A Locally Adaptive Transformation Method of Boundary Correction in Kernel Density Estimation

Transcription

1 A Locally Adaptive Transformation Metod of Boundary Correction in Kernel Density Estimation R.J. Karunamuni a and T. Alberts b a Department of Matematical and Statistical Sciences University of Alberta, Edmonton, Alberta, Canada, T6G 2G1; b Courant Institute of Matematical Sciences, New York University, New York, NY, USA Abstract Kernel smooting metods are widely used in many researc areas in statistics. However, kernel estimators suffer from boundary effects wen te support of te function to be estimated as finite endpoints. Boundary effects seriously affect te overall performance of te estimator. In tis article, we propose a new metod of boundary correction for univariate kernel density estimation. Our tecnique is based on a data transformation tat depends on te point of estimation. Te proposed metod possesses desirable properties suc as local adaptivity and non-negativity. Furtermore, unlike many oter transformation metods available, te proposed estimator is easy to implement. In a Monte Carlo study, te accuracy of te proposed estimator is numerically analyzed and compared wit te existing metods of boundary correction. We find tat it performs well for most sapes of densities. Te teory beind te new metodology, along wit te bias and variance of te proposed estimator, are presented. Results of a data analysis are also given. Keywords: density estimation; mean squared error; kernel estimation; transformation. Sort Title: Kernel Density Estimation AMS Subject Classifications: Primary: 62G07; Secondary: 62G20 Corresponding Autor: R.J.Karunamuni@ualberta.ca 1

2 1 Introduction Nonparametric kernel density estimation is now popular and in wide use wit great success in statistical applications. Kernel density estimates are commonly used to display te sape of a data set witout relying on a parametric model, not to mention te exposition of skewness, multimodality, dispersion, and more. Early results on kernel density estimation are due to Rosenblatt (1956) and Parzen (1962). Since ten, muc researc as been done in te area; see, e.g., te monograps of Silverman (1986), and Wand and Jones (1995). Let f denote a probability density function wit support [0, ) and consider nonparametric estimation of f based on a random sample X 1,..., X n from f. Ten te traditional kernel estimator of f is given by f n (x) = 1 n ( ) x Xi K, (1.1) were K is some cosen unimodal density function, symmetric about zero wit support traditionally on [, 1], and is te bandwidt ( 0 as n ). Te basic properties of f n are well-known, and under some smootness assumptions tese include, for x 0, E f n (x) = f(x) K(t)dt f (1) (x) tk(t)dt f (2) (x) Var f n (x) = f(x) ( ) 1 K 2 (t)dt + o, n n t 2 K(t)dt + o( 2 ), were c = min(x/, 1). For x, te so-called interior points, te bias of f n (x) is of order O( 2 ), wereas at boundary points, i.e. for x [0, ), f n is not even consistent. In nonparametric curve estimation problems tis penomenon is referred to as te boundary effects of (1.1). To remove tese boundary effects a variety of metods ave been developed in te literature. Some well-known metods are summarized below: 2

3 Te reflection metod (Cline and Hart, 1991; Scuster, 1985; Silverman, 1986). Te boundary kernel metod (Gasser and Muller, 1979; Gasser et al., 1983; Jones, 1993; Muller, 1991; Zang and Karunamuni, 2000). Te transformation metod (Marron and Ruppert, 1994; Ruppert and Cline, 1994; Wand et al., 1991). Te pseudo-data metod (Cowling and Hall, 1996). Te local linear metod (Ceng et al., 1997; Ceng, 1997; Zang and Karunamuni, 1998). Oter metods (Zang et al., 1999; Hall and Park, 2002). Te reflection metod is specifically designed for te case f (1) (0) = 0, were f (1) denotes te first derivative of f. Te boundary kernel metod is more general tan te reflection metod in te sense tat it can adapt to any sape of density. However, a drawback of tis metod is tat te estimates migt be negative near te endpoints; especially wen f(0) 0. To correct tis deficiency of boundary kernel metods some remedies ave been proposed; see Jones (1993), Jones and Foster (1996), Gasser and Muller (1979) and Zang and Karunamuni (1998). Te local linear metod is a special case of te boundary kernel metod tat is tougt of by some as a simple, ard-to-beat default approac, partly because of optimal teoretical properties (Ceng et al., 1997) in te boundary kernel implicit in local linear fitting. Te pseudo-data metod of Cowling and Hall (1996) generates some extra data X (i) s using wat tey call te tree-point-rule, wic are ten combined wit te original data X i s to form a kernel type estimator. Marron and Ruppert s (1994) transformation metod consists of a tree-step process. First, a transformation g is selected from a parametric family so tat te density of Y i = g(x i ) as a first derivative tat is approximately equal 3

4 to 0 at te boundaries of its support. Next, a kernel estimator wit reflection is applied to te Y i s. Finally, tis estimator is converted by te cange-of-variables formula to obtain an estimate of f. Among oter metods, two very promising recent ones are due to Zang et al. (1999) and Hall and Park (2002). Te former metod is a combination of te metods of pseudo-data, transformation and reflection; wereas te latter metod is based on wat tey call an empirical translation correction. Te boundary kernel and related metods usually ave low bias but te price for tat is an increase in variance. It as been observed tat approaces involving only kernel modifications witout regard to f or data, suc as te boundary kernel metod, are always associated wit larger variance. Furtermore, te corresponding estimates tend to take negative values near te boundary points. On te oter and, transformation-based boundary correction estimates are non-negative and generally ave low variance, possibly due to non-negativity. It as gradually gotten troug to researcers tat tis non-negativity property is important in practical applications and te approaces producing non-negative estimators are well wort exploring. In tis article, we develop a new transformation approac for boundary correction. Our metod consists of a straigtforward transformation of data. It is easy to implement in practice compared to oter existing transformation metods (compare wit Marron and Ruppert, 1994). A distinct feature of our metod is tat it is locally adaptive; tat is, te transformation depends on te point of estimation. Oter desirable properties of te proposed estimator are: (i) it is non-negative everywere, and (ii) it reduces to te traditional kernel estimator (1.1) at te interior points. In simulations, we compared te proposed estimator wit oter existing metods of boundary correction and observed tat it performs quite well for most sapes of densities. Section 2 contains te metodology and te main results. Section 3 and 4 present simulation studies and a data analysis, respectively. Final comments are given in Section 5. 4

