EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 4, 2019, 1612-1642 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Density and Risk Function of Circular Kernel Study Didier Alain Njamen Njomen1,∗, Hubert Clovis Yayebga1 1 Department of Mathematics and Computer’s Science, Faculty of Science, University of Maroua, Maroua, Cameroon Abstract. In this article, based on the works of [18], [22] and [26] on the estimation of the survival function and the function of risk in independent cases and identically distributed with and without censorship, We manage to established the bias and variance of the density of the circular kernel. In addition, we determined the optimal window b∗n of this estimator after having first established the mean square error (MSE) and mean integrated squared error (MISE) which are necessary conditions for obtaining the optimal window. Finally, we have established the asymptotic expression of the bias of the risk function of the circular kernel estimator. 2010 Mathematics Subject Classifications: 97K80, 62G07, 11R45 Key Words and Phrases: Nonparametric estimator, Density function, Risk function, Circular kernel, Bias, Mean squared error, Mean integrated squared error, Optimal window 1. Introduction Survival analysis refers to a set of statistical techniques for processing censored data which is unknown for some of them, that a lower or higher bound and not a precise value. It represents the study of the delay of the occurrence of an event. It is called survival anal- ysis or lifetime data analysis which is present in several areas of life among which we can mention the medicine to evaluate the effectiveness of a treatment. In demography to build life tables where these are used by actuaries to determine the amount of life insurance and life annuities, among others; actuarial tables are used when data is grouped into intervals. It is also in engineering to estimate the reliability of machines and electronic components. Survival analysis is also useful in astrophysics. Imagine that sources have been detected at a wavelength λ and that we observe at the same positions another wavelength λ′. Some sources are not detected at λ′ because the ratio signal/noise (l′/b) is too low. However, how to calculate the color distribution m − m′ = 2, 5 log10(l′/l) + constant? hence the interest in physics survival analysis. This analysis is linked to an event that is the subject of a study and that is how we will be able to study its lifespan or survival. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3519 Email addresses: didiernjamen1@gmail.com (D. A. N. Njamen), hyayebga@yahoo.com (H. C. Yayabga) http://www.ejpam.com 1612 c© 2019 EJPAM All rights reserved. D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1613 The term survival time refers to the time elapsed until the occurrence of a specific event. The event studied (commonly called death) is the irreversible transition between two states (commonly called living and death). The terminal event is not necessarily death: it can be the appearance of a disease (for example, the time before a relapse or rejection of a transplant), a cure (time between the diagnosis and the cure), the breakdown of a machine (duration of operation of a machine, in reliability) or the occurrence of a disaster (time between two disasters, in actuarial). In survival analysis, the estimation of the hazard function is an interesting problem that appears in the statistical analysis of useful life to study several types of events. Many methods have been proposed to study the estimation survival functions and risk functions in independent and identically distributed (i.i.d) cases with and without censor- ship. We can mention among others: [19], [17], [9], [18], [22], [20], [3], [21], [26], [8], [23], and [16]. However, the study of the determination of the mean squared error (MSE) and the mean integrated squared error (MISE) to obtain the optimum window of the circular kernel remains unclear. This article brings a relief to this problematic. This article is subdivided into six parts: The first part is devoted to the development of the different notions studied in the density and risk functions. The second part deals with the estimation of kernel density function. The third part deals with the risk function estimation. The fourth part reminds us of the notion of mean squared error (MSE). In the fifth part, it’s about the probability and risk function of the circular kernel. Finally in the last part, through simulated and real data, we show that our estimator is efficient. 