Testing goodness-of-fit with interval data

The main terms of interval data analysis was initially founded the measurement theory in metrology where an interval uncertainty is naturally introduced. It is expected that every observation is measured by an instrument with absolute error Д. Thus, if the precise value of an observed response is x, measurement error is e e [-Д, Д], then the measurement is equal to x = x + e . In this case, we deal with a usual complete sample X_n = {Xj,...,X_n}. Nevertheless, the measurement can be represented as an interval (x -Д, x + Д) = (L, R). In this case, for the sample of n observations we obtain an interval sample of the form In ={( A, R1),..., (Ln, Rn)}. The main idea of nonparametric estimation of the distribution function with interval data is based on maximization of the loglikelihood function ln L (In )=£ ln (F (R) - F (L_t)) i =1 at the boundary points of observations L_i, R_i, i = 1, n, under condition of monotonicity of the distribution function. The Turnbull and ICM algorithms are used for calculation of the nonparametric estimate of the distribution function with interval data. The accuracy of the estimates calculation is the same for both algorithm, but the computing time is less for the ICM algorithm. Unknown distribution parameters can be estimated by the maximum likelihood method, which is based on the maximization of likelihood function by parameter 8 : L (Inl8) = n (F(Ri | 8) - F(Li | 8)). i=1 Thus, the maximum likelihood estimates can be written as 8 = arg max ln L (I_n | 8) . 8e© In this paper, the modifications of the classical goodness-of-fit tests for composite hypothesis H₀ : F(t) e {F₀(t; 8), 8 e ©} have been proposed. The main idea of this modification is based on the usage of nonparamet- ric estimate of the distribution function, obtained by the ICM algorithm, instead of the empirical distribution function. In this case, we have the test statistic of the Kolmogorov type as Dn = sup \F_n(t)-F₀(t,8), <ffl2 = jf (Fn (t)- F0 (t, 8)) (t, 8 ) 0 and the statistic of Anderson-Darling type test as *=<«>-0 where F_n(t) is the nonparametric estimate of the distribution function by the interval data, 0 < x₀ < x₁ <... < x_m are ordered different values L_i and R_i, i = 1, n . We have formulated the sequence of steps for estimation of the p-value for the proposed tests. The hypothesis is not rejected if the obtained p-value is larger than the significance level а . As an example, we have tested the normality hypothesis by the interval sample of consumer demand prices for bio-energy drink SPC "SAVA". It has been shown that there is no reason for rejecting the hypothesis of normality of consumer demand prices.

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