5 2 Locally Adaptive Transformation Estimator 2.1 Metodology For convenience, we sall assume tat te unknown probability density function f as support [0, ), and consider estimation of f based on a random sample X 1,..., X n from f. Our transformation idea is based on transforming te original data X 1,..., X n to g(x 1 ),..., g(x n ), were g is a non-negative, continuous and monotonically increasing function from [0, ) to [0, ). Based on te transformed data, we now define for x = c, c 0, f n (x) = 1 n ( ) x g(xi ) / K K(t)dt, (2.1) were is te bandwidt (again 0 as n ), K is a non-negative symmetric kernel function wit support on [, 1], satisfying K(t)dt = 1, tk(t)dt = 0, and 0 t 2 K(t)dt <. We can now state te following lemma, wic exibits te bias and variance of (2.1) under certain conditions on g and f. Te proof is given in te Appendix. Lemma 2.1. Let f n be defined by (2.1). Assume tat f (2) and g (2) exist and are continuous on [0, ), were f (i) and g (i) denote te i t derivative of f and g, respectively, wit f (0) = f, g (0) = g. Furter assume tat g(0) = 0 and g (1) (0) = 1. Ten for x = c, 0 c 1, we 5

6 ave E f n (x) f(x) = + K(t)dt {f(0)g (2) (0) 2 2 K(t)dt { f (2) (0)c 2 (c t)k(t)dt + f (1) (0) K(t)dt + (t c) 2 K(t)dt } tk(t)dt [ f (2) (0) f(0)g (3) (0) 3g (2) (0)(f (1) (0) f(0)g (2) (0)) ] } + o( 2 ), (2.2) and Var f n (x) = = f(0) c n( K(t)dt)2 f(x) n( K(t)dt)2 ( 1 K 2 (t)dt + o n ( 1 K 2 (t)dt + o n ) ), (2.3) Te last equality follows since f(0) = f(x) cf (1) (x)+(c) 2 f (2) (x)/2+o( 2 ) for x = c (see (A.3)). Note tat te leading term of te variance of fn is not affected by te transformation g. Wen c = 1, Var f n (x) = f(x)(n) 1 K2 (t)dt + o((n) ), wic is exactly te expansion of te interior variance of te traditional estimator (1.1). We sall use our transformation g so tat te first order term in te bias expansion (2.2) is zero. Assuming tat f(0) > 0, it is enoug to let / g (2) (0) = f (1) (0) tk(t)dt f(0) (c t)k(t)dt. (2.4) Note tat te rigt-and side of (2.4) depends on c; tat is, te transformation g depends 6

7 on te point of estimation inside te boundary region [0, ). We call tis property local adaptivity. In order to display tis local dependence we sall denote g by g c, 0 c 1, in wat follows. Combining (2.4) wit te additional assumptions in Lemma 2.1, g c sould now satisfy te following tree conditions for eac c, 0 c 1: (i) g c : [0, ) [0, ), g c is continuous, monotonically increasing and g (i) c exists, i = 1, 2, 3, (ii) g c (0) = 0, g c (1) (0) = 1 (iii) g (2) c (0) = f (1) (0) / tk(t)dt f(0) (c t)k(t)dt. (2.5) Functions satisfying conditions (2.5) can be easily constructed, wit te simplest candidates being polynomials. In order to satisfy property (iii) of (2.5) and maintain g c as an increasing function we require at least a cubic (for te case f (1) (0) < 0). So for 0 c 1 we use te transformation g c (y) = y + d 2 l cy 2 + d 2 l 2 cy 3, (2.6) were / l c = tk(t)dt (c t)k(t)dt (2.7) and d = f (1) (0) / f(0). (2.8) 7

8 Wen c = 1, g c (y) = y. Tis means tat f n (x) defined by (2.1) wit g c of (2.6) reduces to te usual kernel estimator f n (x) defined by (1.1) for interior points, i.e., for x, f n (x) coincides wit f n (x). Higer order polynomials could be used to obtain te same properties but tey are increasingly more complicated to deal wit and offer very little marginal advantage. For g c defined by (2.6), te bias term of (2.2) takes te form E f n (x) f(x) = 2 2 K(t)dt + 6 (f (1) (0)) 2 f(0) { f (2) (0) (t 2 2tc)K(t)dt } (t c) 2 K(t)dt(lc 2 + l c ) + o( 2 ), (2.9) for x = c, 0 c 1. Note tat wen c = 1, E f n (x) f(x) = 2 2 f (2) (x) 1 t2 K(t)dt + o( 2 ), wic is exactly te same expression of te interior bias of te traditional kernel estimator (1.1), since f (2) (x) = f (2) (0) + o(1) for x = c (see (A.5)). 2.2 Estimation of g c In order to implement te transformation (2.6) in practice, one must replace d = f (1) (0)/f(0) wit a pilot estimate. Tis requires te use of anoter density estimation metod, suc as semiparametric, kernel or nearest neigbour metod. Our experience is tat proposed estimator of f given below is insensitive to te fine details of te pilot estimate of d, and terefore any convenient estimate can be employed. A natural estimate would be a fixed kernel estimate wit bandwidt cosen optimally at x = 0 as in Zang et al. (1999). Oter possible estimators of d are proposed in Coi and Hall (1999) and Park et al. (2003). Following Zang et al. (1999), in tis paper we use te kernel type pilot estimate of d given 8