2. General results on density and risk functions In this section, we acquire various basic concepts that allow a good understanding of our work. For this purpose, we have been inspired by the founding works, articles and even courses read online. This among others include the works by: [1], [22], [3], [5] and [23]. The survival time T refers to the time elapsed since an initial moment (beginning of treatment, diagnosis, unemployment, ...) until the occurrence of an event of final interest (death of the patient, relapse, remission, healing, work, ...). It is said that the individual survives the time t if at this moment the event of final interest has not yet taken place. The variable studied is called survival time and will be marked T . 2.1. Expression of the density and risk functions in survival analysis In what follows, it is assumed that an individual suffers once and only once a certain type of events. Observations of the occurrence (censored or uncensored) of this event in individuals constitute the samples to identify survival patterns. D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1614 Let T be the survival time that is assumed to be a positive variable. Its probability law can be defined by one of the following functions: 2.1.1. Probability density function Definition 1. The probability density function of T, denoted f(t) is defined by: f(t) = lim ∆t→0+ P ( t ≤ T ≤ t+ ∆t ∆t ) . (1) For fixed t, the probability density is interpreted as the probability that the event occurs in the short time interval ]t, t+ ∆[. 2.1.2. Risk function Definition 2. The risk function of T denoted h(t) is defined by: h(t) = { lim∆t→0 P(tt) ∆t , if t > 0 and such that P(T > t) > 0; +∞ if not. (2) For fixed t, the risk function h is the instantaneous risk that the event occurs in the interval ]t, t + ∆[ knowing that it did not take place before time t (being in ”operation” or be ”alive” before time t). Remark 1. The risk function is also called hazard function, chance rate, failure rate or survival rate. The risk function may have different forms but is necessarily positive on R. Suppose now that T is a continuous variable, we observe that the risk function is defined by: h(t) = lim ∆t→0 P(t < T ≤ t+ ∆t|T > t) ∆t = lim ∆t→0 1 ∆t P(t < T ≤ t+ ∆t) P(T > t) = f(t) S(t) , (3) where S = P(T > t) is the survival function. The Hazard function characterizes the law of T because of the relation: S(t) = exp(− ∫ t 0 h(u)du). 2.1.3. Cumulative risk function Definition 3. The cumulative risk function Λ(t) is defined by: Λ(t) = ∫ t 0 h(u)du. (4) D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1615 2.2. Different forms of the risk function In this subsection we give the main forms of law used in survival data. The usual forms of the function h are constant, monotonous (increasing or decreasing) and in the form of ∩. 2.2.1. Constant risk The only distribution that has a constant h risk function is the exponential law. The probability density and the risk function h are respectively defined by the exponential law denoted ε(λ): f(t, λ) = λe−(λt), t ≥ 0; h(t) = λ. 2.2.2. Monotonous risk There are several life-time distributions with a monotonic risk function. • Weibull Law W (λ, α) There are several life-time distributions with a monotonic risk function. f(t, α, λ) = αλαtα−1e−(λt)α ; h(t) = αλαtα−1. The parameter α is a form parameter of h and λ is a scale parameter. Remark 2. (i) If α = 1, we obtain the exponential law W (λ, 1) = ε(λ) (ii) If 0 < α < 1, the risk function h is decreasing from ∞ to 0 (iii) If α > 1, the risk function h is increasing from (0 to ∞). • Gamma Law Γ(λ, β) The probability density and the risk function are respectively defined by the Gamma law: f(t, λ, β) = λβ 1 Γ(β) tβ−1e−λt, λ, β > 0, t ≥ 0; h(t, λ, β) = f(t, λ, β) 1− F (t, λ, β) . The parameter β is the form parameter of h and λ is scale parameter. Remark 3. (i) If β = 1, we obtain the exponential law with parameter λ: Γ(λ, 1) = ε(λ) (ii) If β > 1, the risk function h is increasing from 0 to λ. (iii) If 0 < β < 1, the risk function h is decreasing from ∞ to 1 λ . D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1616 2.2.3. Risk in ∩ • Log-normal law LN(λ, v) The probability density and the risk function are respectively defined by the Log- Normal law LN(µ, σ): f(t, µ, σ) = 1 σt ϕ ( lnt− µ σ ) ; h(t, µ, σ) = f(t, µ, σ) S(t, µ, σ) . Remark 4. The risk function h is in the form ∩: it increases from 0 to its maximum value and then decreases to 0. • Log-logistic Law LL(λ, v) The probability density and the risk function are defined respectively by the log- logistic law: f(t, λ, v) = (λv)λtv−1 (1 + (λt)v)−2 ; h(t, λ, v) = (λv)λtv−1 (1 + (λt)v)−1 . Remark 5. The v parameter is a form parameter of h. For v > 1, the risk function h increases from 0 to its maximum value and then decreases to 0. 