Numerical Method for Obtaining a Predictive Estimator for the Geometric Distribution

Transcription

1 British Journal of Matheatics & Couter Science 19(5): 1-13, 2016; Article no.bjmcs ISSN: SCIENCEDOMAIN international Nuerical Method for Obtaining a Predictive Estiator for the Geoetric Distribution Kunio Takezawa 1 1 Division of Inforatics and Inventory, Institute for Agro-Environental Sciences, National Agriculture and Food Research Organization, Kannondai 3-1-3, Tsukuba, Ibaraki , Jaan. Author s contribution The sole author designed, analyzed and interreted and reared the anuscrit. Article Inforation DOI: /BJMCS/2016/29941 Editor(s): (1) Morteza Seddighin, Indiana University East Richond, USA. Reviewers: (1) Radosaw Jedynak, Kaziierz Pulaski University of Technology and Huanities, Poland. (2) Jong-Wuu Wu, National Chiayi University, Taiwan. (3) Louis Asiedu, University of Ghana, Ghana. (4) S. B. Munoli, Karnatak University, India. Colete Peer review History: htt:// Received: 5 th October 2016 Acceted: 5 th Noveber 2016 Original Research Article Published: 10 th Noveber 2016 Abstract An otial estiator in the light of future data (i.e., a redictive estiator) is obtained using nuerical siulations. The redictive estiator is assued to be one of various functions of the axiu likelihood estiator. We then forulate an estiator that yields better results than the axiu likelihood estiator when the araeters are located within a secific range. Using this ethod, we derive a redictive estiator for the geoetric distribution. This rocedure leads to a redictive estiator that outerfors the axiu likelihood estiator in ters of the exected log-likelihood when the araeter is known to be located within a certain range. Keywords: Exected log-likelihood; future data; geoetric distribution; axiu likelihood estiator; redictive estiator Matheatics Subject Classification: 60G25, 62F10, 62M20. *Corresonding author: E-ail: nonara@gail.co, takezawa@affrc.go.j;

2 1 Introduction The ain urose of statistical inference lies in finding estiates that redict future data within the desired degree of accuracy. The axiu likelihood estiator, however, erfors estiations by fitting available data (i.e., data fro the ast). As this ethod can suffer fro the overfitting henoenon, the resultant estiates do not always redict future data sufficiently well. To coe with this roble, we require an estiator that gives a good fit to future data. We call such an estiator a redictive estiator (e.g., age 35 in 1]). The exact solution of the redictive estiator for the variance of the noral distribution is already known. This is called the third variance (2], section 5.5 in 3]). The exact solution of the redictive estiator for the araeter of the exonential distribution has also been derived 4], although another redictive estiator for the exonential distribution ay be available. In addition, the characteristics of redictive estiators for the binoial distribution and the geoetric distribution have been exained 5, 6]. We exect that further research on the analytical characteristics of redictive estiators will be conducted fro various ersectives. However, if the analytical features of a redictive estiator are largely unexlored, an aroxiate redictive estiator given by nuerical siulations can be a ractical alternative to the axiu likelihood estiator. Such nuerically derived redictive estiators are useful for analytical studies. As redictive estiators for estiating the variance of the noral distribution and the araeter of the exonential distribution are described as functions of the axiu likelihood estiators, we assue here that the redictive estiator is a function of the axiu likelihood estiator. We then consider diverse functions as redictive estiators to calculate the exected log-likelihood. The results of this calculation using R version allow us to choose an estiator that realizes better redictions than the axiu likelihood estiator when the araeters are located within a secific range. Using this ethod, we construct the redictive estiator for the araeter of the geoetric distribution. Moreover, sile nuerical siulations using R language to coare the redictive estiator with the axiu likelihood estiator indicate that the redictive estiator obtained here is of great ractical use. 2 Construction of a Predictive Estiator Using a Nuerical Method We assue that the nuber of araeters is one (θ). The true value of θ is tered θ. The robability density function is reresented as f(x θ), and the rando variable obeying this robability density function is called X. The realizations of X, that is, {x 1, x 2,..., x n }, are assued to be available. In this setting, the likelihood function can be written as L(θ {x 1, x 2,..., x n }) = The log-likelihood function is therefore described as n f(x i θ). (2.1) l(θ {x 1, x 2,..., x n }) = log(l(θ {x 1, x 2,..., x n })) = n log(f(x i θ)). (2.2) The value of θ that axiizes l(θ {x 1, x 2,..., x n }) is reresented as ˆθ. This ˆθ is the axiu likelihood estiator. The estiator ˆθ fits well with the data at hand ({x 1, x 2,..., x n}). However, an estiator that fits well with future data ay be slightly different fro ˆθ. Hence, we assue that the axiu 2