9 by ˆd = (log f n( 1 ) log f n(0))/ 1, (2.10) were f n( 1 ) = 1 n 1 ( ) 1 X i K + 1 (2.11) n 2 1 and f n(0) = max { 1 n 0 ( ) } Xi K (0), 1, (2.12) 0 n 2 were 1 = o() wit as in (2.1), K is a usual symmetric kernel function as in (1.1), K (0) is a so-called endpoint order two kernel satisfying K (0) (t)dt = 1, tk (0) (t)dt = 0, and 0 < t 2 K (0) (t)dt <, and 0 = b(0) 1 wit b(0) given by We now define b(0) = { ( 1 t2 K(t)dt) 2 ( 0 K2 (0) (t)dt) ( 0 t2 K (0) (t)dt) 2 ( 1 K2 (t)dt) }. (2.13) ĝ c (y) = y + ˆd 2 l cy 2 + ˆd 2 l 2 cy 3, (2.14) for 0 c 1, as our estimator of g c (y) defined by (2.6), were ˆd is given by (2.10). Note tat ĝ c and ˆd depend on n but tis dependence is suppressed for notational convenience. 9

10 It as been argued in te literature (see, e.g., Park et al., 2003) tat one may use a bandwidt 1 different from for te pilot estimator of d, and tat 1 sould be of te same order as or faster tan to inerit te full advantages of te proposed density estimator. Tis adjustment also improves te asymptotic teoretical results of our proposed density estimator ˆf n (x) given below at (2.15), and sould act as te basis for te coice of 1 in (2.10) and (2.11). 2.3 Te Proposed Estimator Our proposed estimator of f(x) is defined as, for x = c, c 0, ˆf n (x) = 1 n ( ) x ĝc (X i ) / K K(t)dt, (2.15) were ĝ c is given by (2.14), K is as given in (2.1), and is te bandwidt used in (2.1). For x, ˆfn (x) reduces to te traditional kernel estimator f n (x) given by (1.1). Tus ˆf n (x) is a natural boundary continuation of te usual kernel estimator (1.1). An important adjustment in te estimator (2.15) is tat it is based on a locally adaptive transformation ĝ c given by (2.14), since ĝ c transforms te data depending on te point of estimation. Furtermore, it is important to remark ere tat te adaptive kernel estimator (2.15) is non-negative (provided K is non-negative), a property sared wit reflection based estimators (see Jones and Foster, 1996; Zang et al. 1999) and transformation based estimators (see Marron and Ruppert, 1994; Zang et al. 1999; Hall and Park, 2002), but not wit most boundary kernel approaces. Te next teorem establises te bias and variance of (2.15). Teorem 2.1. Let ˆfn (x) be defined by (2.15) wit a kernel function K as given in (2.1) and a bandwidt = O(n /5 ). Suppose 1 in (2.10) is of te form 1 = O(n /4 ). Furter 10

11 assume tat K (1) exists and is continuous on [, 1], and tat f(0) > 0 and f (2) exists and is continuous in a neigbourood of 0. Ten for x = c, 0 c 1, we ave E ˆf n (x) f(x) = 2 2 K(t)dt + 6 (f (1) (0)) 2 f(0) { f (2) (0) (t 2 2tc)K(t)dt } (t c) 2 K(t)dt(lc 2 + l c ) + o( 2 ), (2.16) were l c is given by (2.7), and Var ˆf n (x) = f(0) c ( ) 1 n( K 2 (t)dt + o. (2.17) K(t)dt)2 n Remark 2.1 As one of te referees correctly pointed out, te coice of bandwidt is very important for te good performance of any kernel estimator. Te optimal bandwidt for (2.15) can be derived as follows. From (2.16) and (2.17), te leading terms of te asymptotic MSE of ˆf n (x) at x = c, 0 c 1 are 4 4 ( ) 2 { K(t)dt f (2) (0) (t 2 2tc)K(t)dt +6(f(0)) (f (1) (0)) 2 ( ) 2 +(n) f(0) K(t)dt K 2 (t)dt. } 2 (t c) 2 K(t)dt(lc 2 + l c ) 11

12 Minimizing te preceding expression wit respect to yields ] [ c = n {[4f(0) /5 K 2 (t)dt f (2) (0) (t 2 2tc)K(t)dt } c 2 1/5 + 6(f(0)) (f (1) (0)) 2 (t c) 2 K(t)dt(lc 2 + l c )] (2.18) wic is te local optimal bandwidt at x = c, 0 c 1, for te transformation estimator (2.15). Expression (2.18) is quite similar to tat of te adaptive variable location kernel estimator given by Park et al. (2003). In order to see tis more clearly we find from teir paper tat te leading terms of te asymptotic MSE of teir adaptive location kernel estimator at x = c, 0 c 1, are 4 4 ( 1 ) 2 { [ 1 K(t)dt f (2) (x) t 2 K(t)dt + 2ρ(c) c c 1 c ] tk (1) (t)dt 1 +(f(x)) (f (1) (x)) 2 ρ(c) 2 K (2) (t)dt ( 1 ) 2 1 +(n) f(x) K(t)dt K 2 (t)dt, c c were ρ(c) = (K(c)) 1 tk(t)dt. Minimizing te preceding expression wit respect to c yields 1 ] [ ( 1 1 ) c = n {[4f(x) /5 K 2 (t)dt f (2) (x) t 2 K(t)dt + 2ρ(c) tk (1) (t)dt c c c 1 2 } 1/5 + (f(x)) (f (1) (x)) 2 ρ(c) 2 K (t)] (2), (2.19) c c } 2 wic is te local optimal bandwidt at x = c, 0 c 1, for Park et al. s (2003) adaptive estimator. It is easy to see te similarity between (2.18) and (2.19). In applications neiter expression can be employed directly, owever, due to teir dependence on te unknown 12