3. Density function estimation 3.1. Kernel estimator: case of uncensored data The concepts used in this section come from the following documents: [19], [17], [11], [24], [9], [20], [3], [20] and [23]. Let T1, T2, ..., Tn distribution function survival durations F and let Vx0 =]t0− bn 2 , t0+ bn 2 [ a neighborhood of t0, where (bn)n≥0 is a sequence of positive parameters called window. An estimator of the density f at the point t0 is given by: fn(t0) = Fn(t0 + bn 2 )− Fn(t0 − bn 2 ) bn = number of events on ]t0 − bn 2 , t0 + bn 2 [ bn , where Fn is the empirical distribution function. We can also write it in the following form: fn(t0) = 1 nbn n∑ i=1 11]t0− bn2 ,t0+ bn 2 [(Ti) D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1617 = 1 nbn n∑ i=1 11]− 1 2 , 1 2 ] ( t0 − Ti bn ) . This writing shows a function K called kernel, which is in the form: K(.) = 11]− 1 2 , 1 2 ](.). Hence the definition of a kernel estimator of f , for t ∈ ] −1 2 , 1 2 ] : f̂n(t) = 1 nbn n∑ i=1 K ( t− Ti bn ) , (5) where the kernel K is an application of R on R+, bounded, of integral equal to 1, sym- metrical and of more bn → 0 when n→ +∞. 3.1.1. Examples of kernels The various kernels most used in estimating the density are: • Gaussian kernel ∀u ∈ R,K(u) = 1√ 2π e− u2 2 (6) • Uniform kernel ∀u ∈ R,K(u) = 1 2 11|u|≤1 (7) • Epanechnikov kernel ∀u ∈ R,K(u) = 3 4 (1− u2)11|u|≤1 (8) • Triangular kernel ∀u ∈ R,K(u) = (1− |u|)11|u|≤1 (9) • Quadratic kernel ∀u ∈ R,K(u) = 15 16 (1− u2)211|u|≤1 (10) Definition 4. A kernel K is called Parzen-Rosemblatt if K is symmetric and if lim|u|→∞ |u|K(u) = 0. If we consider Kbn(y) = 1 bn K( ybn ), kernel estimator fn is written: fn(t) = ∫ +∞ −∞ Kbn(t− s)dµn(s) = (Kbn ∗ µn)(t), where µn = 1 n ∑n i=1 δTi . D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1618 3.1.2. Optimal choices of kernel K and window bn The convergence results of the kernel estimator fn require conditions on K and bn, from which the problem of the optimal choice of (K, bn). We pose: ∆n(t) = E [fn(t)− f(t)]2 = E [fn(t)− Efn(t) + Efn(t)− f(t)]2 = [Efn(t)− f(t)]2 + E [fn(t)− Efn(t)]2 . It is assumed that the kernel support K is [−1, 1]. The following theorem gives the asymptotic behavior of the bias and the variance of the estimator fn. Theorem 1. ([3]) If f is of class C2 and K is a kernel of Parzen-Rosemblatt. So if bn → 0 and nbn → ∞, we have: • Bias(fn(x)) = b2n 2 f ′′ (x) ∫ R t2K(t)dt+O(b2n). (11) • Var(fn(x)) = 1 nbn f(x) ∫ R K2(t)dt+O ( 1 nbn ) . (12) The choice of the optimal window b∗n is done if the kernel K is given. The smoothing parameter bn is a positive real whose choice is preponderant over that of the symmetrical continuous kernel K. Choosing a value of bn too big leads to a curve that is too smooth. On the other hand, by choosing a very small smoothing parameter than the one previously adopted, the shape of the distribution changes. Theorem 2. ([3]) Under the conditions of the theorem 1, the optimal window is: b∗n = ( f(t) f ′′2(t) [∫ u2K(u)du ]2 ∫ 1 −1 K2(u)du ) 1 5 n− 1 5 . (13) 3.2. Kernel estimator: case of censored data The purpose of this subsection is to estimate the density of survival times T in the context of censored data. Let T1, T2, ..., Tn be a sequence of positive random variables i.i.d of distribution function F representing survival times and C1, C2, ..., Cn is the sequence of random variables i.i.d representing censoring, of distribution function G. We assume that the sequence (Ti) is independent of the sequence (Ci) and we observe: Xi = Ti∧Ci, Di = 11{Ti≤Ci}, i = 1, ..., n. D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1619 Kaplan-Meier estimator [6] of the survival function S is given by: ŜKM (t) = ∏ t(i)≤t ( 1− 1 n− i+ 1 )D(i) . (14) It can also be written in the form: ŜKM (t) = { ∏ X(i)≤t ( 1− D(i) n−i+1 ) si t ≤ X(n) 1− F̂n(X(n)) t > X(n), where X(1) X(n) we considered 1−F̂n(t) = 1−F̂n(X(n)) since there is no more observations after t. This estimator (ŜKM (t)) of survival function was studied by [13], [14], [15] in the context of competing risks. We assume that F admits a density f in relation to Lebesgue’s measure proposes to estimate using the observations (Xi, Di), I = 1, ..., n. Based on the Kaplan-Meier estimator, [2] proposed an estimator of the density f by the kernel method given by: fn(t) = 1 bn ∫ +∞ 0 K ( t− s bn ) dF̂n(s), (15) where F̂n is an empirical distribution function, (bn)n≥1 is the window with bn → 0 when n→∞ and K a kernel of support [−1, 1]. This estimator fn has been studied in particular by [2], [4] and [10]. 