Hazard Rate Function Estimation Using Erlang Kernel

Transcription

1 Pure Matematical Sciences, Vol. 3, 04, no. 4, 4-5 HIKARI Ltd, ttp://dx.doi.org/0.988/pms Hazard Rate Function Estimation Using Erlang Kernel Raid B. Sala Department of Matematics Te Islamic University of Gaza Palestine Hazem I. El Sek Amed Department of Matematics Al Quds Open University Palestine Iyad M. Aloubi University College of Science and Tecnology Palestine Copyrigt c 04 Raid B. Sala, Hazem I. El Sek Amed and Iyad M. Aloubi. Tis is an open access article 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 In tis paper, we define te Erlang kernel and use it to nonparametically estimation of te probability density function (pdf) and te azard rate function for independent and identically distributed (iid) data.te bias, variance and te optimal bandwidt of te proposed estimator are investigated. Moreover, te asymptotic normality of te proposed estimator is investigated. Te performance of te proposed estimator is tested using simulation study and real data. Keywords: Erlang kernel, azard rate function, kernel estimation, asymptotic normality

2 4 Raid B. Sala, Hazem I. El Sek Amed and Iyad M. Aloubi. Introduction Hazard rate functions can be used for several statistical analyses in medicine, engineering and economics. For instance, tey are commonly used wen presenting results in clinical trials involving survival data. Several metods for azard function estimation ave been considered in te literature. Hazard function estimation by nonparametric metods as an advantage in flexibility because no formal assumptions are made about te mecanism tat generates te sample order tan te randomness. Estimators of te azard function based on kernel smooting ave been studied extensively. For instance, see Sala 8, 7, O. Scaillet 5, Watson and Leadbetter and Rice and Rosenblatt 6. Te performance off te estimator at boundary points differs from te interior points due to so-called boundary effects tat occur in nonparametric curve estimation problems. More specifically, te bias off te estimator at boundary points. To remove tose boundary effects in kernel density estimation, a variety of metods ave been developed in te literature. Some well-known metods are summarized below: (i) Te reflection metod (Cline and Hart 3; Scuster 0; Silverman,. (ii) Te local linear metod (Ceng ; Zang and Karunamuni 3). Cen solved tis problem by replacing te symmetric kernels by asymmetric Gamma kernel wic never assigns weigt outside te support. A lot of people wo care of estimation do many specific distributions, suc as normal, log-normal, gamma and inverse-gamma distributions, and tis made us pose a question wic is: Is tere any oter distributions wic can be used as a kernel in te estimation and ten give us acceptable results? In tis paper, we propose te Erlang kernel wic also never assigns weigt outside te support. Te Erlang distribution is a continuous probability distribution wit wide applicability primarily due to its relation to te exponential and Gamma distributions. Te Erlang distribution was developed by A. K. Erlang to examine te number of telepone calls wic migt be made at te same time to te operators of te switcing stations. Tis paper is organised in six sections. In te first section, we present some information about kernel smooting, azard functions and te proposed kernel. In te second section, we introduce some definitions and relations, and state te conditions under wic te results of te paper will be proved. In te tird section, we investigate te bias, variance and optimal bandwidt of te Erlang kernel estimator. In te fourt section, we investigate te asymptotic

3 Hazard rate function estimation using Erlang kernel 43 normality of te Erlang kernel estimator of te pdf and of te azard function estimator. In te fift section, te performance of te proposed estimator will be tested via two applications, simulated and real life data. In te sixt section, we introduce comments and te conclusion.. Preliminaries In tis section, we state te conditions under wic te results of te paper will be proved. Also, we mention te definition of gamma function and some relations related to it, ten we will define Erlang kernel. Conditions () Let {x i } n i= be a random sample from a distribution of unknown probability density function f defined on 0, ) suc tat f as a continuous second derivative. () 0 < x 4 ( f) (x)dx < and f(x) dx < x (3) is smooting parameter satisfying + 0 as n. n 5 Te Gamma function is defined to be te improper integral: Γ(x) = Te Taylor expansion about zero of Γ(x) Γ(x) = x + γx + (γ π 6 )x3 + o(x 3 ) x. 0 t x exp( t)dt, x is available and given by: Also, troug tis paper we will approximate te folloing: Te beta function B(, ) = (Γ( )) Γ( ) 4 3, ( ) + 4, +. Tis approximation because we concern wit values of close to zero. In tis paper we consider te following Erlang kernel function: K E (x, )(t) = Γ( + ) x ( + + ) t t exp( x ( + )), t 0 If a random variable Y as a pdf K E (x, )(t), ten E(Y ) = x, and te variance is var(y ) = x. + We propose te following estimator of te probability density function f(.),