13 functional quantities f(x), f (1) (x) and f (2) (x). An alternative approac is to apply te bandwidt variation function tecnique tat is frequently implemented in boundary kernel metods. Adaptive bandwidt or variable window widt kernel density estimators are generally of te form f n (x) = 1 n ( ) 1 x α(x i ) K Xi, (2.20) α(x i ) were α is some non-negative function; see, e.g. Abramson (1982), Hall et al. (1995) and Terrell and Scott (1992), among oters. It is easy to sow tat te estimator (2.20) also exibits boundary effects wen te unknown density f is supported on [0, ) or [0, 1] and a kernel K wit support on [, 1] is used. To te best of our knowledge tere are no boundary adjusted adaptive bandwidt or variable window widt kernel estimators tat exist in te literature. A transformation tecnique based on suc an estimator would be f n (x) = 1 n K(t)dt 1 α(g(x i )) K ( ) x g(xi ), (2.21) α(g(x i )) were g is a suitable transformation, like (2.6), tat removes te boundary effects. However, te details of estimators of te form (2.21) are beyond te scope of te present paper. Remark 2.2 Smooting metods based on local linear and local polynomial fitting metods are now popular. In te density estimation context te local polynomial metod as been implemented in Ceng et al. (1997), Ceng (1997) and Zang and Karunamuni (1998), among oters. For example, Zang and Karunamuni (1998) ave studied te following local polynomial density estimator (a very similar estimator is given in Ceng (1997) and Ceng et 13

14 al. (1997)): f n(x) = 1 n K o ( ) x Xi (2.22) were K o(t) = e T o S (1, t,..., t p ) T K(t), S = (s j,l ) is a p p matrix wit s j,l = t j+l K(t)dt, 0 j, l p, and e T = (0, 1,..., 0) T is a p 1 vector wose second element is 1 and te oters are zero. Ten te bias and variance of (2.22) at x = c, 0 c 1, are Bias(f n(x)) = 1 (p + 1)! (b(c))p+1 f (p+1) (x) /b(c) t p+1 K o,c/b(c)(t)dt and Var(f n(x)) = 1 nb(c) f(x) /b(c) ( K o,c/b(c) (t) ) 2 dt, were b : [0, 1] R + wit b(1) = 1 is a bandwidt variation function and Ko,c is called an equivalent boundary kernel ; see Zang and Karunamuni (1998) for more details. Te asymptotic MSE of fn(x) is typically less tan tat of ˆfn (x) given by (2.15) in a weak minimax sense; see Ceng et al. (1997). However, local polynomial estimators are based on te so-called equivalent boundary kernels Ko,c wic are not always positive. As a result, local polynomial estimators may take negative values, especially were te unknown density is very small. At te crucial point x = 0 if f(0) 0 ten fn(0) may be negative; see, e.g., Figure 1. Tis unattractive feature of local polynomial density estimators as been criticized by many autors; see, e.g., Zang et al. (1999) and Hall and Park (2002). On te oter and, te proposed estimator ˆf n (x) given by (2.15) is always non-negative wenever te kernel K is non-negative. Tus, as far as applications are concerned, we recommend te use of ˆf n (x) over fn(x), even toug te latter possesses some good teoretical properties. 14

15 3 Simulations and Discussion To test te effectiveness of our estimator we simulated its performance against tat of oter well-known metods. Among tese were some relatively new tecniques, suc as a recent estimator due to Hall and Park (2002) wic is very similar to our own, and te transformation and reflection based metod given by Zang et al. (1999). We also competed against some more classical estimators; namely te boundary kernel and its close counterpart te local linear fitting metod, Jones and Foster s (1996) nonnegative adaptation estimator, and Cowling and Hall s (1996) pseudo-data metod. Te first of te competing estimators is due to Hall and Park (2002) (ereafter H&P). It is defined as, for x = c, 0 c 1, ˆf HP (x) = 1 n ( ) x Xi + ˆα(x) / K K(t)dt, (3.1) were ˆα(x) is a correction term given by ˆα(x) = f 2 (x) ( x ) f(x) ρ, and f(x) = 1 n ( ) x Xi / K K(t)dt, (3.2) wic is referred to as te cut-and-normalized kernel estimator, f (x) is an estimate of f (1) (x), and ρ(u) = K(u) v u vk(v)dv. To estimate f (1) (x), we used te endpoint kernel 15

16 of order (1,2) (see Zang and Karunamuni, 1998) given by K 1,c (t) = 12(2c(1 + t) 3t2 4t 1) (c + 1) 4 I [,c], and te corresponding kernel estimator f (x) = 1 n 2 c ( ) x Xi K 1,c/b(c), c were c = b(c) wit te bandwidt variation function b(c) = 2 1/5 (1 c) + c for 0 c 1. Note tat Hall and Park s estimator is a special case of te estimator (2.1), but wit ĝ c (y) = y ˆα(c), were x = c for 0 c 1. Tis ĝ, owever, does not necessarily satisfy te conditions of (2.5). Te metod of Zang et al. (1999) (ereafter Z,K&J) applies a transformation to te data and ten reflects it across te left endpoint of te support of te density. Te resultant kernel estimator is of te form ˆf ZKJ (x) = 1 n { K ( ) ( )} x Xi x + gn (X i ) + K (3.3) were g n (x) = x + d n x 2 + Ad 2 nx 3. Note tis g n is very similar to our own, te main difference being it is not dependent on te point of estimation. Te only requirement on A is tat 3A > 1, and in practice we used te recommended value of A =.55. However, d n is again an estimate of d = f (1) (0)/f(0), and te metodology used is te same as tat given by (2.10) to (2.13). 16