4. Risk function estimation 4.1. Kernel estimator: case of uncensored data In this part, we will study an estimator of the risk function in the case of uncensored data, by the method of kernel by developing the article written by [18]. In 1983, Henrik Ramlau-Hansen had proposed a methodology, cited by [7], to con- struct an estimator ĥn of the risk function by smoothing. This smoothing occurs through convolution by a kernel and by use of a window bn. Let T1, T2, ..., Tn the random variables i.i.d of density f and F distribution function. The kernel estimator of the density proposed by [19] is defined by: f̂n(t) = 1 bn ∫ +∞ −∞ K ( t− s bn ) dFn(s), (16) where Fn is the empirical distribution function of Ti and K a Parzen-Rosenblatt kernel of support [−1, 1] (K is bounded, integrable, symmetric, ∫ 1 −1Kdµ = 1 where µ is a measure D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1620 on R and lim|x|→+∞ |x||K(x)| = 0). Recall that the risk function h(t) = f(t) 1− F (t) , (17) with t ≥ 0 and F (t) < 1. [25] studied a kernel estimator of risk function h given by: ĥn(t) = 1 n− i+ 1 n∑ i=1 1 bn K ( t− Ti bn ) , (18) where T(1), T(2), ..., T(n) is the order statistic of Ti. If we pose Nn(t) = nFn(t) = n∑ i=1 11{Ti≤t} and Y n(t) = n−Nn(t)(t−) = n∑ i=1 11{Ti≥t}, then, the expression (18) can be written in the form: ĥn(t) = 1 bn ∫ +∞ 0 K ( t− s bn ) dĤNA(s), (19) where ĤNA(t) = ∫ t 0 dNn(s) Y n(s) (20) is the Nelson-Aalen estimator of the cumulative hazard rate (see [12], [1] and [13]). Definition 5. Let K be a bounded function of integral 1 and bn a positive window. The kernel estimator corresponding to the h hazard rate is given by: ĥn(t) = 1 bn ∫ 1 0 K ( t− s bn ) dĤNA(s). (21) The instants of process N jumps are T(1), T(2), ..., T(n). So, ĥn(t) = 1 bn ∑ iK( t−T(i) bn ) Y n(T(i)) . (22) D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1621 4.2. Mean and variance of the kernel estimator ĥn We assumed that the support of the kernel K is [−1, 1] and 0 < bn < 1 2 . Let consider t ∈ [bn, 1− bn]. Then we have: h̃n(t) = 1 bn ∫ 1 0 K ( t− s bn ) dH̃n(s) = ∫ 1 −1 K(u)h(t− bnu)J(t− bnu)du, with J(t) = 11{Y (t)>0}. Proposition 1. ([8]) By taking, Kbn(s) = 1 bn K ( s bn ) and jn(s) = E(Jn(s)) then for all t ≥ 0, we have: E(ĥn(t)) = E(h̃n(t)) = (Kbn ∗ hjn)(t), (23) and, σ2 n(t) = E ( ĥn(t)− h̃n(t) )2 = 1 bn ∫ 1 −1 K2(u)h(t− bnu)E ( Jn(t− bnu) Y n(t− bnu) ) du, (24) with jn(t)) = 11{Y n(t)>0}. Proposition 2. ( [8]) For all t ∈ [bn, 1− bn], an unbiased estimator of σ2 is σ̂2(t) = 1 b2n ∫ 1 −1 K2( t− s bn ) ( Jn(s) Y 2 n(s) ) dNn(s), (25) with jn(t)) = 11{Y n(t)>0}. 4.3. Kernel estimation of the risk function: case of censored data In this subsection, we drew on the work of [22]. Let the survival durations T1, T2, ..., Tn of distribution function F and density f and C1, C2, ..., Cn the censorship random variables and FC their distribution function. It is assumed that Ti are independent of Ci for all i = 1, ..., n and we observe Xi = Ti ∧Ci and Di = 11{Ti≤Ci}. We denote fX the density of Xi and FX their distribution function. Recall that the kernel estimator of the risk function of Tn is given by: ĥn(t) = n∑ j=1 D(j) n− j + 1 Kbn(t−X(j)) = n∑ i=1 D(i) n−Ri + 1 Kbn(t−Xi), (26) D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1622 where X(1), X(2), ..., X(n) is the statistic of order and Ri the rank of Xi. This estimator appeared for the first time in the work of [25] and developed [18]. For the rest, we assume that: (i) The risk function h is continuous and 0 < F (y) < 1. (ii) The kernel K is symmetric, positive and K(t) = O(t−1) when t → +∞ and∫ K(t)dt = 1. 