A nonparametric confidence interval for At-Risk-of-Poverty-Rate: an example of application

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1 A nonparametric confidence interal for At-Ris-of-Poerty-Rate: an example of application Wojciech Zielińsi Department of Econometrics and Statistics Warsaw Uniersity of Life Sciences Nowoursynowsa 159, PL--776-Warszawa Abstract In the European Commision Eurostat document Doc IPSE/65/4/EN page 11, the at-ris-of-poerty rate ARP R is defined as a percent of population with income smaller than 6% of population median Zielińsi 8 proposed a distribution free confidence interal for ARP R In the paper, an example of application of the constructed confidence interal is shown Keywords: binomial distribution, confidence interal, ARP R JEL: C14, C13 1 Introduction In the European Commision Eurostat document Doc IPSE/65/4/EN page 11, the at-ris-of-poerty rate ARP R is defined as follows Let EQ INC i denote the euialised disposable income of the i th person and let weight i denote the weight of person i The at-ris-of-poerty threshold ARP T is calculated as 6% of calculated median alue, ie where and EQ INC MEDIAN = ARP T = At ris of poerty threshold = 6%EQ INC MEDIAN, { 1 EQ INC j + EQ INC j+1, if j i=1 weight i = W, EQ INC j+1, if j i=1 weight i < W < j+1 i=1 weight i, W = All persons weight i Then the at-ris-of-poerty rate is calculated as the percentage of persons oer the total population with an euialised disposable income below the at-ris-of-poerty threshold ie the euialised disposable income of each person is compared with at-ris-of-poerty threshold The cumulated weights of persons whose euialised disposable income is below the at-ris-of-poerty threshold, is diided by the cumulated weights of the total population ie sum of all the personal weights: ARP R = All persons with EQ INC<at ris of poerty threshold W weight i 1 1

2 Let X 1,, X n be a sample of disposable incomes of randomly drawn n persons and Med denotes the sample median The natural estimator ARP R is defined: ARP R = 1 n #{X i 6 Med} The properties of ARP R were inestigated by Zielińsi 6, 7 Howeer, the problem is in interal estimation of ARP R Zielińsi 8 proposed a nonparametric confidence interal for ARP R This confidence interal is presented in Chapter In Chapter 3 an example of application of the confidence interal is shown Confidence interal Let F denotes the cdf of a distribution of population income It is assumed that F is continuous We are interested in estimation of the parameter θ = F α Q, for gien α,, 1, where Q denotes the uantile function Qx = F 1 x For α = 6 and = 5 parameter θ is ARP R We are interested in constructing a confidence interal for θ Let X 1, X,, X n be a sample from F and let X 1:n X n:n be order statistics As an estimator of θ we tae θ = 1 n #{X i α X M:n }, where M = n + 1 a is the greatest integer not grater than a Here X M:n is an estimator of uantile Q of the F distribution Let ξ be the number of obserations not grater than α X M:n : ξ = #{X i α X M:n } The distribution of ξ is P F {ξ = } = P F {ξ } P F {ξ + 1} = P F {X :n α X M:n } P F {X +1:n α X M:n }, =,, M 1 I may be checed that Daid and Nagaraja 3, Zielińsi 8 P F {X :n α X M:n } = F αq = B,M + F αq B,M [ F αq B,M b M,n M+1 d ] F αq B,M b M,n M+1 d Here B a,b and b a,b denotes cdf and pdf of beta distribution with parameters a, b, respectiely