17 Te boundary kernel wit bandwidt variation is defined (see Zang and Karunamuni, 1998) as ˆf B (x) = 1 n c ( ) x Xi K (c/b(c)) c (3.4) wit c = min(x/, 1). On te boundary te bandwidt variation function c = b(c) is employed, ere b(c) = 2 c. We used te boundary kernel K (c) (t) = { } 12 (1 + t) (1 2c)t + 3c2 2c + 1 I (1 + c) 4 [,c]. (3.5) 2 Zang and Karuamuni ave sown tat tis kernel is optimal in te sense of minimizing te MSE in te class of all kernels of order (0,2) wit exactly one cange of sign in teir support. A simple modification of te boundary kernel gives us te local linear fitting metod (LL). We used te kernel K (c) (t) = 12(1 t 2 ) (1 + c) 4 (3c 2 18c + 19) {8 16c + 24c2 12c 3 + t(15 30c + 15c 2 )}I [,c] (3.6) in (3.4) and call te resulting estimator ˆf LL (x). In tis case te bandwidt variation function is b(c) = c. Te Jones and Foster (1996) nonnegative adaptation estimator (ereafter J&F) is defined as ˆf JF (x) = f(x) exp { } ˆf(x) f(x) 1 (3.7) were f(x) is te cut-and-normalized kernel estimator of (3.2). Tis version of te J&F estimator takes ˆf to be some boundary kernel estimate of f, ere we use ˆf(x) = ˆf B (x) as defined by (3.5). 17

18 Cowling and Hall s (1996) pseudo-data estimator generates data beyond te left endpoint of te support of te density. Teir estimator (ereafter C&H) is defined as [ ˆf CH (x) = 1 K n ( ) x Xi + m ( ) ] x X( i) K (3.8) were m is an integer of larger order tan n but smaller order tan n; we cose m = n 9/10. Furter X ( i) is given by teir tree-point rule X ( i) = 5X (i/3) 4X (2i/3) X (i) were, for real t > 0, X (t) linearly interpolates among te values 0, X (1),..., X (n), were X (i) represents te i t order statistic. Trougout our simulations, we used te Epanecnikov kernel K(t) = 3(1 4 t2 )I [,1] wenever a kernel of order two was required. Note tat outside of te boundary, te majority of te estimators presented reduce to te regular kernel metod given by (1.1) wit te Epanecnikov. Te only exception is Cowling and Hall s, wic is a close approximation to te kernel estimator away from te origin. Eac estimator was tested over various sapes of density curves, but we ave collapsed our results into four specific densities wic are representative of te different possible combinations of te beaviour of f at 0. Density 1 andles te case f(0) = 0 and Densities 2, 3, & 4 satisfy f(0) > 0 but f (1) (0) = 0, f (1) (0) > 0 and f (1) (0) < 0, respectively. In all simulations a sample size of n = 200 was used. Te bandwidt was taken to be te optimal global bandwidt of te regular kernel estimator (1.1), given by te formula = { 1 K(t)2 dt [ 1 t2 K(t)dt] 2 [f (2) (x)] 2 dx } 1/5 n /5, (3.9) 18

19 (see Silverman, 1986, C. 3). Tis particular coice was made so tat it is possible to compare te different estimators witout regard to bandwidt effects. In our estimation of d we used te bandwidt 1 = n /20. For eac density we ave calculated te bias, variance and MSE of te estimators at zero, along wit te MISE over te region [0, ). Te results are all averaged over 1000 trials and are presented in Table 1. We ave also graped te performance of ten typical performances of eac estimator, along wit te regular kernel estimator, over te boundary region and part of te interior. Tese are presented in Figures 1 troug 4. Table 1 and Figures 1-4 about ere. In Density 1, wit f(0) = 0, our estimator beaves quite well at zero. It is only sligtly beind Hall and Park s in terms of MSE at zero. In terms of MISE te boundary kernel puts in te best sowing, followed closely by te LL metod and J&F. It sould be noted toug tat te boundary kernel and LL metod are usually negative near zero. Despite it s good sowing at zero te H&P metod comes in fourt in MISE as it sligtly underestimates te density over [0, ). In fift and sixt place are te Z,K&J and C&H metods. In last place in te MISE column is our estimator, wic, from Figure 1, is evidently te result of severely underestimating f to te rigt of zero. We believe tat error in te estimate ˆd of d is te reason beind tis. Wen f(0) = 0, ˆd may be inaccurate and introduce error into our metod. Future work may be done on creating better estimates of ˆd in tis situation. For Density 2, wit f (1) (0) = 0 and f(0) > 0, our estimator is te clear winner in terms of MSE at zero and te MISE over te boundary. Tis is also obvious from Figure 2. It is somewat surprising tat Hall and Park s estimator is muc weaker for tis density, especially given tat it and our estimators are all special cases of te same general form (2.1). 19

20 Density 3 proves to be difficult for all of te estimators. Tis is most evident from Figure 3, in wic we see a large amount of variability. In terms of MISE all te estimators perform similarly except for H&P, wic does sligtly worse tan te rest, and C&H, wic does poorly over te wole boundary. At zero, our estimator is te best overall but only by a slim margin tis time. In terms of MISE it is te LL and J&F metods tat are te winners, but again only by a small amount. Our estimator as its poorest sowing on te exponential sape of Density 4. At zero our estimator produces an extremely ig MSE value relative to te oters, and even in terms of MISE it is andily beaten (except by C&H). From Figure 4 we see te underlying reason is te inability of our estimator to accurately capture te steep growt of te exponential density. Te Z,K&J estimator is quite accurate at zero and over te entire boundary, and te performance of te boundary kernel, LL metod, J&F and H&P estimators do not lag far beind. Even Cowling and Hall s estimator does well at zero, but pays te price for it in MISE over te rest of te boundary region. 4 Data Analysis We tested our proposed estimator over two datasets. Te first consists of 35 measurements of te average December precipitation in Des Moines, Iowa from 1961 to Te bandwidt = was produced by cross-validation (see Bowman and Azzalini, 1997). Figure 5 sows our proposed estimator (solid line) along wit Hall and Park s (dased line). Over most of te boundary region tey are similar, but around zero tey disagree on te sape of te density. Hall and Park s estimator seems to believe tere s a local minimum to te rigt of zero, but te istogram and te rug sow oterwise. A larger value for would probably diminis tis beavior. We ave observed tat te boundary kernel, LL metod and J&F 20