4.4. Mean and variance of the estimator ĥn Let m(y) = f(y) [1− FC(y)] fX(y) with fX(y) > 0. To determine the mean and the variance of ĥn, we need the following lemma: Lemma 1. ([22]) For all j, we have: E(D(j)/X(j) = y) = m(y), (27) and for all r < s, y < z, we have: E(D(r)D(s)/X(r) = y,X(s) = z) = m(y)m(z). (28) The following theorem gives the mean and the variance of ĥn. Theorem 3. ([22]) We have: E ( ĥn(t) ) = ∫ (1− FnX(y))h(y)Kbn(t− y)dy, (29) and, V ( ĥn(t) ) = ∫ In (FX(y))h(y)K2 bn(t− y)dy + 2 ∫ ∫ y≤z {FnX(z)− FnX(y)FnX(z) (30) − 1− FX(y) FX(z)− FX(y) [FnX(z)− FnX(y)]}h(y)h(z)Kbn(t− y)Kbn(t− z)dydz,(31) where In(FX) = n∑ k=0 (n− k)−1CknF k X(1− F )n−k. D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1623 5. Mean Squared Error In statistics, the mean squared error (MSE) of an estimator θ̂ with a parameter θ of dimension 1 is a measure characterizing the precision of this estimator. It is more often called quadratic risk. Definition 6. The mean squared error is defined by: MSE(θ̂) = E [ (θ̂ − θ)2 ] . (32) Mean squared error can be expressed as a function of bias and the variance of the estimator hence the following theorem: Theorem 4. ([5]) MSE(θ̂) = Biais(θ̂)2 + V ar(θ̂). (33) Remark 6. As the name suggests, the mean integrated squared error (MISE) is in fact the integration made on the mean squared error. 6. Probability Density and Risk Function of the Circular Kernel In this section, we consider a sequence (xi) n i=1 of random variables according to an un- known probability density f and such that f has a continuous second derivative. Moreover, we work within the strict case of non censored data. 6.1. Circular Kernel The circular kernel still called the cosine kernel is defined by: K(t) = π 4 cos (π 2 t ) .11{|u|≤1}. (34) The expectation and variance of the circular kernel, were defined by [3]. Proposition 3. Let X be a random variable following a circular law then its expectation and its variance are given by: • E(X) = 0 • V(X) = 1− 8 π2 . 6.2. Circular kernel estimator The density estimator probability f associated to a circular kernel is given by: f̂(x) = 1 nbn n∑ i=1 K ( xi − x bn ) = 1 nbn n∑ i=1 π 4 cos [ π 2 ( x− xi bn ) ] . (35) D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1624 6.3. Risk function estimation of the circular kernel estimator In this section, we are talking about estimating the risk function of the circular kernel. Definition 7. The kernel estimator of the hazard function is defined by: ĥ(x) = f̂(x) 1− F̂ (x) = f̂(x) Ŝ(x) , where F (.) and S(.) respectively designate the distribution and survival functions; F̂ (.) and Ŝ(.) their respective estimators. 7. Main results 7.1. Case of the circular kernel estimator 7.1.1. Bias and Variance of the circular kernel estimator The bias and variance of the circular kernel estimator is given by the following proposition and is the first result of this article. Proposition 4. The bias and variance of the circular kernel estimator f̂ of the relation (35) is given respectively by: E ( f̂(x)− f(x) ) = b2n 2 f ′′(x) [ 1− 8 π2 ] +O(b2n). (36) (37) and V(f̂(x)) = 1 nbn f(t) π2 16 +O ( 1 nbn ) . (38) Proof. Let X1, X2, ..., Xn be random variables that are i.i.d. We then have: • E(f̂(x)) = E { 1 n n∑ i=1 1 bn K ( x−Xi bn )} = 1 n n∑ i=1 E { 1 bn K ( x−Xi bn )} = E { 1 bn K ( x−X1 bn )} = ∫ R 1 bn K ( x− x1 bn ) f(x1)dx1 We perform the following variable change: −t = x−x1 bn . D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1625 Hence, x1 = bnt+ x. From the expression of x1 above, we obtain: E(f̂(x)) = ∫ R K(−t)f(x+ bnt)dt. Thus, Bias(f̂(x)) = E(f̂(x)− f(x)) = ∫ R K(−t)f(x+ bnt)dt− f(x) = ∫ R K(t)f(x+ bnt)dt− f(x) because K is symmetrical. In an aim to obtain a simpler form, which depends only on the parameter bn, we approximate the bias formula using the Taylor-Lagrange formula: f(x+ bnt) = f(x) + bntf ′(x) + b2nt 2 2 f ′′(x) +O(b2nt 2). Thus, replacing the expression of f(x+ bnt) in the expression of Bias(f̂(x)) above, we get: Bias(f̂(x)) = ∫ R K(t) ( f(x) + bntf ′(x) + b2nt 2 2 f ′′(x) +O(b2nt 2) ) dt− f(x) = f(x) ∫ R K(t)dt+ bnf ′(x) ∫ R tK(t)dt + 1 2 b2nf ′′(x) ∫ R t2K(t)dt− f(x) +O(b2n). And according to our assumptions on the K kernel, we finally get: Bias(f̂(x)) = b2n 2 f ′′(x) ∫ R t2K(t)dt+O(b2n), so, Ef̂(x)− f(x) = b2n 2 f ′′(x) ∫ R t2K(t)dt+O(b2n). Yet, ∫ R t2K(t)dt = ∫ 1 −1 t2K(t)dt = 1− 8 π2 , so, by replacing it in our expression above, we obtain: E ( f̂(x)− f(x) ) = b2n 2 f ′′(x) [ 1− 8 π2 ] +O(b2n). D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1626 Hence, Bias(f̂(x)) = E ( f̂(x)− f(x) ) = b2n 2 f ′′(x) [ 1− 8 π2 ] +O(b2n). Hence the bias of f̂ . • Starting from the assumption of independence between the Xi, we have: Var(f̂(x)) = Var { 1 n n∑ i=1 1 bn K ( x−Xi bn )} = 1 n V ar { 1 bn K ( x−X1 bn )} = 1 n E [{ 1 bn K ( x−X1 