In this paper it is intended to apply the inequality
82
-9 - > P(I~I > IXil)
-4x ~ 1 1-
to test the examined probability variable C has a distribution function F(x)

Barabás, B.

Sebestyén, L.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

GOODNESS OF FIT WITH THE HELP OF A SHARPENED BERNOULLI INEQUALITY B. BAR.AK,(S and L. SEBESTYEN Department of Civil Engineering Mathematics, Technical University, H·1521 Budapest Received December 28, 1984 Presented by Prof. Dr. J. Reimann Ahstract In this paper it is intended to apply the inequality 82 -9 - > P(I~I > IXil) -x~ 1 4 1-to test the examined probability variable C has a distribution function F(x). It is intended to test the hypothesis that the observed probability va-riable e has distribution function F(x). The sample concerning probability variable e having element n yielded by n number of experiments be denoted by Xl' X 2' ••• Xn; while the corre-sponding ordered statistics by xi, x~, ... x~. On the basis of this sample the empirical distribution function Fn(x) of probability variable e is determined 0, if X:;;;; xI Fn(x) =~, if n x~ < x:;;;; Xf+l (1) 1, if x~<x. The value of the empirical distribution function Fn(x) of probability variable e in case of arbitrary X is a probability variable whose possible values 1 2 n are number O~ -, -, ..• - and the corresponding probabilities, i.e. the proba-n n n bility distribution of probability variable ~ = Fn(x) P fe =;) = (:) [F(x)]k [1 - F(x)]n-k, (k = 0,1, ... n). (2) k Namely, the value of Fn(x) in the arbitrary x point is-, if the interval n (-ex>. x) contains exactly k element, of the sample on the other hand, the 140 B. BA.RABAs-L. SEBESTYEN probability of the event that C has a value smaller than x, is P( ~ < x) = F(x). Accordingly, probability variable n~ = nFn(x) (-00 < x < +00) is a proba-bility variable of bionominal distribution having parameters n, p = F(x). Con-sidering this fact its ex-pected value is 1 1 m(~) = -E(n Fn(x») = -nF(x) = F(x), n n (3) and the variance 1 1 a2 = - nF(x) (1 - F(x» = - F(x) (1 - F(x». n2 n (4) If ~ is a probability variable of binomial distribution, then in case of arbitrary positive e we have P(IC - ml > e) -9-' -e2 (5) 4 where m is the expected value of probability variable ~, and a2 is the variance, therefore, here P( /Fn(x) - F(x) I > e) ~ F(x) ~1 - F(x» - n . e2 (6) 4 Applying the inequality to the complementary event P(/Fn(x) - F(x) I < e) > 1 _ F(x) ~ - F(x» -ne2 (7) 4 This statement is proved in the paper of L. Sebestyen entitled "The Chebishev inequality in case of probability variable of special distribution function" . According to the above mentioned, in case of the assumed distribution function F(x) and of certainty 1 - 0; for e the follo,dng critical value is obtain-ed in case of arbitrary x e~~f = ~ V F(x) (1 - F(x») 3 no; 1 1f 4F(x) (1 - F(x» 3~Yn (8) Thereafter, the hypothesis is accepted at a probability level 1 - 0; if the deviation of the assumed distribution function F(x) from the empirical distri-CLEAR FITI'ING EXAMINATION 141 ~iPSl? -1 1 "2 F(x) Fig. 1 bution function Fn(x) is smaller than etl, in an opposite case this hypothesis is rejected. On the basis of equation (8) the number of experiments needed to the prescribed accuracy e, as "well as to the similarly prescribed certainty 1 - ex: level can be determined in advance. Finally, the above mentioned probe has been compared to the Kolmogo-rov-Smirnov probe falling nearest. Be e.g. the certainty-level of 1 - ex:: 90% and the number of experiments n. In this case ex: = 0.1 and 1 1.054 vo.r V V 4F(x) (1 - F(x» = --V 4F(x) (1 - F(x» 3 0.1 n Yn (9) From this equation it can be seen that ekr is depend on x and F(x), resp. and the highest deviation allowed is presented on Fig. 1. In the case of the same level and accuracy, the Kolmogorov-Smirnov 1.23 probe allows a maximal deviation vn at any value of x. In Fig. 1 this has been marked by broken line. If the certainty level is 95 %, i.e. ex: = 0.05, then e1/;05) = 1.39 V 4F(x)(1 - F(x» , Yn (10) 1.36 and the highest deviation allowed by the Kolmogorov- Smirnov probe is vn' Comparison of the two test is given in Fig. 2. Alfred Renyi, in his book entitled "Probability theory" remarks: the Kolmogorov-Smirnov probe investigates the deviation of Fn(x) from F(x) in-dependently of the value of F(x); i.e. the same importance has been attributed to deviation IFn(x) - F(x) I = 0.01 in a point Xl where F(Xl) = 0.5, i.e. the deviation 0.01 means a relative deviation of 2 % and in a point x 2' where F(xz)= = 0.01, i.e. where the absolute deviation 0.01 means a relative deviation of 142 B.BARABA5-L.SEBESTyEN ~ f------I I I I o 1 2 Fig. 2 1 F(x) 100%. Therefore, the boundary distribution of probability variable ,r.;; I Fn(x) - F(x) I rn sup F(x) (ll) is determined and this is the basis of the so-called Kolmogorov-Smirnov-Renyi probe which is an improved version of the Kolmogorov- Smirnov probe. Now, our probe is compared to that of Kolmogorov-Smirnov-Renyi. The certainty level 1 - cc be 90 % and the number of experiments n. As already known, the critical value based on the probe suggested by us is 2 Vn e}2,:ll = 1.054 V 4F(x) (1 - F(x)) • Yn 1~~~~_~ __ ~ __ __ !n0,1 I 0,1 + Fig. 3 F(x) CLEAR FITTING EXAlvI1NATION 143 At the same time, using the Kolmogorov-Smirnov-Renyi probe and involv-ing exclusively those x values for which F(x) 0.1, the highest deviation allow-ed is 6 -F(x) }In The certainty range corresponding to both probes is presented in Fig. 3. The certainty range of Kolmogorov-Smirnov-Renyi probe is demon-strated by the area below the straight line, while that of our probe is to be seen as the area under the ellipse. References 1. PREKOPA; A.: Probability Theory ,~ith Technical Application. * Muszaki Konyvkiad6, Buda-pest, 1972. 2. RENYI, A.: Theory of Probability.* Tankonyvkiad6, Budapest, 1954. Bela BARAB-.\S } H-1~')1 B d D L k ' S ' ;)~ u apest r. u acs EBESTYEN ,. In Hungarian. 

GOODNESS OF FIT WITH THE HELP OF A SHARPENED BERNOULLI INEQUALITY

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GOODNESS OF FIT WITH THE HELP OF A SHARPENED BERNOULLI INEQUALITY

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Periodica Polytechnica (Budapest University of Technology and Economics)