21 estimators tend to linearly approximate te density over te boundary region, wit little regard for curvature. Figure 5 about ere. Te second dataset is comprised of 365 wind speed measurements taken daily in 1993 at 2 AM in Eindoven, te Neterlands (source: Royal Neterlands Meteorological Institute). Intuition would suggest tat te underlying density f sould be similar to Density 3, wit f(0) > 0 and f (1) (0) > 0, so tat tere is substantial mass at zero but te mode is sligtly to te rigt. We used bandwidt = 2.5 wic we cose subjectively. Values suggested by cross-validation typically seem to undersmoot tis data, ence te subjective coice. We ave observed tat for tis dataset te sape of te estimators is fairly insensitive to te coice of bandwidt, provided tat it is large enoug, and so we believe our coice is justified. Figure 6 sows our proposed estimator (solid line) and te LL metod (dased line) applied to te wind data. Bot estimators capture te ypotesized sape of te density and are similar over te boundary region. However our estimator clearly produces a smooter curve. Figure 6 about ere. 5 Concluding Remarks Marron and Ruppert s (1994) excellent work on kernel density estimation was among te first to introduce a transformation tecnique in boundary correction (also see Wand et al., 1991). Teir sopisticated transformation metodology, owever, as been labelled complicated by some researcers; see, e.g., Jones and Foster (1996). In tis paper, we presented an easy-to-implement, general transformation metod given by (2.1). Tis class of estimators possesses certain desirable properties, suc as being everywere non-negative and aving 21

22 unit integral asymptotically, and can be tailored specifically to combat te boundary effects found in te traditional kernel estimator. Te proposed estimator defined by (2.15) and te boundary corrected estimator of Hall and Park (2002) are special cases of te preceding general tecnique. Beyond tis, bot estimators are locally adaptive in te sense tat te amount of transformation depends upon te point of estimation, and, even furter, bot transformations are estimated from te data itself. We believe te latter point is especially important, as it allows for all-important reductions in bias wile, at te same time, maintaining low variance. Te transformation tat we ave cosen at (2.14) is a convenient and computationally easy one, but it is by no means te result of an exaustive searc of functions satisfying te conditions (2.5). In fact, we conjecture tat it may be possible to construct different transformations satisfying tese conditions but wit muc improved performance, in te sense tat tey will be more robust to te sape of te density. Acknowledgements: Tis researc was supported by a grant from te Natural Sciences and Engineering Researc Council of Canada. We wis to tank te Editor, te Associate Editors and te referees for very constructive and useful comments tat led to a muc improved presentation. We also tank Drs. B.U. Park and S. Zang for some personal correspondence. 22

23 A Appendix: Proofs Proof of Lemma 2.1: For x = c, 0 c 1, we ave from (2.1) E f n (x) K(t)dt = 1 ( ) x E K g(xi ) = 1 ( x g(y) K = 0 ) f(y)dy K(t) f(g ((c t))) g (1) (g ((c t))) dt. (A.1) Letting q(t) = g ((c t)) we ave q (1) (t) = g (1) (q(t)), and q(2) (t) = 2 g (2) (q(t)), g (1) (q(t)) 3 so ten a Taylor expansion of order 2 on te function f(q( ))/g (1) (q( )) at t = c gives f(q(t)) g (1) (q(t)) = f(q(c)) [ ] f (1) (q(c))g (1) (q(c)) f(q(c))g (2) (q(c)) (t c) g (1) (q(c)) [g (1) (q(c))] (t c) 2 {[ f (2) (q(c)) [g (1) (q(c))] 3 g (1) (q(c)) f (1) (q(c)) g(2) (q(c)) [g (1) (q(c))] 2 ] f(q(c)) g(3) (q(c))g (1) (q(c)) g (2) (q(c)) 2 [g (1) (q(c))] [g (1) (q(c))] 3 ( 2 f (1) (q(c)) f(q(c)) g(2) (q(c)) g (1) (q(c)) = f(0) (t c)[f (1) (0) f(0)g (2) (0)] ) } g (2) (q(c)) + o( 2 ) g (1) (q(c)) (t c)2 [f (2) (0) f(0)g (3) (0) 3g (2) (0)(f (1) (0) f(0)g (2) (0))] + o( 2 ), (A.2) 23

24 wit te last equality following from q(c) = g (0) = 0 and g (1) (q(c)) = g (1) (0) = 1. Also, by te existence and continuity of f (2) ( ) near 0, for x = c we ave f(0) = f(x) cf (1) (x) + (c)2 2 f (2) (x) + o( 2 ), (A.3) f (1) (x) = f (1) (0) + cf (2) (0) + o(), (A.4) and f (2) (x) = f (2) (0) + o(1). (A.5) Now from (A.1) to (A.5), we obtain for x = c, 0 c 1, E f n (x) K(t)dt = K(t)dt(f(x) cf (1) (x) + (c)2 2 f (2) (x) + o( 2 )) (t c)k(t)dt(f (1) (0) f(0)g (2) (0)) (c) 2 f (2) (0) (t c) 2 K(t)dt { f (2) (0) f(0)g (3) (0) 3g (2) (0)(f (1) (0) f(0)g (2) (0)) } = f(x) K(t)dt K(t)dt + o( 2 ) { f (1) (0) f(0)g (2) (0) tk(t)dt } + f(0)g (2) (0) ck(t)dt {( + 2 [f (t c) K(t)dt) 2 (2) (0) f(0)g (3) (0) 2 3g (2) (0)(f (1) (0) f(0)g (2) (0)) ] } 2c 2 f (2) (0) K(t)dt + o( 2 ). (A.6) 24

