Bickel Rosenblatt test

Transcription

1 University of Latvia

2 A classical Let X 1,..., X n be i.i.d. random variables with a continuous probability density function f. Consider a simple hypothesis H 0 : f = f 0 with a significance level α and completely specified f 0. Given the kernel density estimate f n (x) = 1 nh n n ( x Xi K i=1 h n ), where h n = h(n), a test statistic can be defined. The classical statistic [Bickel and Rosenblatt(1973)] = nh n ˆT br n [f n (x) f 0 (x)] 2 a(x)dx.

3 A smoothed modification To avoid bias problems a smoothed version of f 0, namely ) ( z (K hn f 0 )( ) = hn 1 K f 0 (z)dz, where is a convolution operator, is employed. And a(x) 1 is used as the arbitrary weight function, which leads to a modification of the statistic T n = nh d/2 n h n [f n (x) (K hn f 0 )(x)] 2 dx, and for composite hypothesis [fn T = n,ˆθ nhd/2 n (x) (K hn fˆθ )(x)] 2 dx.

4 Graphical representation conv f n diff Figure: Convolution, f n and the squared error.

5 An absolutely regular weakly dependent process Let (X t ) t Z, X t R be a strictly stationary process on a probability space (Ω, F, P). For any two σ-fields A and B F define the following measure of dependence β(a, B) := sup 1 2 I J P(A i B j ) P(A i )P(B j ), i=1 j=1 such that A i A i and B j B j, where i, j A i, B j Ω. Define F L J := σ(x k, J k L), when J L. Definition (X t ) t Z is called absolutely regular or β-mixing if β(n) = sup J Z β(f, J FJ+n ) 0, when n.

6 Asymptotic distribution of T n Theorem ([Neumann and Paparoditis(2000)]) If certain assumptions are fulfilled, then under H 0, (T n µ) d N(0, σ 2 ), where µ and σ 2 are σ 2 = 2 µ = hn d/2 [ f0 2 (x)dx K 2 (u)du, K(u)K(u + v)du] 2 dv.

7 Motivation (+) The test statistic can be used for: simple as well as composite hypothesis, independent and dependent identically distributed data without modification. ( ) No procedure for selecting the bandwidth h n.

8 Simulation study We define by f u the probability density function of the uniform U[0, 1] distribution f u = F u, F u = U[0, 1]. For the process (X t ) t Z we test the single hypothesis H 0 : f = f u versus H 1 : f f u. Suppose that a random variable X has a continuous cumulative distribution function F X, then F X (X ) = Y U[0, 1].

9 Alternatives close to U[0, 1] [Kallenberg and Ledwina(1995)] uses g 1 (x) = 1 + ρ cos(jπx), k g 2 (x) = exp θ j π j (x) ψ k (θ), j=1 with {π j } the orthonormal Legendre polynomials on [0, 1], ψ k (θ) = log 1 0 exp(θ φ(x))dx, θ Rk g g 2

10 Simulated power for dependent (AR(1), θ = 0.3) data Table: T n percentage rejections of the true H 0 at 5% significance level with n = 20, 50, 100, 500, 1000 for AR(1) case with φ = 0.3 made with 10,000 replications; h = h 0n 1/4 ; kernel U(0, 1). h 0 n Table: AR(1) case with φ = 0.3. Simulated power for alternatives g 1 and g 2 with n = 50 and 10,000 replications; h = h 0n 1/4, kernel U(0, 1). h 0 ρ j θ (0, 3) (0,-0. 4) (0.25,-0. 35)

11 Bandwidth selection for nonparametric kernel tests [Gao and Gijbels(2008)] consider a statistic ˆT n (h), similar to T n, for regression fit and derive Edgeworth expansions for size and power functions: α n (h) = P( ˆT n (h) > l α H 0 ) β n (h) = P( ˆT n (h) > l α H 1 ), and where l α is a simulated critical value of ˆT n (h). The Edgeworth expansions of α n (h) and β n (h) are then used to choose a suitable bandwidth β n (h ew ) = max β n(h), h H n(α) with H n (α) = {h : α c min < α n (h) < α + c min } for a small 0 < c min < α.

12 Gao an Gijbels used Edgeworth expansions for quadratic forms. [Bachmann and Dette(2005)] states that under H 0 (T n /nh) is a degenerate U statistic. For i.i.d. random variables and fixed bandwidth [Tenreiro(2005)] states that statistic I 2 n (h) = T n /h is a V statistic, In 2 (h) = 1 n Q h (X i, X j ), n i,j=1 Q h (u, v) = k(x, u, h)k(x, v, h)dx... [Fan and Linton(2003)] have derived Edgeworth expansions for a regression model specification test statistic, that is also a degenerate U statistic.

13 [Bachmann and Dette(2005)] T n nh 1 nh K 2 (x)dx [H h (f f 0 )] 2 (x)dx = U n + 2 n where Y i = (K h g h )(Z i ) E[K h g h (Z i )] and U n = 2 n 2 H n (Z i, Z j ) i<j i<j n ( ) 1 Y i + O P, n = 2 n 2 [K h (x Z i ) K h f (x)] [K h (x Z j ) K h f (x)] dx and U n is a degenerate U statistic. i=1

14 Bibliography I D. Bachmann and H. Dette. A note on the Bickel - Rosenblatt test in autoregressive time series. Stat. Probab. Lett., 74(3): , P. J. Bickel and M. Rosenblatt. On some global measures ot the deviations of density function estimates. The Annals of Statistics, 1(6): , Y. Fan and O. Linton. Some higher-order theory for a consistent non-parametric model specification test. J. Stat. Plann. Inference, 109(1-2): , J. Gao and I. Gijbels. Bandwidth selection in nonparametric kernel testing. Journal of the American Statistical Association, 103(484): , W. C. M. Kallenberg and T. Ledwina. Consistency and monte carlo simulation of a data driven version of smooth goodness-of-fit tests. Annals of Statistics, 23(5): , M. H. Neumann and E. Paparoditis. On bootstrapping L 2 -type statistics in density testing. Statistics & Probability Letters, 50(2): , C. Tenreiro. On the role played by the fixed bandwidth in the bickel-rosenblatt goodness-of-fit test. SORT, 29(2): , 2005.

15 Thank you!