4 44 Raid B. Sala, Hazem I. El Sek Amed and Iyad M. Aloubi te Erlang estimator ˆf(x) = n n K E (x, )(X i) i= 3. Bias, Variance and optimal bandwidt In tis section, te bias and variance will be evaluated. Ten we will discuss te optimal bandwidt of te Erlang kernel estimator. Proposition. Te bias of te proposed estimator is given by: Proof: Bias( ˆf(x)) = x f(x) + o() () + E( ˆf(x) = 0 K E (x, )(y)f(y) dy = E(f(ξ x) were ξ x follows an Erlang distribution wit mean µ x = E(ξ x ) = x and variance, V x = V ar(ξ x ) = x. + Te Taylor expansion about µ x for f(ξ x ) is: f(ξ x ) = f(µ x ) + f(µ x )(ξ x µ x ) + f(µ x )(ξ x µ x ) + o() So, E( ˆf(ξ x )) = f(x) + f(x)var(ξ x ) + o() E( ˆf(ξ x )) = f(x) + x f(x) + o() () + Hence, Bias( ˆf(x)) = x f(x) + o() + Proposition. Te variance of te proposed estimator is given by: V ar( ˆf(x)) = ( + )( ) + f(x) + o((n ) ) (3) nx 3 Proof: V ar( ˆf(x)) = V ar K n E (x, )(t) ( = n E K E (x, )(t)) E(K E (x, )(t)) were KE (x, )(t) = ( + ) + t Γ(+ exp( t ( + )) ) x x ) + = t Γ( +). ( + ) + t + Γ( + ) exp( t ( + )) x x = R x, t. Γ(+ ) ( Γ( + ) x ( + ) + t + exp( t x ( + ))

5 Hazard rate function estimation using Erlang kernel 45 were R x, = (+)( ) +. 3 Terefore, EKE (x, )(t) = ER x,ηx f(η x ) were η x follows as Erlang distribution wit mean µ x variance V x = V ar(η x ) = x. (+) Te Taylor expansion of η f(η x ) about µ x is as follows: ηx f(η x ) = µ x f(µ x )+ (η x µ x ) + o() So, µ x E ηx f(η x ) = x f(x) + Tis implies tat x f(µ x ) + µ x `f(µx ) x 3 f(x) x `f(x) + x + (η x µ x )+ f(x) + 8(+) 3 E R x, ηx f(η x ) = (+)( ) f(x)+ (+)x ( ) 3 x o( ) Terefore, V ar( ˆf(x)) = (+)( ) = + nx 3 (+)( ) + µ 3 x = E(η x ) = x and f(µ x ) `f(µ µ x ) + f(µx x µ x ) x (+) + o() f(x) + o(n ) (f(x) + o()) n nx 3 f(x) + o((n ) ) x 3 f(x) x `f(x) + x f(x) + Optimal Bandwidt: First of all, we will define MSE and Mean Integrated Squared Error (MISE) as follows: MSE( ˆf) = (+)( ) + nx 3 f(x) + o((n ) ) + + = (+)( ) f(x) + o((n ) ) + nx (+)( ) f(x) + nx 3 4 Terefore, x4 (+) f(x) MISE( ˆf) = ( + ) ( ) + were, I = f(x) dx and I x = x 4 f(x) dx Terefore, we can approximate MISE to be: x f(x) + o() + x4 (+) f(x) + o() I n ( + ) I (4) MISE( ˆf) = I 4n + 4 I (5) We will now find te optimal bandwidt by minimizing (5) wit respect to, so we ave

