ON THE MODIFICATION OF THE EMPTY CELLS TEST

Transcription

1 A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA OECONOMICA 228, 2009 G rzegorz Kończak* ON THE MODIFICATION OF THE EMPTY CELLS TEST Abstract. In the paper the proposition of the nonparaetric test to verify the hypothesis on the distribution of the rando variable is presented. The proposed test is the odification of well known epty cells test. In the epty cells test the area of variability of the rando variable is divided into soe fixed cells. In the proposed odification the cell is oving over the whole area of variability of the rando variable. The analysis of testing the hypothesis of norality is presented. The table with critical values of the test statistic and the coparison of the epty cells test and the proposed odification is presented. Key words: test, epty cells test, Monte Carlo. I. INTRODUCTION Aong the other goodness-of-fit tests that are described and discussed in nonparaetric statistic books is David s epty cells test (David F.N. 1950, Sheskin D. J., 2004). This test can be used to test the hypothesis of the distribution of rando variable. The area of variability of rando variable is divided into cells and the nuber of eleents in each cell is counted. Then the nuber of epty cells is deterined. This nuber of epty cells is copared to the critical value. The proposition of the odification of the epty cells is presented in the paper. In the proposed odification the cell is oving over the whole area of variability of the rando variable. The analysis in the case of verifying the hypothesis of norality is presented. The table with critical values of the test statistic and the coparison o f the epty cells test and the proposed odification is presented. The Monte Carlo study for coparison properties o f the classical for of the epty cells test and the proposed odification were ade. The results o f this siulation were presented. *Ph.D Departent of Statistics, Katowice University of Econoics, koncz@ae.katowice.pl.

2 II. THE EMPTY CELSS TEST Let X be the continuous rando variable and let F о be the distribution function of this rando variable. Let Xi, *2, > *n be an и-eleent siple saple. We will test the hypothesis that the saple is taken fro the F0 distribution. Let S denotes the area of variability of the rando variable X. In the classical version of the epty cells test the area of variability S of the rando variable X is divided into cells Sb S2, Swhich fulfill conditions: i- s = U s í /=1 2. S; n Sj = 0 for i Ф j 3..P(xeSj) = for i = 1, 2,. For each cell Sb S2, Swe deterine the nuber of eleents in the cell. The nuber of eleents in the i'-th cell we denote as,. Let K be the nuber of epty cells. The statistic Kn can be written as follows K n = card{i: /и, = О} (1) where t is the nuber of eleents in the /'-th cell. The probability function o f nuber of epty cells is known and can be written as follows (Hellwig Z., 1965, Csorgo M. and Guttan I., 1962) p{kn =k) l H ) ' - k \ r У - k - r (2) where к = h, h and h = ax(0, - r i ) The cuulative distribution function of the statistics Kn can be written as follows k í \ / \ - s - s - r w - I < - > ' (3) s = 0 \ J V r - 0 V r The statistic K can be used to test the hypothesis H 0 :F(x) = F0(x) H l : F ( x ) * F 0(x)

3 For the assued significance level a the rejection region can be written as follows Q = { k : k > K n a } (5) Where K na is taken fro the tables (eg. Hellwig Z., 1965, Doański Cz., Pruska K. 2000). Ш. THE MODIFICATION OF THE EMPTY CELLS TEST In the classical for of the epty cells test the cells are fixed. Let us consider the case that the cells are not fixed. In the proposed odification there is one cell which is oving over the whole area S of variability of the rando variable X. The probability that *, (/= 1, 2,..., n) is in the cell under H0 is constant. The idea of the proposed odification is presented in the Fig. 1. There are = 4 fixed cells (the classical for of the epty cells test) and the cell Sx (the odification of the epty cells test) which is oving over the set [a, b]. Let us consider a set S* of cells Sx v/hich satisfy following two conditions: 1. x e ql q,.l S r = qß-l qß+l Where qa is the quantile of order a of the rando variable X, ß denotes the order of the quantile of л- ^ < ß < 1 - у and 0 < /> < 1.

4 We can notice that Sx is a cell in which л: is a id-point. Let *1, xi,..., xn be an i.i.d. saple. The hypothesis (4) will be tested. For each x e under H0 we have P{xt e S x) = p = const (/= 1, 2, n). Therefore the probability that the cell Sx is epty can be written as follows: P(card{Sx} = 0) = / ((.x, Sx)a{x2 i S)A...A(xn S)) (6) Under the assuption that jti, x2, follows x are independent it can be written as P(card{Sx} = 0) = P(x{ t Sx) P(x2 «Sx)-... P(x Sx) = (7) Let us consider the function h? ;V 2 2 {0,1} given as follows [0 if card Sx > 0 h(x) = 11 if card Sx = 0 (8) The forula (8) can be written equally as follows [0 if 3 Xie[qp _pi2;qßi+pl2] h(x) = if Vx, [Яр1-р/2\Яр,+р/2] where Д <, ß i <, 1 - у is given as follows

5 That s ean that the value h(x) is equal to 1 if and only if the cell Sx is epty. The statistic K fro the classical for of the epty cells test (1) can be rewritten as follows K n = ^ГА(л:(i)) where is the nuber of cells and x<j) is the i=i id-point of the /-th cell. Therefore the proposed odification can be treated as a generalization of the classical for of the epty cells test. The function h(x) is equal to 1 if and only if the corresponding to x cell is epty, that s ean h(x) = 1 о card{sx} = 0. It can be written as follows P(h(x) = \)= P(card{Sx} = 0) = ( \ - p Y for each x e qp ',q p 2 '"7. Fig 2. Function h(x) for 3 eleent saple. The idea of the function h{x) is presented in the Fig. 2. The n - 3 eleent saple is taken. Function h{x) is equal to 1 if and only if x, ŕ S x for /' = 1, 2, 3. To test the hypothesis (4) it can be used following statistic T = 9\-pl2~qp/2 p/2 1 P/2 r * ^h{x)dx ( 10) It can be notice that 0 <,T < 1. The value of the statistic T represents the relative length of the epty cells area and is equal to the area under h(x). We reject the hypothesis if T > T a.