3 likelihood estiator in the light of future data (i.e., the redictive estiator, denoted as θ + ) is reresented as θ + = g k (ˆθ) (k = 1, 2,..., G). (2.3) Then, we consider G functions of g k ( ) to obtain a redictive estiator. When we assue that the future data are {x 1, x 2,..., x }, the log-likelihood of θ + in the light of future data is written as l(θ + {x 1, x 2,..., x } = n log(f(x i θ + )). (2.4) Dividing both sides of the above equation by and letting, we have 1 li l(θ+ {x 1, x 2,..., x 1 } = li n log(f(x i θ + )) = E x f(x θ + ) ]. (2.5) E x f(x θ + ) ] is the average of f(x θ + ) when x is saled an infinite nuber of ties. That is, this function gives the exectation of f(x θ + ). The exectation in the sense of Eq. (2.5) is used to derive AIC (Akaike Inforation Criterion). For exale, E G(xn ) on age 55 in 1] is the exectation in this sense. When the axiu estiator is adoted, Eq. (2.5) is relaced by 1 li l(ˆθ {x 1, x 2,..., x 1 } = li Hence, when a secific value is set as θ and n log(f(x ] i ˆθ)) = E x f(x ˆθ). (2.6) E x f(x θ + ) ] > E x f(x ˆθ) ], (2.7) we conclude that the redictive estiator (θ + ) is better than the axiu likelihood estiator (ˆθ) in ters of rediction on the assution that the araeter of the oulation is θ. However, in ractice, the araeter of the oulation is unknown. Even so, we often have soe knowledge, such as θ is between θ in and θ ax, fro an exerient or survey. Therefore, when we know that θ is between θ in and θ ax, θ + is a better estiator than ˆθ in ters of rediction if Eq. (2.7) holds on the assution that θ is located between θ in and θ ax. Hence, various functions g k ( ) (1 k G) are used to calculate Eqs. (2.5) and (2.6) with θ between θ in and θ ax. One or ore g k ( ) that satisfy Eq. (2.7) are then selected. If no functions satisfy Eq. (2.7), other functions are used for g k ( ). If ore than one g k ( ) are selected, the values of E y f(y θ + ) ] are calculated using equisaced values between θ in and θ ax as θ, and the resultant values are averaged. This calculation is carried ] out with the selected g k ( ). These averages are then coared with the average of E y f(y ˆθ) with the sae setting. The function gk ( ) that axiizes the difference between the average given by a redictive estiator and that given by the axiu likelihood estiator is regarded as the otial g k ( ). 3 Construction of a Predictive Estiator for the Geoetric Distribution A redictive estiator for the geoetric distribution is now constructed using the ethod described in the revious section. The robability density function of the geoetric distribution is written as f(ξ) = (1 ) ξ 1, ξ = 1, 2, 3,.... (3.1) 3

