Logistic Kernel Estimator and Bandwidth Selection. for Density Function

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1 International Journal of Contemporary Matematical Sciences Vol. 13, 2018, no. 6, HIKARI Ltd, ttps://doi.org/ /ijcms Logistic Kernel Estimator and Bandwidt Selection for Density Function Sama M. Abo-El-Hadid Department of Matematics, Insurance and Applied Statistics Faculty of Commerce and Business Administration Helwan University, Egypt Copyrigt 2018 Sama M. Abo-El-Hadid. Tis article is distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. Abstract Kernel density estimator is a non-parametric way to estimate te density function f(x) of a random variable X given a set of data. Tis density estimator is calculated by weigting te distances of all te sample data points, tese weigts are given by a function. Te estimated density is also affected by a parameter called te smooting parameter or te bandwidt. Tis smooting parameter affects ow smoot te resulting density is. In tis paper, te standard logistic distribution is suggested as a function. Te asymptotic bias, variance, mean squared error (MSE), and integrated mean squared error (IMSE) of tis proposed logistic estimator are investigated. Te optimal smooting parameter of te suggested logistic estimator is estimated. A simulation study is introduced to compare te proposed estimator wit oter estimators. We also apply te proposed logistic estimator to a real data set of stock market index for securities in Egypt wic is EGX 30 index. Keywords: logistic distribution, Kernel Density Estimation, EGX Introduction Te density estimator metod was introduced by Rosenblatt (1956). Rosenblatt s density estimator of te unknown density f(x) is given by (Härdle et. al. 2004): f (x) = 1 n K (x x i i=1, x (1) n )

2 280 Sama M. Abo-El-Hadid were n is te sample size; K( ) and are te function and te bandwidt respectively, were te function K( ) is assumed to be symmetric density function. Te statistical properties of density estimator in equation (1) and its optimal bandwidt are given by Parzen (1962). Te rest of tis paper is organized as follows: In Section 2 te proposed logistic estimator and its statistical properties are introduced. In Section 3 te optimal smooting parameter is obtained. Section 4 provides te results of a simulation study in wic te beaviour of te standard logistic estimator is compared wit te oter density estimators. Real data application is introduced in section 5. Finally, in section 6, a brief conclusion is provided. 2. Te standard logistic Kernel In tis paper, we propose te use of standard logistic for density estimation. Te standard logistic function is defined as (Evans et. al. 2004): Were K(u) = e u (1+e u ) E(u) = 2, u (2) uk(u) du = 0 (3) var(u) = u 2 K(u) du = π2 3 Ten te standard logistic density estimator is given by: f (x) = 1 (4) ( x x i n e ) n i=1 [1+e (x x i ) 2, x (5) ] Te expectation of te density estimator is(silverman 1986): E[f (x)] = E [ 1 n K (x x i i=1 ] = 1 K (x x i ) f(x)dx (6) n Let u = x x i ten E[f (x)] = 1 K(u) f(x u)du (7) Using Taylor expansion for f(x u) yields tat: Bias[f (x)] f (x) uk(u) du f (x) u 2 K(u) du (8) Substitute from equations (3) and (4) into equation (8) yields: ) Bias[f (x)] 2 f (x) ( π2 ) = 2 π f (x) (9) Also, it can be sown tat te variance of te density estimator is (Wand and Jones 1995): Var[f (x)] = 1 {E [ K n 2 (x x i )]2 [EK ( x x i )]2 } (10) = 1 { [ K n 2 (x x i )]2 f(x i )dx i [ K ( x x i ) f(x i)dx i ] 2 } let u = x x i, ten Var[f (x)] = 1 { n 2 K2 (u)f(x u)du [ K(u) f(x u)du] 2 }