6 IV. THE CASE OF NORMAL DISTRIBUTION Let us assue that X ~ N(ju,cr) and x u x2, xn is the n eleent i.i.d. saple and let p =. To obtain the critical values for test the hypothesis that n rando variable X is norally distributed the Monte Carlo siulation were ade. For saple sizes of n = 3, 4,..., 15 there were found quantilies of the statistic T. They were found for the significance levels a 0.10, 0.05 and There are following steps in coputer siulations: 1. The values xi, x2, (и = 3, 4,..., 15) were generated fro noral distribution with ean ju = 100 and standard deviation cr= For each saple the value of the T statistic was calculated. 3. The steps 1-2 were repeated ties. 4. The epirical quantilies 0.90, 0.95 and 0.99 were accepted as estiates of quantiles of the statistic T. Table 1. The estiates quantiles of the test statistic T Saple size n Quantil <7(1-a) Source: Monte Carlo study The results of Monte Carlo study are presented in table I. For saple size fro 3 to 15 there are presented estiates quantiles of the statistic Г(10).

7 V. MONTE CARLO STUDY - COMPARISON OF THE EMPTY CELLS AND THE PROPOSED MODIFICATION To copare the classical for of the epty cells test and the proposed odification the series of coputer siulations was ade. The saples of size n was taken fro noral distribution N(105, 5). There were test the hypothesis H 0 : F (x ) = F0(x) against tf, : F (x) = F{( x ), where F0(x) is the cuulative distribution function o f the rando variable X~ N(100,5) and F](*) is the cuulative distribution function of the rando variable X~ N(105,5). For every n saples were generated and for each saple the value of the statistic T was calculated. The critical values of the statistic T was taken fro table 1. The estiates o f probabilities o f rejection the hypothesis #o are presented in table 2. Saple size Table 2. The estiates probabilities of rejection H0 hypothesis under H\ The proposition Epty cells test «= 0.10 я = 0.05 a = 0.01 «= 0.10 a = 0.05 «= Source: The results of the Monte Carlo study. As we can see it is ipossible to reject the null hypothesis in classical for of epty cells test for n = 3 eleent saple (for a = and 0.01). The proposed odification can be used for sall saple.

8 Hlłie proposition a = 0.10 cad-ha = 0.10 n Fig. 3. The estiates probabilities of rejection H0 hypothesis under Я, (a = 0.10) The results for the significance level a = 0.10 of the Monte Carlo study are presented in the Fig. 3. It can be noticed that use of the odification of the epty cells test leads ore often to rejection the # 0 hypothesis (under #,). VL CONCLUDING REMARKS The proposed odification of the epty cells test can be used to test the hypothesis in statistical control quality procedures. It can be especially used in process onitoring using Shewhart s control chart to test the hypothesis o f norality distribution in sall saple cases. The Monte Carlo study have been ade. In the first part of the siulation the critical values of the proposed statistic have been derived. In the second part the coparison of the classical epty cells test and the proposed odification has been done. If the null hypothesis is false then the proposed odification ore often leads to the rejection o f the null hypothesis. The proposed odification of the epty cells is natural enhanceent of the classical for of this test and is easy to use. REFERENCES Csorgo M., Guttan I. (1962) On the Epty Cell Test, Technoetrics, vol. 4, No. 2, p David F.N. (1950) Order Statistics. J. Wiley & Sons Inc.. New York. Doański Cz. Pruska К. (2000) Nieklasyczne etody statystyczne. PWE Warszawa. Hellwig Z. (1965) Test zgodności dla alej próby. Przegląd Statystyczny. 12. p Sheskin D. J. (2004) Handbook of Paraetric and Nonparaetric Statistical Procedures, Chapan & Hall /CRC, Boca Raton.

9 Grzegorz Kończak O PEWNEJ MODYFIKACJI TESTU PUSTYCH CEL W artykule przedstawiono propozycję nieparaetrycznego testu do weryfikacji hipotezy o postaci rozkładu badanej ziennej. Proponowany test jest odyfikacją znanego testu pustych cel. W teście pustych cel obszar zienności jest dzielony na ustalone cele i sprawdza się w ilu celach nie a żadnego eleentu z próby. W proponowanej odyfikacji położenie celi jest zienne. Wyznaczana jest funkcja podająca czy dla danego położenia celi jest ona pusta, a następnie na podstawie przebiegu tej funkcji podejowana jest decyzja odnośnie weryfikowanej hipotezy. Przedstawiono rozważania dla szczególnego przypadku gdy testowana jest hipoteza o noralności rozkładu. Wyznaczone zostały wartości krytyczne dla proponowanego testu oraz porównania tej etody z teste pustych cel. Proponowana odyfikacja została porównana z klasyczny teste pustych cel.