6 46 Raid B. Sala, Hazem I. El Sek Amed and Iyad M. Aloubi d MISE( ˆf) = I d n + 3 I, (6) d MISE( ˆf) = 3I d n + 4 I > 0. (7) Setting (6) equal zero yields te optimal bandwidt opt for te given pdf and kernel: ( I opt = (8) ni In addition, (7) proves tat tis value minimize (5). Substituting (8) for in (5) gives te minimum MISE for te given pdf and kerne wic is given by: ( ) I I MISE opt = (9) n Note tat opt depend on te sample size n, te kernel and on te unknown pdf. ) 4 4. Asymptotic normality In tis section, we state te main two teorems talking about te asymptotic normality for te proposed estimator and an important lemma wic we will use in te second main teorem. Definition. A azard rate function is defined as te probability of an event appening in a sort time interval. More precisely, it is defined as: P (X x + x X > x) r(x) = lim, x > 0. x 0 x Te azard rate function can be written as te ratio between te pdf f(.) and te survivor function S(.) = F (.) as follows: r(x) = f(x) S(x) Te kernel estimator of S(.) is Ŝ(x) = ˆF (x) were, x ˆF (x) = ˆf(u)du = n x K E (u, n )(X i)du 0 Definition. we defined te proposed estimator for te azard rate function to be: i= 0 ˆr(x) = ˆf(x) Ŝ(x)

7 Hazard rate function estimation using Erlang kernel 47 Teorem. Under conditions,, and 3, te following olds ( n 5 + ( ˆf(x) d f(x)) N 0, f(x) ) 8x (0) Proof: Let V ni = K E (x, )(X i), i =,,..., n. Ten ˆf(x) = n n i= V ni, were V ni, i =,,..., n are independent and identically distributions (iid) and σ ni <. We sow now tat te Liapounov condition is satisfied, tat is for some δ > 0, E V n E(V n ) +δ lim n n δ σ +δ (V n ) First of all, we ave: V n +δ = ( + ) + (+δ) t +δ Γ(+ exp( ( + δ) t ( + )), t 0 )+δ x x = x (+ )δ = x (+ )δ = αt δ. x (+ ) + t δ exp( δ t x (+ + ))Γ( ) = 0 x (+ + ) t exp( t x (+ )) Γ( + ). Γ(+ )+δ + t δ exp( δ t x (+ ))t ( ) + Γ( + ). x (+ + ) Γ(+ )+δ + t + exp( t x (+ )) Γ( + ) were, α = x (+ + )δ ( ) + Γ( + E( V n +δ ) = 0 αy δ Γ(+ )+δ + x (+ ) )exp( δ t x (+ )) t + exp( t x (+ )) Γ( + ). Ten we ave: t + exp( t x (+ )) Γ( + ) f(y)dy = E(αξ δ x f(ξ x )), were, ξ x follows an Erlang distribution wit mean µ x = E(ξ x ) = x and variance var(ξ x ) = x (+) Te Taylor expansion of ξ δ x ξ δ f(ξ x ) about te mean µ x = E(ξ x ) = x as follows: ( δ )x δ f(x) + x δ f(x) (ξ x x) + x f(ξ x ) = x δ f(x) + ( δ )( δ )x δ 3 f(x) + ( δ )x δ δ f(x) + x f(x) (ξ x x) + o( ) E(ξ δ x f(ξ x )) = x δ f(x) + (δ )(δ )x δ 3 f(x) + (δ )x δ f(x) + x δ f(x) = x δ f(x)+ x δ (+) o( ) x (+) +o( ) (δ )(δ )f(x) + (δ )x f(x) + x f(x) +