3 Logistic estimator and 281 Again by Taylor expansion: f(x u) = f(x) uf (x) + u2 2 f (x) + 2! Ten Var[f (x)] 1 f(x) K 2 (u)du (11) n Using te standard logistic function: 0 let z = 1 + e u, ten: e u = z 1, K 2 (u) du = ( e u 0 (1+e u ) 2)2 du = e 2u (1+e u ) 4 e 2u = (z 1) 2 dz du = e u, du = dz K 2 (u) e 2u (1+e u ) 4 z 4 = z 1 du = du (12) = dz e u (z 1) (z 1)2 dz z 4 (z 1) 1 z 3 z 4 1 2(1+e u ) 2 3(1+e u ) 3] 1 = dz = = [ 1 1 2z 2 3z 3] 1 = [ du = [ 3(1+e u ) 2 ] 6(1+e u ) 3 = [ 1+3e u 6(1+e u ) 3] (13) dz (14) 6 (15) (16) Substitute equation (16) into equation (11), ten te asymptotic variance of te logistic density estimator is: Var[f (x)] = 1 f(x) (17) 6n Combining (9) and (17), te mean squared errors for f (x) is (Simonoff 1996): MSE[f (x)] = Var[f (x)] + Bias 2 [f (x)] = 1 f(x) + π 2 6n (2 6 f (x)) 2 (18) = 1 f(x) + 4 π 4 ( f (x)) 2 (19) 6n 36 Also, te asymptotic integrated mean squared error IMSE for f (x) is: IMSE[f (x)] = π 4 ( 6n 36 f (x)) 2 dx (20) 3. Te optimal bandwidt Te optimal bandwidts wic minimize te IMSE for f (x) is obtained as follows: IMSE[f (x)] = 1 π 4 (x)) 2 6n 2 36 dx = 0 (21) 1 Ten π 4 ( f (x)) 2 dx 6n 2 9 and ence: opt = [ 2 3 nπ4 ( f (x)) 2 dx] 1 5 (22) Now let us replace te unknown term (f (x)) 2 dx in (22) by te standard logistic density as a reference distribution. Let: f(x) = e x (1+e x ) 2, x (23)

4 282 Sama M. Abo-El-Hadid ten f (x) = e x (1+e x ) 2 + 2e 2x (1+e x ) (1+e x ) 4 (24) = 2e 2x e x (1+e x ) 3 (1+e x ) 2 (25) f (x) = 4e 2x (1+e x ) 3 + 6e 3x (1+e x ) 2 [ e x (1+e x ) 2 + 2e 2x (1+e x ) ] (1+e x ) 6 (1+e x ) 4 (26) f (x) = 6e 3x (1+e x ) 4 Put terms over a common denominator and ence 6e 2x (1+e x ) 3 + e x (1+e x ) 2 (27) f (x) = 6e 3x 6e 2x (1+e x )+e x (1+e x ) 2 (1+e x ) 4 (28) f (x) = e x [(1+e x ) 2 6e x ] (1+e x ) 4 (29) (f (x)) 2 dx = e 2x [(1+e x ) 2 6e x ] 2 dx (1+e x ) 8 Again let z = 1 + e x, ten: e x = z 1, e 2x = (z 1) 2 dz dx = e x, dx = dz e x = dz (z 1) (f (x)) 2 dx = (z 1)2 [z 2 6(z 1)] 2 = (z 1)(z4 +36z z 3 +12z 2 72z) z 8 z 8 dz (z 1) = [ 21 z z z z z+216 ] 42z 7 Undo substitution z = 1 + e x = [ 21(1+e x ) (1+e x ) 4 630(1+e x ) (1+e x ) 2 756(1+e x )+216 (30) (31) dz (32) 42 (1+e x ) 7 ] (f (x)) 2 dx = 1 42 Substituting (33) into (22), te optimal smooting parameter is: opt = [ 2 3 nπ4 ( 1 42 )] 1 5 = [ nπ4 63 ] 1 5 (33) (34) 4. Simulation In tis section a simulation study designed to investigate te performance of te proposed logistic estimators. Tree estimators were compared in tis simulation study: te estimator K(u) = 1 e ( 1 2 u2) ; te 3 σ 2π u2 (1 ); and te proposed logistic estimator Random samples were generated from logistic (-1,1), logistic (0,1) and logistic (1,1). Te size of te random samples are n {10, 100, 1000}. Ten, te actual and te estimated densities are plotted; and te following errors measures are computed:

5 Logistic estimator and 283 Mean squared error (MSE) = n (f(x i ) f (x i )) 2 i=1 Mean absolute error (MAE) = i=1 f(x i ) f (x i ) n n n (35) (36) n Mean absolute percentage error (MAPE) = f(x i ) f (x i ) i=1 n f(x i ) (37) Te values of te above goodness of fit measures are given in te tables below: Table (1): Goodness of fit measure's of te difference between te actual density and te estimated densities wit logistic (-1,1) Sample Estimated density Measure Size Logistic MSE n = 10 MAE MAPE MSE n=100 MAE MAPE MSE n=1000 MAE MAPE Table (2): Goodness of fit measure's of te difference between te actual density and te estimated densities wit logistic (0,1) Sample Estimated density Measure Size Logistic MSE n = 10 MAE MAPE MSE n=100 MAE MAPE MSE n=1000 MAE MAPE Table (3): Goodness of fit measure's of te difference between te actual density and te estimated densities wit logistic (1,1) Sample Estimated density Measure Size Logistic MSE n = 10 MAE MAPE MSE n=100 MAE MAPE MSE n=1000 MAE MAPE Te above tables sows tat te estimated densities get closer to te actual density

6 284 Sama M. Abo-El-Hadid as te sample size increases, and te suggested logistic always outperforms te oters s, wile te is te worst one. Figures (1), (2) and (3), present te actual density wit te estimated densities using: te estimator; te ; and te proposed logistic estimator at te different values of parameters and different samples sizes. (a) Logistic (b) Logistic (c) Logistic 10 Fig. (1): Te actual logistic (-1,1) density wit its estimation using different s wit sample sizes (a) n=10, (b) n=100, and (c) n=1000 (a) (b) Logistic Logistic 10 Logistic 10 (c) Fig. (2): Te actual logistic (0,1) density wit its estimation different s wit sample sizes (a) n=10, (b) n=100, and (c) n=1000

7 Logistic estimator and 285 (a) Logistic (b) Logistic (c) Logistic 10 Fig. (3): Te actual logistic (1,1) density wit its estimation using different s wit sample sizes (a) n=10, (b) n=100, and (c) n=1000 Te above figures sow tat te tree functions gets closer to te original density function as te sample size increases. Also, among te tree functions, te suggested logistic always outperforms te oters for different values of n wile te is te worst one. 5. Application In tis section, te density function of te montly excange rate of EGX 30 index is estimated using te proposed logistic estimator. EGX 30 is a stock market index for securities in Egypt. It is includes te top 30 companies in terms of liquidity and activity. EGX 30 dataset is obtained from (ttps:// com/indices/egx30-istorical-data). Tis data set contains information on montly excange rate of EGX 30 index from January 2010 to August Fig. (4) sows te estimated distribution of montly excange rate of EGX 30 index using bot te parametric logistic distribution and te nonparametric logistic distribution using te optimal bandwit according to equation (34) logistic Parametric logistic Fig. (4): Te istogram; parametric and nonparametric logistic estimator of montly excange rate of EGX 30 index

8 286 Sama M. Abo-El-Hadid Te above figure indicates te flexibility of logistic in modelling te montly excange rate of EGX 30 index distributions. It sows tat te nonparametric logistic estimator agrees well wit te EGX 30 index data. 6. Conclusion In tis paper, te nonparametric estimation of te density function wit a support on (, ) using te standard logistic function is introduced. Te teoretical asymptotic properties of te proposed density estimator are derived. A simulation study and real data application were introduced. From te simulation study, te,, and logistic s get closer to te original density function as te sample size increases, but te suggested logistic always outperforms te oters wile te is te worst one. For furter studies, te beaviour of te s can be studied in te case of multivariate densities. Also, te new considered ere can also be exploited in regression curve estimation or azard rate estimation. References [1] Caterine Forbes Merran Evans Nicolas Hastings Brian Peacock, Statistical Distributions, 4t edition, Jon Wiley and Sons Inc., New York, ttps://doi.org/102/ [2] Wolfgang Härdle, Axel Werwatz, Marlene Müller, Stefan Sperlic, Nonparametric and Semiparametric Model, Springer-Verlag, New York, ttps://doi.org/107/ [3] E. Parzen, On estimation of a probability density function and mode, Te Annals of Matematical Statistics, 33 (1962), ttps://doi.org/ /aoms/ [4] M. Rosenblatt, Remarks on some nonparametric estimates of density function, Te Annals of Matematical Statistics, 27 (1956), ttps://doi.org/ /aoms/ [5] B. W. Silverman, Density Estimation for Statistics and Data Analysis, Taylor & Francis Group, New York, ttps://doi.org/ / [6] J. S. Simonoff, Smooting Metods in Statistics, Springer-Verlag, New York, ttps://doi.org/107/ [7] M. P. Wand, Kernel Smooting, Taylor & Francis Group, New York, ttps://doi.org/ /b14876 Received: December 9, 2018; Publised: December 24, 2018