3 It is well nown, that if S n is a random ariable distributed as binomial with parameters n and p, then We obtain: M 1 P F {ξ = } = [ M 1 + where P n,p {S n } = j= p 1 p M 1 p 1 p M 1 n p j 1 p n j = B n,+1 1 p j M 1 p = F αq Hence, the distribution of ξ is almost binomial with parameters M 1 and F αq Let γ, 1 Consider an interal see Appendix p 1 p M 1 ] b M,n M+1 d B 1 ξ, M ξ + 1; 1 γ ; B 1 ξ + 1, M ξ; 1 + γ, where B 1 a, b; δ is the δ uantile of beta distribution with parameters a, b This interal may be considered as a confidence interal for θ Zielińsi 8 It appears that the probability P F {θ B 1 ξ, M ξ + 1; 1 γ ; B 1 ξ + 1, M ξ; 1 + γ } of coering the true alue of θ strongly depends on the underlying distribution 3 An example Theoretical results from Chapter were applied to estimation of ARP R in Poland in 3 As a sample, there were 39 data of the euialised disposable income data were used with ind permission of Dr Hanna Dude, Warsaw Uniersity of Life Sciences The empirical cdf of the most interesting part of the data are shown in the Picture It was calculated α = 6, = 5: M = 16148, Med = Q = , ξ = 5576, ARP R = 1767 Those calculations are illustrated in the Picture The confidence interal for ARP R taes on the form γ = 95 5B , 1573; 5 ; 5B , 157; 975 = 1693; The uestion is: what is the confidence leel of the aboe confidence interal? There are at least two methods of estimating that leel The first one relies on the fitting a theoretical distribution to gien data This 3

4 ARP R 1 6Med Med Picture Empirical cdf method is rather useless in the case It is because, there are many different distributions F which may model considered data and for each such distribution calculations are numerically complicated and time consuming The second method maes use of the well-nown bootstrap techniue Chernic 1999 This method is simply and gies almost true results So bootstrap method was applied in estimation of the confidence leel of obtained confidence interal All data were numbered from 1 up to 39 There were generated, according to uniform distribution on the set {1,, 39}, n numbers i 1,, i n From the of data, a sample X i1,, X in was drawn For the sample there were calculated the alue of ξ The procedure was repeated K times, so alues ξ 1,, ξ K were obtained In the next step, mean alue ξ = 1 Kn K i=1 ξ i was calculated For eery ξ i, i = 1,, K, the confidence interal was calculated, and it was checed whether ξ falls into the obtained confidence interal or not The percentage of confidence interals containing ξ may be considered as an estimator of confidence leel of confidence interal for ARP R In our inestigations n = 1 and K = 5 Obtained results are as follows ξ = 17315, 1 K K ξ i ξ = 164 i=1 Estimated confidence leel is 954 and the standard error of that estimate is 94 Hence, it may be said that the obtained confidence interal for ARP R in Poland in 3 is on the confidence leel about 954 4

5 References Brown L D, Cai T T, DasGupta A 1 Interal Estimation for Binomial proportion, Statistical Science, 16, Chernic M R 1999, Bootstrap Methods, A Practitioner s Guide, Wiley Daid H A, Nagaraja H N 3 Order Statistics, Third Edition, Wiley Zielińsi R 6 Exact distribution of the natural ARP R estimator in small samples from infinite populations, Statistics In Transition, 7, Zielińsi R 7 A confidence interal for ARP R at-ris-of-poerty-rate, Statistics In Transition, 8, 17- Zielińsi W 8, A nonparametric confidence interal for At-Ris-of-Poerty-Rate, The Economics Letters submitted preprint: informacje/prace pedefy/s46pdf Appendix: confidence interal for binomial proportion Let η be a binomial random ariable with parameters n and unnown p It is well nown that P p {η } = B n,+1 1 p and P p {η } = B,n +1 p Let δ, 1 be a gien number Confidence interal for p at the confidence leel δ is defined as P p {p L η p p U η} = δ, for all p, 1 For gien n and let p L be the solution of B n +1, 1 p L = 1 + δ or euialently B,n +1 p L = 1 δ We obtain Similarly we obtain p L = B 1, n + 1; 1 δ p U = B 1 + 1, n ; 1 + δ Hence, the confidence interal for p at the confidence leel δ is of the form { P p B 1 η, n η + 1; 1 δ p B 1 η + 1, n η; 1 + δ } δ, for all p, 1 The actual confidence leel is higher than the nominal one because of discreteness of binomial distribution see for example Brown et al 1 5