bn )}2 ] − 1 n [ E { 1 bn K ( x−X1 bn )}]2 = 1 n ∫ R 1 h2 K2 ( x− x1 bn ) f(x1)dx1 − 1 n {∫ R 1 bn K ( x− x1 bn ) f(x1) }2 . By performing the −t = x−x1 bn variable change, we get: V(f̂(x)) = 1 nb2n ∫ R K(−t)2f(x+ bnt)bndt− 1 n {∫ R K(−t)f(x+ bnt)bndt }2 = 1 nbn ∫ R K(t)2f(bnt+ x)dt− 1 n [ Biais(f̂(x)) + f(x) ]2 = 1 nbn ∫ R K(t)2f(bnt+ x)dt− 1 n { O(b2n) + f(x) }2 = 1 nbn ∫ R K2(t) ( f(x) + bntf ′(x) + bn 2 t2f ′′(x) +O(b2nt 2) ) − 1 n { O(b2n) + f(x) }2 = 1 nbn f(x) ∫ R K2(t)dt+ 1 n f ′(x) ∫ R tK2(t)dt+ 1 n bnf ′′(x) ∫ R t2K2(t)dt − 1 n { O(b2n) + f(x) }2 . Yet, ∫ R tK2(t)dt = ∫ 1 −1 t (π 4 cos( π 2 t) )2 (t)dt = 0, We have: V(f̂(x)) = 1 nbn f(x) ∫ R K2(t)dt+ 1 n bnf ′′(x) ∫ R t2K2(t)dt− 1 n { O(b2n) + f(x) }2 . D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1627 Finally, under the condition of having ∫ K(t)2dt < +∞ and for n large enough, we have: V ( f̂(x) ) = 1 nbn f(x) ∫ R K(t)2dt. By elsewhere, ∫ R K2(t)dt = ∫ 1 −1 K2(t)dt = ∫ 1 −1 π2 16 cos2( π 2 t)dt = π2 16 , Hence the variance of f̂ is: V(f̂(x)) = 1 nbn f(x) π2 16 +O ( 1 nbn ) . To determine the optimal window, we first need to find the MSE and the MISE of the estimator f̂ . In statistics, the mean squared error is a way of estimating the difference between an estimator and the actual value of the quantity to be calculated. It provides a way to choose the best estimator. The proposition gives the mean squared error (MSE) and the integrated mean squared error (MISE) of the circular kernel estimator, and is therefore the second result of this paper. 7.1.2. Mean squared error (MSE) and mean integrated squared error (MISE) The mean squared error (MSE) and the mean integrated squared error (MISE) of the circular kernel estimator are given by the following result: Proposition 5. The Mean Square Error (MSE) and the Mean Integrated Squared Error (MISE) of the circular kernel estimator f̂ of the relation (35) are given respectively by: (i) MSE(f̂) ≈ b4n 4 (1− 8 π2 )2f ′′2(x) + π2 16nbn f(x); (ii) MISE(f̂) ≈ b2n 4 (1− 8 π2 )2I1 + π2 16nbn I2, where I1 = ∫ f ′′2(x)dx and I2 = ∫ f(x)dx. Proof. (i) According to the definition of the mean squared error, we have: MSE(f̂) = V(f(x)) +Bias(f̂(x))2 = 1 nbn f(x) π2 16 +O( 1 nbn ) + [ b2n 2 f ′′(x) [ 1− 8 π2 ] +O(b2n) ]2 ≈ 1 nbn f(x) π2 16 + [ b2n 2 f ′′(x) [ 1− 8 π2 ]]2 D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1628 ≈ b4n 4 (1− 8 π2 )2f ′′2(x) + π2 16nbn f(x). So, the MSE(f̂). (ii) We know that MISE(f̂) = ∫ MSE(f̂)dx, then we have: MISE(f̂) = ∫ MSE(f̂)dx ≈ ∫ [ b4n 4 (1− 8 π2 )2f ′′2(x) + π2 16nbn f(x) ] dx ≈ b4n 4 (1− 8 π2 )2 ∫ f ′′2(x)dx+ π2 16nbn ∫ f(x)dx. By posing I1 = ∫ f ′′2(x)dx and I2 = ∫ f(x)dx, we obtain: MISE(f̂) ≈ b4n 4 (1− 8 π2 )2I1 + π2 16nbn I2. 7.1.3. Optimal window of the circular kernel estimator The following result gives the optimal window of the estimator (35) and is the third result of this article: Theorem 5. The optimal window of the kernel estimator f̂ of the relation (35) is given by: b∗n = π 5 √ πI2 16nI1(π2 − 8)2 . (39) Proof. To determine the optimal window, we look for the value of the window which minimizes the integrated mean squared error. For that, let’s look for the value of bn that minimizes the integrated mean squared error. According to the above proposal, we have: MISE(f̂) ≈ b4n 4 (1− 8 π2 )2I1 + π2 16nbn I2. By deriving MISE(f̂), we have: ∂ ∂bn MISE(f̂) = ∂ ∂bn ( b4n 4 (1− 8 π2 )2I1 + π2 16nbn I2 ) D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1629 = b3n ( 1− 8 π2 )2 I1 − π2 16nb2n I2. By canceling the derivative above, we have: ∂ ∂bn MISE(f̂) = 0 ⇔ b3n ( 1− 8 π2 )2 I1 − π2 16nb2n I2 = 0 ⇔ b5n ( 1− 8 π2 )2 I1 = π2 16n I2 ⇔ b5n = π2 16nI2( 1− 8 π2 )2 I1 ⇔ bn = 5 √√√√ π2 16nI2( 1− 8 π2 )2 I1 . Moreover, ∂2 ∂b2n MISE(f̂) = 3b2n ( 1− 8 π2 )2 I1 + π2 8nb3n I2 > 0. As the quantities 3b2n ( 1− 8 π2 )2 and 8nb3n are positive, and since I1 and I2 are also positive according to condition (C.4), then we conclude that the value of bn that minimizes the integrated mean squared error is: b∗n = π 5 √ πI2 16nI1(π2 − 8)2 . Hence the result. 