with

$$\begin{aligned} h^{-2} \{ K_h*f(x) - f(x) \} = \iint _0^1 (1-t) u^2 K(u) f''(x-tuh) \mathrm{d}u \mathrm{d}t, \end{aligned}$$

(32)

is asymptotically normal with zero mean and variance equal to $\mu _2(K)^2 \mathrm {Var}_f(f''(X_1))$.

$\square$

Proposition 5

We have

$$\begin{aligned} J_{n,2} = R\big ( f - g(\cdot ; \theta _1(f),\theta _2(f) ) \big ) + o_p(1). \end{aligned}$$

Moreover, under the null hypothesis we have

$$\begin{aligned} J_{n,2} = O_p\big ( n^{-1} \big ). \end{aligned}$$

Proof

It follows straightforwardly from (27) and (28). $\square$

Proposition 6

We have

$$\begin{aligned} J_{n,3} = O_p\big ( (nh)^{-1/2} \big ). \end{aligned}$$

Moreover, under the null hypothesis we have

$$\begin{aligned} J_{n,3} = - \frac{1}{\sqrt{n}h^{-2}} \big ( W_n + o_p(1) \big ) + O_p \big ( n^{-1} h^{-1/r} + n^{-1/2} \xi _n^\omega \big ), \end{aligned}$$

where $W_n$ is given by (36).

Proof

The first statement follows from Propositions 4 and 5 because $|J_{n,3}| \le J_{n,1}^{1/2} J_{n,2}^{1/2}$ and $R(K_h*f - f)=O(h^4)$. Write

$$\begin{aligned} J_{n,3}&= \int \{ f_{\hat{h}}(x) - K_{\hat{h}}*f(x) \} \hat{\delta }_n(x) \mathrm{d}x + \int \{ K_{\hat{h}}*f(x) - f(x) \} \hat{\delta }_n(x) \mathrm{d}x \nonumber \\&=: J_{n,3}^1 + J_{n,3}^2 , \end{aligned}$$

(33)

where $\hat{\delta }_n(x)=f(x) - g(x; \hat{\theta }_1, \hat{\theta }_2 )$. From (22) and (23) we have

$$\begin{aligned} J_{n,3}^1&= \sum _{\ell =0}^{\omega -1} \xi _n^\ell \frac{1}{n} \sum _{i=1}^n \int \big \{ K^{\partial (\ell )}_h(x-X_i) - K^{\partial (\ell )}_h\!*\!f(x) \big \} \hat{\delta }_n(x) \mathrm{d}x \nonumber \\&\quad + \xi _n^\omega \frac{1}{n} \sum _{i=1}^n \int \big \{ K^{\partial (\omega )}_h(x-X_i,\hat{h}) - K^{\partial (\omega )}_h(\cdot ,\hat{h})\!*\!f(x) \big \} \hat{\delta }_n(x) \mathrm{d}x, \end{aligned}$$

where from Lemma 1 we get

$$\begin{aligned} J_{n,3}^1 = O_p \big ( n^{-1} h^{-1/r} + n^{-1/2} \xi _n^\omega \big ). \end{aligned}$$

(34)

On the other hand, from (23) we have

$$\begin{aligned} J_{n,3}^{2}&= \int \{ K_{h}*f(x) - f(x) \} \hat{\delta }_n(x) \mathrm{d}x \nonumber \\&\quad + \sum _{\ell =1}^{\omega -1} \xi _n^\ell \int K^{\partial (\ell )}_h\!*\!f(x) \hat{\delta }_n(x) \mathrm{d}x + \xi _n^\omega \int K^{\partial (\omega )}_h(\cdot ,\hat{h})\!*\!f(x) \hat{\delta }_n(x) \mathrm{d}x, \end{aligned}$$