25 Te proof of (2.2) is now completed by dividing bot sides of (A.6) by K(t)dt. In order to prove (2.3) we first note tat from (2.1), ( { c 2 Var f 1 ( ) n (x) = K(t)dt) } x g(xi ) Var K n ( ) { 2 ( ) ( ( )) } 2 1 x = K(t)dt E K 2 g(xi ) x g(xi ) E K n 2 ( 2 = K(t)dt) (I 1 + I 2 ), (A.7) were I 1 = 1 ( ) x n E g(xi ) 2 K2 = 1 K 2 (t) f(g ((c t))) n = 1 n f(0) g (1) (g ((c t))) dt ( 1 K 2 (t)dt + o n ), (A.8) from (A.2). Similarly, I 2 = 1 ( E K n 2 = 1 ( = o n ( 2 1 n ( )) 2 x g(xi ) K(t) f(g ((c t))) g (1) (g ((c t))) dt ), (A.9) ) 2 again from (A.2). By combining (A.7) to (A.9) we now complete te proof of (2.3). Lemma A.1. Let ˆd be defined by (2.10) wit = O(n /5 ) and 1 = O(n /4 ). Suppose 25

26 tat f(x) > 0 for x = 0, and tat f (2) is continuous in a neigbourood of 0. Ten [ ] 3 E ˆd d Xi = x i, X j = x j = O( 3 1) for any integer i, 1 i, j n, were d is given by (2.8). Proof. Similar to te proof of Lemma A.2 of Zang, Karunamuni and Jones (1999). Proof of Teorem 2.1: For x = c, 0 c 1, we write E ˆf n (x) f(x) = I 3 + I 4, (A.10) were I 3 = E ˆf n (x) E f n (x) and I 4 = E f n (x) f(x), wit f n (x) given by (2.1). From Lemma 2.1, we obtain I 4 = o( 2 ) wen g is cosen to satisfy (2.4). By an application of Taylor s expansion of order 1 on K, we see I 3 satisfies I 3 ( K(t)dt) n = ( K(t)dt) n ( ) ( ) E x K ĝc (X i ) x gc (X i ) K ( ) ( ) E gc (X i ) ĝ c (X i ) x K (1) gc (X i ) + ε(g c (X i ) ĝ c (X i )), (A.11) were 0 < ε < 1 is a constant. Note tat for any constants d and l c, we ave for any y 0, g c (y) = y + d 2 l cy 2 + (dl c ) 2 y 3 = y [ ( dl c y + 1 ) ] y. Tus g c (y) for y 16. Terefore, εĝ 15 c(x i ) + (1 ε)g c (X i ) for X i 16. Since K 15 26

27 vanises outside [, 1], from (A.11) we ave I 3 ( K(t)dt) n C n 2 { ( ) E gc (X i ) ĝ c (X i ) ( ) x εĝc (X i ) (1 ε)g c (X i ) K(1) I[0 X i ρ]} E ĝ c (X i ) g c (X i ) I[0 X i ρ], (A.12) were ρ = 16/15 and C = sup t 1 K (1) (t). Now by te definitions of g c and ĝ c (see (2.6) and (2.14)) we obtain E ĝ c (X i ) g c (X i ) I[0 X i ρ] = E c l 2 ( ˆd d)xi 2 + lc( 2 ˆd 2 d 2 )Xi 3 I[0 X i ρ] l c 2 2 ρ 2 E ˆd d I[0 X i ρ] + l 2 c(ρ) 3 E ˆd 2 d 2 I[0 X i ρ]. (A.13) Te Caucy-Scwartz inequality and Lemma A.1 yield tat E [ ˆd d k ] X i = x i = O( k 1), for 1 k 3. From tis we ave E ˆd { d I[0 X i ρ] = E E[ ˆd d I[0 X i ρ] } X i = x i ] { = E I[0 X i ρ] E[ ˆd d } Xi = x i ] O( 1 ) E I[0 X i ρ] = o( 2 ), (A.14) were te last equality follows from te fact tat lim n E I[0 X i ρ] = f(0) for 27

28 eac 0 c 1. From te same expression we also obtain tat E ˆd 2 d 2 I[0 X i ρ] = E ˆd d ˆd + d I[0 X i ρ] = E ˆd d ˆd d + 2d I[0 X i ρ] E ˆd d 2 I[0 X i ρ] + 2 d E ˆd d I[0 X i ρ] = o( 2 ). (A.15) By combining (A.12) to (A.15), we now ave I 3 = o( 2 ). Te proof is now completed by te preceding result, te fact tat I 4 = o( 2 ), and (A.10). We now prove (2.17). First write ( ) { 2 K(t)dt Var ˆf n (x) = 1 ( ) } x (n) Var ĝc (X i ) K 2 { = 1 [ ( ) ( )] x (n) Var ĝc (X i ) x gc (X i ) K K 2 ( ) } x gc (X i ) + K = I 5 + I 6 + I 7, (A.16) were g c is given by (2.6), I 5 = 1 (n) 2 Var { [ K ( ) ( )] } x ĝc (X i ) x gc (X i ) K, (A.17) I 6 = 1 ( ) x (n) Var gc (X i ) K, (A.18) 2 28

29 and I 7 = 2 (n) 2 Cov ( [ K ( ) ( x ĝc (X i ) x gc (X i ) K )], ( ) ) x gc (X i ) K. (A.19) From Lemma 2.1, we ave I 6 = f(0) n ( ) 1 K 2 (t)dt + o. (A.20) n Now consider I 5. Note tat I 5 1 (n) 2 E = 1 (n) (n) 2 { E [ K 1 i<j n ( ) ( )] } 2 x ĝc (X i ) x gc (X i ) K ( ) ( )] 2 x ĝc (X i ) x gc (X i ) K [ ( ) ( )] x ĝc (X i ) x gc (X i ) E K K [ ( ) ( )] x ĝc (X j ) x gc (X j ) K K [ K = I 8 + I 9. (A.21) 29