4 The exectation of Eq. (3.1) is reresented as ξf(ξ) =. (3.2) 1 ξ=1 The rando variable obeying f(ξ) is denoted by X. The realizations (data) of this rando variable are naed {x i } (1 i n). The log-likelihood (l( {x i })) of this data is described as n l( {x i}) = nlog() + log(1 ) (x i 1). (3.3) To derive the value of that axiizes the above function, we differentiate Eq. (3.3) with resect to and set it equal to 0. Then, we have n n (x i 1) = 0. (3.4) 1 Hence, the following axiu likelihood estiator is obtained: n ˆ = n x. (3.5) i This ˆ is the axiu likelihood estiator in the light of the data at hand ({x i }, 1 i n). Next, we assue that the redictive estiator + is a function of the axiu likelihood estiator ˆ. Thus, Eq. (2.3) becoes + = g k (ˆ) (k = 1, 2,..., G). (3.6) When we reresent future data as {x 1, x 2,..., x }, the log-likelihood of + in the light of these data is written in a siilar for to Eq. (3.3): l( + {x i }) The + that axiizes this value (denoted as ) is reresented as ( = log( + ) + log(1 + x ) i ) 1. (3.7) =. (3.8) x i When we use infinite nuber of future data to exaine the validity of an estiate, the evaluation is ore reliable than that given by finite nuber of future data. Then, we assue that is infinite. In this setting, when is denoted as, Eq. (3.2) shows that the following holds: = li x i =. (3.9) That is, the axiu likelihood estiator using infinite future data is the true araeter of the oulation,. Thus, the log-likelihood in the light of an infinite nuber of future data is written as l ( + ) ( ) li = log(+ ) + log(1 + ) 1 1. (3.10) l ( + ) Next, we consider the average of li given by saling x an infinite nuber of ties. Equivalently, we consider the exectation of l ( + ). However, the saling ust be conducted again if all of the data {x i} are equal to 1. The exectation is written as l ( + ) E x li ] = E x log( + ) + log(1 + )( 1 1 ) ]. (3.11) 4

5 When X i obeys the geoetric distribution, n X i obeys the negative binoial distribution (e.g., 87 age in 7]). Hence, Eq. (3.11) is transfored into E x li ] l ( + ) = j=n+1 ( ( ) ) log( + ) log(1 + ) j 1C n 1 n (1 ) j n j=n+1 j 1C n 1 n (1 ) j n. (3.12) (j 1)! j 1C n 1 is defined as j 1 C n 1 = (! reresents a factorial. ). The suation in (j n)!(n 1)! the nuerator of the right-hand side of Eq. (3.12) is instead of. Moreover, the right-hand side is noralized with resect to j=n+1 j=n+1 j=n j 1C n 1 n (1 ) j n. These odifications account for the resaling when all {x i } are equal to 1. When + in Eq. (3.12) is relaced by ˆ, we have E x li ] l (ˆ) = j=n+1 ( ( ) ) log(ˆ) log(1 ˆ) j 1C n 1 n (1 ) j n j=n+1 j 1C n 1 n (1 ) j n. (3.13) We reresent the difference between the value of Eq. (3.12) and that of Eq. (3.13) as : = E x li ] l ( + ) E x li ] l (ˆ). (3.14) If a secific value is given as and is ositive, + (= g k (ˆ)) is a redictive estiator that outerfors the axiu likelihood estiator. However, in ractice, the exact value of is unknown. Instead, we often have inforation such as is between in and ax. Therefore, + is a better estiator than ˆ in ters of rediction if is always ositive (i.e., Eq. (2.7) holds) on the assution that θ is between θ in and θ ax. Nuerical siulations were conducted by setting the nuber of data n to 20 and using the three equations below as g k (ˆ). g 1 (ˆ, a 1 ) = ( + a 1 ˆ) 0.9, (3.15) g 2 (ˆ, a 1 ) = + a 1 ˆ, (3.16) g 3(ˆ, a 1) = ( + a 1 ˆ) 1.1. (3.17) 5

6 a 1 =0.7 a 1 = a 1 = a 1 =1 Fig. 1. The value of for various values of a 1 for each value of. + = ( + a 1 ˆ) 0.9 is adoted. a 1 = 0.7 (to-left), a 1 = 0.8 (to-right), a 1 = 0.9 (botto-left), a 1 = 1 (botto-right). The contour is drawn in the region of ositive. The contour is not drawn in the region of negative and incoutable a 1 =0.7 a 1 = a 1 =0.9 a 1 = Fig. 2. The value of for various values of a 1 for each value of. + = + a 1 ˆ is adoted. a 1 = 0.7 (to-left), a 1 = 0.8 (to-right), a 1 = 0.9 (botto-left), a 1 = 1 (botto-right). The contour is drawn in the region of ositive. The contour is not drawn in the region of negative and incoutable 6