8 48 Raid B. Sala, Hazem I. El Sek Amed and Iyad M. Aloubi Terefore, E( V n +δ ) = x ( + ) δ + ( ) + ( + )exp( δ t x ( + )) 4 5 Γ( + ) δ Now substituting δ = 0, te following is old: x δ f(x)+o( 7 ) () V ar(v n ) = ( ) + ( + ) 4 5 x f(x) + o( 7 ). Hence, E V n E(V n) +δ n δ σ +δ (V n) E Vn +δ n δ σ +δ (V n) + x (+ )δ n δ Γ(+ )δ 4 5 ( ) + (+)exp( δ t x (+ ))x δ f(x) ( ) + (+) 4 5 +δ x f(x) = x δ + δ + ( ) δ exp( tδ x (+ ))(+)f(x) n δ Γ(+ )δ 4 5 δ (+)x f(x) δ+ = x δ + δ + 45 δ exp( tδ x (+ )) (+)f(x) n δ ( δ + Γ(+ )δ (+)x f(x) δ+ 4) δ 0 Te last term by condition 3 vanises as n 0. Also, te remaining component of te last term are bounded. Lemma. Under conditions, and 3 te following olds n ˆF (x) F (x) p 0 () Proof: First of all, we ave from te definition of ˆF (x) te following relations: E ˆF (x) = x k 0 0 E(u, )(y)duf(y)dy = x k 0 0 E (u, )(y)f(y)dydu = x E(f(ξ 0 x)))du = x (f(u) + 0 u f(u))du + o() + = F (x) + o() were ξ x is defined as in Proposition. Tis implies tat, E ˆF (x) F (x) = o() Terefore, (n) E ˆF (x) F (x) = o((n) ) p 0 (3) On te oter and, ˆF (x) can be written as follows: ˆF (x) = n x n i= k 0 E(u, )(y)du = n n i= W i(x) were W i (x) = x k 0 E(u, )(y)du

9 Hazard rate function estimation using Erlang kernel 49 Now, given ε > 0, δ > 0, ten we ave: P (n) ˆF (x) E ˆF (x) > ε E ˆF (x) E ˆF (x) +δ ε +δ (n) (+δ) = ε δ (n) +δ E ˆF (x) E ˆF (x) +δ = ε δ (n) +δ n δ E n i= W i(x) EW i (x) +δ ε δ (n ) +δ n i= E W i(x) +δ + ε δ (n ) +δ n i= EW i(x) +δ Te second term vanises as n and 0, since from we ave n (n ) +δ EW i (x) +δ = o((n ) +δ ), i= Furter, by replacing δ wit δ in () and assuming tat τ,δ = + ( ( + )δ + ) ( + ) Γ( + ) δ we ave: P (n) ˆF (x) E ˆF (x) > ε ε +δ (n ) +δ n i= E W i(x) +δ = ε δ n δ +δ n x ke (u, )(y) +δ 0 0 duf(y)dy = ε δ n δ +δ x ke (u, )(y) +δ 0 0 f(y)dydu = ε δ x τ,δ exp( δ t u (+ )) 0 4n δ δ+ δ u δ f(u) du +4 +o((n ) δ 6 ) 0 (n) ˆF (x) E ˆF (x) p 0. (4) Since ˆF (x) F (x) ˆF (x) E ˆF (x) + E ˆF (x) F (x), ten by (3) and (4), we ave: (n) ˆF (x) F (x) (n) ˆF (x) E ˆF (x) + (n) E ˆF p (x) F (x) 0 Tis complete te proof of te lemma.. Teorem. Under conditions,, and 3, te following olds n 5 d (ˆr(x) r(x)) + N(0, r(x) 8xS(x) ) (5)

10 50 Raid B. Sala, Hazem I. El Sek Amed and Iyad M. Aloubi Proof: n 5 + (ˆr(x) r(x)) = = = Ŝ(x) = Ŝ(x) n 5 n 5 ( ˆf(x) f(x) ) + Ŝ(x) S(x) ( ˆf(x) f(x) f(x) + f(x) + Ŝ(x) Ŝ(x) S(x) ) Ŝ(x) n 5 ( ˆf(x) f(x) f(x)ŝ(x) + f(x)) + S(x) n 5 ( ˆf(x) f(x)) + f(x) n 5 S(x)Ŝ(x) + d N(0, r(x) 8xS(x) ) (Ŝ(x) S(x)) Note tat te second term vanises by (), and te first term is asymptotically normally distributed by (0). Moreover, from (0) and (5) too we ave: E(ˆr(x)) Terefore, = E(ˆ(f)(x)) E(ˆ(S)(x)) = f(x)+ x f(x) + S(x) f(x) = r(x) + x + S(x) + o() + o() and, Bias(ˆr(x)) = x f(x) + + o() S(x) V ar(ˆr(x)) = ( + )( ) + nx 3 5. Applications r(x) S(x) + o((n ) ) In tis section, te performance of te proposed estimator in estimating te pdf and azard rate function is tested upon two applications using a simulated and real life data. 5.. A Simulation Study. A sample of size 00 from te exponential distribution wit pdf f(x) = exp( x) is simulated. We computed te bandwidt using te relation opt = 0.79Rn 5, (6) see page 47 and it equals ( ). Te density and te azard functions were estimated using te Erlang estimator. Te estimated values and te true exponential pdf are plotted in Figures (a), tis figure sows tat te performance of te Erlang estimator is acceptable at te boundary near te zero. In te interior te beavior of te pdf estimator becomes more similar for large values. Also figure (b)