30 By an application of Taylor s expansion of order 1 on K, we obtain I 8 = 1 (n) 2 = 1 (n) 2 1 n 2 4 C n 2 4 [ E K ( ) ( )] 2 x ĝc (X i ) x gc (X i ) K [ gc (X i ) ĝ c (X i ) E K (1) ( x gc (X i ) + ε(g c (X i ) ĝ c (X i )) )] 2 [ ( )] 2 x E (g c (X i ) ĝ c (X i ))K (1) gc (X i ) + ε(g c (X i ) ĝ c (X i )) E(ĝ c (X i ) g c (X i )) 2 I[0 X i ρ], (A.22) using an argument similar to (A.12) above, were 0 < ε < 1, ρ = 16/15 and C > 0 are all constants independent of n. Similar to (A.13) we can write E(ĝ c (X i ) g c (X i )) 2 I[0 X i ρ] [ ] 2 lc = E 2 ( ˆd d)xi 2 + lc( 2 ˆd 2 d 2 )Xi 3 I[0 X i ρ] l2 c 2 (ρ)4 E( ˆd d) 2 I[0 X i ρ] + 2l 4 c(ρ) 6 E( ˆd 2 d 2 ) 2 I[0 X i ρ] O( 4 2 1) + O( 6 2 1) = o( 7 ). (A.23) Now combining (A.21) and (A.23), we obtain I 8 = o (n 3 ) = o ((n) ). An argument similar to (A.22) yields tat I 9 C n i j n E ĝ c (X i ) g c (X i ) ĝ c (X j ) g c (X j ) I[0 X i ρ, 0 X j ρ], (A.24) were ρ = 16/15, and C > 0 is a constant independent of n. As in (A.23), we obtain using 30

31 Lemma A.1, E ĝ c (X i ) g c (X i ) ĝ c (X j ) g c (X j ) I[0 X i ρ, 0 X j ρ] = E c l 2 ( ˆd d)xi 2 + lc( 2 ˆd 2 d 2 )Xi 3 c l 2 ( ˆd d)xj 2 + lc( 2 ˆd 2 d 2 )Xj 3 I[0 X i ρ, 0 X j ρ] { C 1 E 2 ˆd d + 3 ˆd 2 2 d } 2 I[0 Xi ρ, 0 X j ρ] 2C 1 {E 4 ˆd d 2 I[0 X i ρ, 0 X j ρ] + E ˆd } 2 d 2 2 I[0 X i ρ, 0 X j ρ] C 2 4 { 2 1 E I[0 X i ρ, 0 X j ρ] } = O( ) = o( 8 ), (A.25) were C 1, C 2 are positive constants independent of n. Now combine (A.24) and (A.25) to conclude tat I 9 = o( 4 ) = o((n) ). Similarly, it is easy to sow tat I 4 = o((n) ) using te covariance inequality. Tis completes te proof of (2.17). 31

32 References [1] Abramson, I. (1982). On bandwidt variation in kernel estimates - A square root law. Te Annals of Statistics, 10, [2] Bowman, A.W. and Azzalini, A. (1997). Applied Smooting Tecniques for Data Analysis: te kernel approac wit S-plus illustrations, Oxford University Press. [3] Ceng, M.Y. (1997). Boundary-aware estimators of integrated density derivative. Journal of te Royal Statistical Society Ser. B, 59, [4] Ceng, M.Y., Fan, J. and Marron, J.S. (1997). On Automatic Boundary Corrections. Te Annals of Statistics, 25, [5] Cline, D.B.H and Hart, J.D. (1991). Kernel Estimation of Densities of Discontinuous Derivatives. Statistics, 22, [6] Coi, E. and Hall, P. (1999). Data sarpening as a prelude to density estimation. Biometrika, 86, [7] Cowling, A. and Hall, P. (1996). On Pseudodata Metods for Removing Boundary Effects in Kernel Density Estimation. Journal of te Royal Statistical Society Ser. B, 58, [8] Gasser, T. and Müller, H.G. (1979). Kernel Estimation of Regression Functions. In Smooting Tecniques for Curve Estimation, Lecture Notes in Matematics 757, eds. T. Gasser and M. Rosenblatt, Heidelberg: Springer-Verlag, pp [9] Gasser, T., Müller, H.G. and Mammitzsc, V. (1985). Kernels for Nonparametric Curve Estimation. Journal of te Royal Statistical Society Ser. B., 47,

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35 Table 1: Bias, variance and MSE of te indicated estimators at 0, and teir MISE over [0, ). Density 1: f(x) = x2 2 e x, x 0, Density 2: f(x) = 2/πe x2 /2, x 0, wit n = 200, = wit n = 200, = Bias Var MSE MISE Bias Var MSE MISE New Metod H&P metod Z,K&J metod Boundary Kernel LL metod J&F metod C&H metod Density 3: f(x) = (x 2 + 2x )e 2x, Density 4: f(x) = 2e 2x, x 0, x 0, wit n = 200, = wit n = 200, = Bias Var MSE MISE Bias Var MSE MISE New Metod H&P metod Z,K&J metod Boundary Kernel LL metod J&F metod C&H metod

36 New Metod Kernel H&P Z,K&J Boundary LL J&F C&H Figure 1: Ten typical estimates of density 1, f(x) = x2 2 e x, x 0, wit te optimal global bandwidt =

37 New Metod Kernel H&P Z,K&J Boundary LL J&F C&H Figure 2: Ten typical estimates of density 2, f(x) = 2/πe x2 /2, wit te optimal global bandwidt =

38 New Metod Kernel H&P Z,K&J Boundary LL J&F C&H Figure 3: Ten typical estimates of density 3, f(x) = 1 2 (2x2 + 4x + 1)e 2x, wit te optimal global bandwidt =

39 New Metod Kernel H&P Z,K&J Boundary LL J&F C&H Figure 4: Ten typical estimates of density 4, f(x) = 2e 2x, wit te optimal global bandwidt =

40 Figure 5: Density estimates for 35 measurements of average December precipitation in Des Moines, Iowa from , sown on te rug, wit = Te solid line is our proposed estimator (witout BVF) and te dased line is Hall and Park s.

41 Figure 6: Density estimates for 365 measurements of wind speed at Eindoven, te Neterlands, taken at 2 AM every day during Te values are plotted on te bottom. Te solid line is our proposed estimator (witout BVF) and te dased line is te LL estimator, bot wit bandwidt = 2.5.