7 The fors of the equations above are not based on dee insight into redictive estiators. They are ere attets to derive a redictive estiator by sile algorith. However, if such rudientary derivation of a redictive estiator is successful, it ilies that nuerical ethods are roising tools to obtain ore useful estiators than the axiu likelihood estiator. We assued that = 0.0, 0.01, 0.02, 0.03,..., 0.19 and a 1 = 0.7, 0.8, 0.9, 1.0. That is, 228 (3 19 4) functions were used to construct redictive estiators by transforing ˆ, and the values of Eqs. (3.12) and (3.13) were calculated to obtain. Furtherore, because we set = 0.05, 0.1, 0.15,..., 0.95, was couted for a total of 4, 332 ( ) settings. When the value of was negative, we set = 0. When could not be couted because the arguent of the logarith was 0 or negative, we again set = 0. Fig. 1,2 and 3 show the results for g 1(ˆ, a 1), g 2(ˆ, a 1), and g 3(ˆ, a 1), resectively. The four grahs in each figure corresond to the settings of a 1 = 0.7, 0.8, 0.9, 1.0. In each grah, the values of are drawn for = 0.0, 0.01, 0.02, 0.03,..., 0.19 and = 0.05, 0.1, 0.15,..., a 1 =0.7 a 1 =0.8 a 1 =0.9 a 1 = Fig. 3. The value of for various values of a 1 for each value of. + = ( + a 1 ˆ) 1.1 is adoted. a 1 = 0.7 (to-left), a 1 = 0.8 (to-right), a 1 = 0.9 (botto-left), a 1 = 1 (botto-right). The contour is drawn in the region of ositive. The contour is not drawn in the region of negative and incoutable 7

8 frequency =0.4 frequency estiate estiate frequency =0.5 frequency estiate estiate Fig. 4. Estiation of when we have inforation: is between 0.4 and 0.7. (To-left) The distribution of the redictive estiator and (botto-left) axiu likelihood estiator when = 0.4. (To-right) The distribution of the redictive estiator and (botto-right) axiu likelihood estiator when = 0.5. We assued that is between 0.4 and 0.7. We assued that we have inforation: is between 0.4 and 0.7. We also assued that = 0.4 was the real araeter in the oulation. Then, using Figs. 1, 2 and 3 we selected the equations that ensured (Eq. (3.15)) was ositive when took values of {0.4, 0.45, 0.5,..., 0.7}. Aong the selected equations, we adoted the otial equation that axiized the average value of given by setting to {0.4, 0.45, 0.5,..., 0.7}. This rocedure found that the following equation was the otial redictive estiator. g 1 (ˆ, a 1 ) = ( ˆ) 0.9. (3.18) Next, 5, 000 datasets (each consisting of 20 data) were generated according to the geoetric distribution with = 0.4. Fig. 4 (to-left and botto-left) coares the distributions of the estiates given by Eq. (3.5) and the estiates given by Eq. (3.18). The ean of the redictive estiator is , and its unbiased variance is In contrast, the ean of the axiu likelihood estiator is , and its unbiased variance is That is, the axiu likelihood estiator is closer to an unbiased estiator. The variance of the redictive estiator, however, is saller. Hence, to deterine the referred estiator, we ust exaine the goodness of fit to future data (i.e., exected log-likelihood). Thus, we derived the values of the exected log-likelihood using the following two equations based on Eq. (3.7). l( + j {x i }) ( = log( + j ) + log(1 + x ) i j ) 1. (3.19) l(ˆ j {x i }) ( = log(ˆ j) + log(1 ˆ j) x ) i 1, (3.20) where + j is the estiate obtained using the j-th siulation datu and the redictive estiator, and ˆ j is the estiate obtained using the j-th siulation datu and the axiu likelihood estiator. { l( + j We then calculated {x i }) } { l(ˆj {x i }) } and using the 5, 000 sets of siulation data with { l( + j = 500, and averaged the 5, 000 values of {x i }) } { l(ˆj {x i }) } and the 5, 000 values of. 8