(35)

where for all $\epsilon \in \,]0,\eta [$ and for $|\hat{h}/h-1| \le \epsilon$ we have

$$\begin{aligned} \bigg | \int K^{\partial (\omega )}_h(\cdot ,\hat{h})\!*\!f(x) \hat{\delta }_n(x) \mathrm{d}x \bigg | \le || K^{\partial (\omega ),\eta }||_1 ||f||_2 || \hat{\delta }_n ||_2. \end{aligned}$$

Moreover, as $\int K^{\partial (\ell )}(u) \mathrm{d}u = \int uK^{\partial (\ell )}(u) \mathrm{d}u=0$ for $\ell \ge 1$, a Taylor expansion of second order leads to

$$\begin{aligned} K^{\partial (\ell )}_h\!*\!f(x) = h^2 \iint _0^1 (1-t) u^2 K^{\partial (\ell )}(u) f^{\prime \prime }(x-tuh) \mathrm{d}t\mathrm{d}u. \end{aligned}$$

Therefore, for $\ell \ge 1$ we have

$$\begin{aligned} \bigg | \int K^{\partial (\ell )}_h \!*\!f(x) \hat{\delta }_n(x) \mathrm{d}x \bigg | \le h^2 \int u^2 |K^{\partial (\ell )}(u)| \mathrm{d}u ||f''||_2 || \hat{\delta }_n ||_2. \end{aligned}$$

Taking into account (27) and the fact that $|| \hat{\delta }_n ||_2 = O_p(n^{-1/2})$ under the null hypothesis, from (35) we get

$$\begin{aligned} J_{n,3}^{2} = \int \{ K_{h}*f(x) - f(x) \} \hat{\delta }_n(x) \mathrm{d}x + O_p\big ( n^{-1/2} ( h^2 \xi _n + \xi _n^\omega ) \big ). \end{aligned}$$

Finally, from (29) and (32), and assumptions (E) and ($G'$), we have

$$\begin{aligned}&\int \{ K_{h}*f(x) - f(x) \} \hat{\delta }_n(x) \mathrm{d}x \\&\quad = h^2 \iiint _0^1 (1-t) u^2 K(u) f^{\prime \prime }(x-tuh) \hat{\delta }_n(x) \mathrm{d}t\mathrm{d}u \mathrm{d}x \\&\quad = - \frac{1}{\sqrt{n}h^{-2}} \big ( W_n + o_p(1) \big ) + O_p\big ( n^{-1} h^2 \big ), \end{aligned}$$

where

$$\begin{aligned} W_n = \frac{1}{\sqrt{n}} \sum _{i=1}^n \sum _k \psi _k(X_i;\theta _1(f),\theta _2(f)) D_k(f), \end{aligned}$$

(36)

with

$$\begin{aligned} D_k(f) = \frac{1}{2} \mu _2(K) \int \bar{f}^{\prime \prime }(x) \frac{\partial g}{\partial \theta _k} (x; \theta _1(f) \theta _2(f)) \mathrm{d}x, \end{aligned}$$

as

$$\begin{aligned} \iiint _0^1 (1-t) u^2 K(u) f^{\prime \prime }(x-tuh) \frac{\partial g}{\partial \theta _k} (x; \theta _1(f) \theta _2(f)) \mathrm{d}t\mathrm{d}u \mathrm{d}x = D_k(f) + o(1), \end{aligned}$$

and

$$\begin{aligned} \bigg | \iiint _0^1 (1-t) u^2 K(u) f^{\prime \prime }(x-tuh) v_n(x) \mathrm{d}t\mathrm{d}u \mathrm{d}x \bigg | \le \mu _2(K) ||f''||_s ||v_n||_r, \end{aligned}$$

with $1/r+1/s=1$. Thus

$$\begin{aligned} J_{n,3}^{2} = - \frac{1}{\sqrt{n}h^{-2}} \big ( W_n + o_p(1) \big ) + O_p\big ( n^{-1} h^2 + n^{-1/2} ( h^2 \xi _n + \xi _n^\omega ) \big ) \end{aligned}$$

(37)