11 Hazard rate function estimation using Erlang kernel 5 sows tat te performance of te Erlang estimator of te azard function is acceptable at te boundary near te zero wic we concern on. Te mean squared error (MSE) of proposed estimator of te density function is equal to and for te azard function is evaluated for te interval 0,0.5 -because we concern about closest values to zero- and is equal to Real Data. In tis subsection, we used te suicide data given in Silverman, to exibit te practical performance of te Erlang estimator. Te data gives te lengts of te treatment spells (in days) of control patients in suicide study. We used te logaritm of te data to draw figures (a) and (b)using bandwidt equals wic computed by (6), tese figures exibit te two estimated functions of te probability density and azard rate functions, respectively. density function Erlang kernel True density function azard rate function Erlang kernel True azard function (a) (b) Figure. (a) Te Erlang kernel estimator of te density function, (b) te azard rate function for te simulated data of te exponential distribution density function azard rate function Time (a) Time (b) Figure. (a) Te Erlang kernel estimator of te density function, (b) te azard rate function for te suicide data.

12 5 Raid B. Sala, Hazem I. El Sek Amed and Iyad M. Aloubi 6. Comments and Conclusion In tis paper, we ave proposed a new kernel estimator of te azard rate function for (iid) data based on te Erlang kernel wit nonnegative support, we sow tat te bias is depends on te smooting parameter and te estimated point x, and it goes to zero as 0, also it gets small for te values of x closed to zero. Te variance is investigated and we noticed tat it depends also on and x. On te oter and, it goes to zero as 0, and gets large at te values of x close to zero. Moreover, te optimal bandwidt and te asymptotic normality were investigated. In addition, te performance of te proposed estimator is tested in two applications. In a simulation study using exponential sample we noticed tat te performance of te proposed estimator is acceptable, and gives a small MSE. Using real data, we exibited te practical performance of te Erlang estimator. References Cen, S. X. (000). Probability density function estimation using gamma kernels. Ann. Inst. Stat. Mat., 5, Ceng, M.Y., Fan, J. and Marron, J.S. (997). On Automatic Boundary Corrections. Te Annals of Statistics, 5, Cline, D.B.H and Hart, J.D. (99). Kernel Estimation of Densities of Discontinuous Deriva-tives. Statistics,, N. D. Singpurwalla and M. Y. Wong (983). Kernel estimation of failure rate function and density estimation: an analogy. J. Amer. Statist. Assoc. 78(3), O. Scaillet (004). Density estimation using inverse and reciprocal inverse Gaussian kernels. Nonparametric Statistics, 6, Rice, J. and Rosenblatt, M. (976). Estimation of te log survivor function and azard function. Sankya Series A 38, Sala, R. (03). Estimating te density and azard rate functions using te reciprocal inverse Gaussian kernel. 5t Applied Stocastic Models and Data Analysis International Conference, Mataro (Barcelona) Spain, June 5-8, Sala, R. (0). Hazard rate function estimation using inverse Gaussian kernel. Te Islamic university of Gaza journal of Natural and Engineering studies, 0(), Sala, R. (009). Adaptive kernel estimation of te azard rate function. Te Islamic univercity - Gaza Journal, 7(), Scuster, E.F. (985). Incorporating Support Constraints Into Nonparametric Estimators of Densities.Communications in Statistics, Part A - Teory and Metods, 4, Silverman, B. W. (986). Density estimation for statisticas and data analysis. Capman and Hall, London. Watson, G. and Leadbetter, M. (964). Hazard analysis I, Biometrika, 5, Zang, S. and Karunamuni, R.J. (998). On Kernel Density Estimation Near Endpoints. Journal of Statistical Planning and Inference, 70, Received: June, 04