9 { l( + j Accordingly, the average of {x i }) } { l(ˆj {x i }) } was found to be , and that of was found to be = Hence, the redictive estiator is better in ters of the exected log-likelihood. However, it is difficult to gain an intuitive feel for how the discreancy in the exected log-likelihood is associated with any ractical advantages. Thus, we coared the axiu likelihood estiator and the redictive estiator by calculating the su of the squared differences between the true value and the redictive estiator, because this squared difference is a ore intuitive etric. The su of squared differences between the redictive estiator and the true value is defined as d( + ) = 1 5,000 ( + j ) 2, (3.21) 5, 000 and that between the axiu likelihood estiator and the true value is defined as j=1 d(ˆ) = 1 5,000 (ˆ j ) 2. (3.22) 5, 000 Using these exressions, we obtain values of d( + ) = and d(ˆ) = Therefore, we can conclude that the difference between the redictive estiator and the axiu likelihood estiator is not negligible. Next, 5, 000 datasets (each consisting of 20 data) were generated with = 0.5. Using these data, a siulation was conducted under the sae conditions as the revious siulation. The results are shown in Fig. 4 (to-right and botto-right). The ean of the redictive estiator is , and its unbiased variance is In contrast, the ean of the axiu likelihood estiator is , and its unbiased variance is That is, the redictive estiator is not only closer to the unbiased estiator, but its variance is also saller. Therefore, the sueriority of the redictive estiator is intuitively clear. In fact, a coarison of the exected { l( + j log-likelihood shows that the average of {x i }) } { l(ˆj {x i }) } is , whereas that of is Calculating the sus of the squared errors between the true value and the two estiators, we obtain d( + ) = and d(ˆ) = These results are as exected. j=1 r like r error Fig. 5. It is assued that we have inforation: is between 0.1 and 0.6. r like ( +, ˆ) (left) and r error ( +, ˆ) (right) are shown for equal to {0.1, 0.11, 0.12,..., 0.6} 9

10 Next, we assued that we have inforation: is between 0.1 and 0.6. We set = 0.1, 0.11, 0.12,..., 0.6, and exained whether the redictive estiator erfored better than the axiu likelihood estiator in ters of the exected log-likelihood. We obtained the following redictive estiator: g 2 (ˆ, a 1 ) = ˆ. (3.23) To coare the exected log-likelihood of the redictive estiator and that of the axiu likelihood estiator, we calculated r like ( +, ˆ) = l(ˆ {x i }). (3.24) l( + {x i }) The following exression was also derived to coare the su of the squared errors between the true value and the redictive estiator and that between the true value and the axiu likelihood estiator: r error ( +, ˆ) = d(+ ) d(ˆ). (3.25) The results of Eqs. (3.24) and (3.25) are illustrated in Fig. 5. This figure shows that when the true value of is around 0.2, r error( +, ˆ) is less than 0.8. In ractical situations such as estiation of issile failure robability, this iroveent indicates that the nuber of exerient to achieve a reliable is reduced to aroxiately 0.89 ( 0.8) ties. It will generate substantial econoic efficiency gain. Therefore, this technique is not just a atheatical curiosity but have iortant consequences for the real econoy. r like r error Fig. 6. It is assued that we have inforation: is between 0.4 and 0.9. r like ( +, ˆ) (left) and r error ( +, ˆ) (right) are shown for equal to {0.4, 0.41, 0.42,..., 0.9}. r like r error Fig. 7. It is assued that we have inforation: is between 0.2 and 0.4. r like ( +, ˆ) (left) and r error ( +, ˆ) (right) are shown for equal to {0.2, 0.21, 0.22,..., 0.4}. 10