7.1.4. Optimal mean integrated squared error (MISEopt) The following result gives the optimal integrated mean squared error (MISEopt) of the estimator (35) and represents the 4th result of this article. Theorem 6. The optimal mean integrated squared error denoted (MISEopt) of the esti- mator (35) is given by: MISEopt(f̂) = 5 ( π4I1I 4 2 (π2 − 8)2 413 × n4 ) 1 5 . (40) Proof. We know that the optimal mean integrated squared error(MISEopt) of the estimator (35) can be written in the form: MISEopt(f̂) = b∗4n 4 (1− 8 π2 )2I1 + π2 16nb∗n I2. D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1630 Then, by replacing the optimal window b∗n by its value, we obtain: MISEopt(f̂) = b∗4n 4 (1− 8 π2 )2I1 + π2 16nb∗n I2 = π4 4 ( 1− 8 π2 )2 I1 5 √( πI2 16n(π2 − 8)2 )4 + π2I2 16nπ 5 √ 16nI1(π2 − 8)2 πI2 = 1 4 (π2 − 8)2π 4 5 I 4 5 2 I 1 5 1 (16n) 4 5 (π2 − 8) 8 5 + π 16n I 4 5 2 I 1 5 1 (16n) 1 5 (π2 − 8) 2 5 π 1 5 = 1 4 (π2 − 8) 2 5π 4 5 I 4 5 2 I 1 5 1 (16n) 4 5 + π 4 5 I 4 5 2 I 1 5 1 (π2 − 8) 2 5 (16n) 4 5 = 5 4 π 4 5 I 4 5 2 I 1 5 1 (π2 − 8) 2 5 (16n) 4 5 = 5 ( π4I1I 4 2 (π2 − 8)2 413 × n4 ) 1 5 . Hence the result. 7.2. Case of the risk function The following theorem gives the bias and variance of the circular kernel risk function estimator and is the 5th result of this article. Theorem 7. The bias and variance of the risk function estimator ĥ of the circular kernel are respectively given by: Biais(ĥ(x)) = b2nf ′′(x)(π2 − 8) 2π2S(x) +O(1), (41) and, V(ĥ(x)) = π2 16 . 1 nbn . h(x) S(x) +O ( (nb2n)−1 ) . Proof. The expectancy of the risk function estimator of the circular kernel is: E(ĥ(x)) = E ( f̂(x) Ŝ(x) ) = E(f̂(x)) E(Ŝ(x)) = f(x) + b2n 2 ( 1− 8 π2 ) f ′′(x) S(x) +O(1) D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1631 = f(x) S(x) + b2n 2 ( 1− 8 π2 ) f ′′(x) S(x) = h(x) + b2n 2 ( 1− 8 π2 ) f ′′(x) S(x) So, Bias ( ĥ(x) ) = b2n 2 ( 1− 8 π2 ) f ′′(x) S(x) +O(1) = b2nf ′′(x)(π2 − 8) 2π2S(x) +O(1). Hence the result. For the calculation of the variance, under Theorem 1 and we have: V(ĥ(x)) = V ( f̂(x) Ŝ(x) ) = 1 nbn (∫ K2(t)dt ) h(x) S(x) +O ( (nb2n)−1 ) . However, according to the above, we have:∫ 1 −1 K2(t)dt = ∫ 1 −1 (π 4 cos( π 2 t) ) dt = π2 16 . So, V(ĥ(x)) = π2 16nbn . h(x) S(x) +O ( (nbn)−1 )2 . Hence the result. 8. Applications In this section, we want to see if our estimator is robust. For this purpose, the efficiency of the circular core density function estimator will be highlighted through the density curves and the kernel estimator which will be performed on the basis of simulated data and actual data. 8.1. Choice of the programming language In this article, we used the programming language R not only for the simulation of our estimator but also, for the realization of our different curves. It is a programming language D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1632 for manipulating data and performing statistical and graphical analysis. This Scheme- inspired language and software, abbreviated as S, is distributed under the GNU (General Public License) terms. This structure is responsible for distributing and developing the R software with many other contributors from around the world. 8.2. Simulation: Case of Simulated Data Let X be a continuous random variable whose law has the density f defined on R by: f(t) = 1√ 2π e− 1 2 t2 . The function f is called density function of the reduced normal centered law denoted N (0, 1). It is a positive function. If X follows a ”model” distribution, it is associated with a curve: D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1633 Figure 1: Density of the Normal Law Centered Reduced The curve above is a symmetrical curve with respect to the origin which has a bell-shaped shape. It is commonly called Gaussian curve. The normal law is used in this subsection because it is the most commonly used prob- abilistic model to describe many phenomena observed in practice. In what follows, we will generate the data from the R software to show the influence of the h smoothing parameter in evaluating the performance of the circular kernel estimator. 8.2.1. Result of the simulation For a sample N = 500 , we assign the values to the smoothing parameter, which allows us to obtain the following curves: For h = 2, we have the following curve: Figure 2: Comparison of the theoretical curve and the kernel density estimator at h = 2 D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1634 For h = 1, we have the following curve: Figure 3: Comparison of the theoretical curve and the kernel density estimator at h = 1 For h = 0.5, we have the following curve: Figure 4: Comparison of the theoretical curve and the kernel density estimator at h = 0.5 For h = 0.3, we have the following curve: Figure 5: Comparison of the theoretical curve and the kernel density estimator at h = 0.3 D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1635 For h = hopt, we have the following