The proposition follows from (33), (34) and (37). $\square$

We can now conclude the proof of Theorem 2. From Proposition 4 and assumption (B’) we have

$$\begin{aligned} J_{n,1}= O_p\big ( (nh)^{-1} + \xi _n^\omega \big ). \end{aligned}$$

Therefore, from expansion (30) and Propositions 5 and 6, we get

$$\begin{aligned}&(nh)^{-1} \big ( J_n(\hat{h}) - R(K) - c_n(f;K) \big ) \\&\quad = R\big ( f - g(\cdot ; \theta _1(f),\theta _2(f) \big ) + O_p\big ( (nh)^{-1/2} + \xi _n^\omega \big ), \end{aligned}$$

which completes the proof of part b). Moreover, from Propositions 4, 5 and 6, under the null hypothesis we also have

$$\begin{aligned} h^{-1/2} \big ( J_n(\hat{h}) - R(K) - c_n(f;K) \big )&= U_n + ( nh^5 )^{1/2} (V_n - 2W_n) \\&\quad + O_p\Big ( h^{-1/2} \xi _n + h^{1/2-1/r} + n h^{1/2} \xi _n^\omega \Big ) + o_p(1). \end{aligned}$$

Taking into account hypothesis (14), this completes the proof of part a) as $r>2$ and, from the central limit theorem for degenerate U-statistics with variable kernels established in Tenreiro (1997, Theorem 1, p. 190), the sum $U_n + ( nh^5 )^{1/2} (V_n - 2W_n)$ is asymptotically normal with zero mean nd variance equal to $\sigma _f^2 = 2R(K*K) R(f) + \lambda _f \mu _2(K)^2 \mathrm {Var}_f(\varphi _f(X))$. $\square$

1.3 Proof of Theorem 3

We consider only the case of the test based on the critical region $\mathscr {C}(J_n(\hat{h}),\alpha )=\{J_n(\hat{h}),\alpha ) > q(J_n(\hat{h}),\alpha ) \}$ given in (15), where $q(J_n(\hat{h}),\alpha )$ is the quantile of order $1-\alpha$ of the null distribution of $J_n(\hat{h})$, but similar arguments can be used to establish the consistency of the test based on $\mathscr {C}(I_n(\hat{h}),\alpha )$. From Theorem 2.a) and for $f\in \mathscr {F}_0$ we have $\upsilon _{f}^{-1} h(f)^{-1/2} \big ( q(J_n(\hat{h}),\alpha ) - R(K) - c_n(f;K) \big ) \rightarrow \Phi ^{-1}(1-\alpha )$. Therefore,

$$\begin{aligned} q(J_n(\hat{h}),\alpha ) \rightarrow R(K)+c(f_0;K), \end{aligned}$$

(38)

because h(f) tends to zero, as $n\rightarrow \infty$, and $c_n(f;K)=c_n(f_0;K)=c(f_0;K) (1+o(1))$, with $c(f;K)=\frac{1}{4} \lambda _{f} \mu _2(K)^2 R(f^{\prime \prime }) (1 + o(1))$ (see Wand and Jones 1995, pp. 19–23, and Bosq and Lecoutre 1987, pp. 80–81). On the other hand, from Theorem 2.b) and for $f\in \mathscr {F}{\setminus }\mathscr {F}_0$ we have $(nh(f))^{-1} \big ( J_n(\hat{h}) - R(K) - c_n(f;K) \big ) {\mathop {\longrightarrow }\limits ^{p}} R\big (f - g(\cdot ; \theta _1(f),\theta _2(f) )\big ) \ne 0,$ which enables us to conclude that

$$\begin{aligned} J_n(\hat{h}) {\mathop {\longrightarrow }\limits ^{p}} + \infty , \; \text{ for } \text{ all } \;f\in \mathscr {F}{\setminus }\mathscr {F}_0, \end{aligned}$$

(39)

as $nh(f) \rightarrow \infty$, and $c_n(f;K)=c(f;K) (1+o(1))$. The consistency of the test based on $\mathscr {C}(J_n(\hat{h}),\alpha )$ follows now from (38) and (39). $\square$