11 r like r error Fig. 8. It is assued that we have inforation: is between 0.6 and 0.8. r like ( +, ˆ) (left) and r error ( +, ˆ) (right) are shown for equal to {0.6, 0.61, 0.62,..., 0.8}. The sae siulation with the inforation the range of is between 0.4 and 0.9 was carried out. We set = 0.4, 0.41, 0.42,..., 0.9. Then, we have Fig. 6. The redictive estiator is g 2 (ˆ, a 1 ) = ˆ. (3.26) Because we defined the redictive estiator as that which is better than the axiu likelihood estiator in ters of the exected log-likelihood, the redictive estiator is inferior to the axiu likelihood estiator in ters of the su of squared errors based on the true value when is close to 0.9. Next, we assued that we have inforation: is between 0.2 and 0.4, and set = 0.2, 0.21, 0.22,..., 0.4. The results are shown in Fig. 7. We also obtained the redictive estiator g 2(ˆ, a 1) = ˆ. (3.27) Furtherore, we assued that we have inforation: is between 0.6 and 0.8, and set = 0.6, 0.61, 0.62,..., 0.8. The results are shown in Fig. 8. Finally, we obtained the redictive estiator g 1 (ˆ, a 1 ) = ( ˆ) 0.9. (3.28) When the ossible range of is narrow, the distance to the true value tends to be shorter. If we know that the ossible range of is narrow, then we have ore inforation on the true value. Hence, the above results are intuitive. 4 Conclusions The ain estiators treated in conventional statistics are the axiu estiator and the unbiased estiator. The reason for the ehasis on these two estiators is not only their ractical utility, but also their atheatical tractability, because they allow a coaratively easy analytical treatent. However, when we consider ore diverse estiators, ore reliable estiations are realized by extracting the axiu inforation fro the available data. In fact, quite a few studies focus on better estiators than the axiu likelihood estiator (e.g., 8], age 414 in 9], 10], age 332 in 11], age 431 in 12], and 13]). 11

12 To show one ore exale of this attet, we derived the redictive estiator for the geoetric distribution, which ais to increase the value of the exected log-likelihood. This was otivated by the goal of obtaining an estiator that rovides a good fit to future data, rather than available data. Fortunately, this olicy yields a redictive estiator with high redictability. This finding will hoefully advance the construction and use of redictive estiators, and encourage further studies on the analytical treatent of redictive estiators. For this urose, araetric equations such as Eqs. (3.15), (3.16), and (3.17) should not be the only fors of redictive estiators. Functions with any araeters should be considered, as these are often used in nonaraetric regression. Moreover, the for of redictive estiators should not be liited to that of Eq. (2.3), which exresses the transforation of the axiu likelihood estiator. Functions that treat the available data directly should also be considered. Construction of the redictive estiator for the geoetric distribution is also of ractical value. For exale, estiation of the araeter of the geoetric distribution lays a critical role in evaluation of syste failure rate. We obtain ore useful estiates of the araeter of the geoetric distribution using the redictive estiator because the redictive estiator develoed here rovides a better fit to future data than the axiu likelihood estiator in any circustances; we are tyically ore interested in the syste in the future than that in the ast. However, the araeter estiation of the geoetric distribution uses truncated data in ost of ractical situations because of liited observation tie. Hence, we should develo the idea of the redictive estiator toward ore flexible ones to treat diverse robles. The silicity of the algorith suggested here ilies that it is not very difficult to extend this ethod for ore general use. The construction of redictive estiators by eans of analytical or nuerical ethods has gathered oentu, and so we should consider urose-oriented estiators instead of liiting our scoe to the axiu likelihood estiator and the unbiased estiator. As the redictive estiator is based on the derivation of estiates that fit well with future data, it directly ursues the goal of rediction, unlike the axiu likelihood estiator and the unbiased estiator. Therefore, the redictive estiator is exected to give better results than conventional ethods in ost ractical situations. However, even if our urose is rediction, we soeties hoe that estiators should be close to unbiased, with soe slight variance. That is, the average error of any redictions should be close to 0 in soe situations if the estiator largely enhances the reliability of rediction. Fro another ersective, it is crucial that the absolute value of the error should not deart fro a secific range, so a suitable estiator for this urose is needed. Furtherore, another estiator is required for situations in which a certain decision is ade on the grounds of redicted results and the robability of aking a wrong decision ust be sall. There are infinite ossibilities for the for and ethod of creating an estiator. We should select an aroriate for and ethod to construct an estiator. Then, we should create an estiator that suits our goal using both analytical and nuerical ethods. This rocedure allows us to extract inforation efficiently fro data in a certain direction; this is essential to ake the best use of oortunities and resources. We are quietly confident that our future studies will develo reliable estiators that deal flexibly with diverse needs. Acknowledgeent The author is very grateful to the referees for carefully reading the aer and for their coents and suggestions which have iroved the aer. Coeting Interests Author has declared that no coeting interests exist. 12

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