curve: Figure 6: Comparison of the theoretical curve and the kernel density estimator at h = hopt For h = 0.15, we have the following curve: Figure 7: Comparison of the theoretical curve and the kernel density estimator at h=0.15 8.2.2. Interpretation As a reminder, bw refers to the window or smoothing parameter that we have noted h in all our work. The density function is estimated using the circular core estimator. This estimator and the function to be estimated for the density function of the reduced normal centered law, if any, are represented according to different values of the smoothing parameter. We noticed that for every larger values of the smoothing window, our estimator has a rounded shape and approaches the line of equation y = 0 (see Figure 2 and Figure 3). On the other hand, for values of the window smaller and smaller, our estimator approaches the true density function of the reduced normal centered law (see Figure 4, Figure 5 and Figure 7). Thus, if the choice of the kernel is considered as having little influence on the D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1636 estimator, it is not the same for the smoothing parameter. A parameter that is too weak causes the appearance of artificial details appearing on the graph of the estimator. For a value of h that is too large (see Figure 2, Figure 3), the majority of the characteristics are on the contrary erased. The choice of h is therefore a central question in estimating density. Hence the importance of determining an optimal smoothing parameter. The calculation of this optimal parameter gives us for a sample of size N = 500, hopt = 0.2587, (see Figure 6). At this value, our estimator is almost similar to the true density, which shows us the efficiency of our estimator and so this estimator can therefore substitute for the density function of the reduced normal centered law. 8.3. Real Data Case In this subsection, we use data collected during a survey to study the concentration of CO2 in the atmosphere around a very active volcano called Mauna Loa in the Hawaiian Archipelago in the United States of America culminating at 4, 100 meters above level. This study was conducted between 1959 and 1997 from January to December of those years of study. The catches below present all these data expressed in ppm (parts per million): Figure 8: Concentration data in CO2 from January to October Figure 9: Concentration data in CO2 from November to December D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1637 In this study, 468 data was collected. The curve of the function of the reduced normal centered law for these real data is obtained through the software R: Figure 10: True density function with real data The curve of the density of the normal law above is positive and is taken in half. In what follows, we will generate the data from the R software 8.3.1. Result of Simulation The simulation was performed for several values of the smoothing parameter. For h = 50, we have the following curve: Figure 11: Curve of the kernel estimator for h = 50 D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1638 For h = 20, we have the following curve: Figure 12: Curve of the kernel estimator for h = 20 For h = 10, we have the following curve: Figure 13: Curve of the kernel estimator for h = 10 For h = 3, we have the following curve: Figure 14: Curve of the kernel estimator for h = 2 D. A. N. Njamen, H. C. Yayebga / Eur. J. Pure Appl. Math, 12 (4) (2019), 1612-1642 1639 For h = 2, we have the following curve: Figure 15: Curve of the kernel estimator for h = 2 For h = 1, we have the following curve: Figure 16: Curve of the kernel estimator for h = 1 8.3.2. Interpretation As in the case of simulated data, according to different values of the smoothing param- eter,we have a curve which has a different look and tends to move away or closer to the true graph of the density function of the reduced normal centered law (see Figure 12 and Figure 13) represented under the basis of our 468 observations. On the other hand, we found that at first, the density curve of the normal law is taken almost half (see Figure 10). And secondly, our different curves have several fluctuations and we observe rounded shapes of their representations (see Figure 14, Figure 15 and Figure 16), this is due to the fact that all the different data we are working on have very small values and so we have used a smaller scale. 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