Skewness, Kurtosis, Estimates of moments,

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In this paper we estimate R = P {X ≤ Y } when X and Y are independent random variables from geometric and Poisson distribution respectively. We find maximum likelihood estimator of R and its asymptotic distribution. This asymptotic distribution is used to construct asymptotic confidence intervals. A procedure for deriving bootstrap confidence intervals is presented. UMVUE of R and UMVUE of its variance are derived and also the Bayes estimator of R for conjugate prior distributions is obtained. Finally, we perform a simulation study in order to compare these estimators.

Normality testing remains an important issue for researchers, despite many solutions that have been published and in use for a long time. There is a need for testing normality in many areas of research and application, among them in Quality control, or more precisely, in the investigation of Shewhart-type control charts. We modified some of our previous results concerning control charts by using the empirical distribution function, proper choice of quantiles and a zone function that quantifies the discrepancy from a normal distribution. That was our approach in constructing a new normality test that we present in this paper. Our results show that our test is more powerful than any other known normality test, even in the case of alternatives, with small departures from normality and for small sample sizes. Additionally, many test statistics are sensitive to outliers when testing normality, but that is not the case with our test statistic. We provide a detailed distribution of the test statistic for the presented test and comparable power analysis with highly illustrative graphics. The discussion covers both the cases for known and for estimated parameters.

We have assembled and assessed the statistical procedure which is capable to objectively explore influence of the Danube's major tributaries (the Rivers Tisa, Sava, and Velika Morava) to its eco-chemical status. Procedure contains several tests for measurement of central tendencies: one-way analysis of variance (ANOVA), repeated measures ANOVA, and nonparametric Kruskal-Wallis and Mann-Whitney tests. Various nuisance factors, (outliers, departures from normality, seasonality, and heteroscedasticity) which are present in large data bases, affect the objectivity of central tendency tests; therefore, it was important not only to estimate their robustness, but also to apply proper procedures for detection of the nuisance factors (Grubbs', generalized ESD-extreme Studentized deviate, Kolmogorov-Smirnov, Shapiro-Wilk, turning point, Wald-Wolfowitz runs, Kendall rank, and Levene's tests) and to mitigate their influence (outlier exclusion, Box-Cox, and logarithmic transformations). The analysis of selected eco-chemical parameters: biological oxygen demand-5, chemical oxygen demand, UV extinction at 254 nm, dissolved oxygen, oxygen saturation, total dissolved solids, electrical conductivity, suspended matter, total phosphorus, phosphates, nitrates, ammonia, pH, total alkalinity, m-2p alkalinity, CO2, and temperature, was performed for 15 years period. The Tisa was the most polluted tributary, but its pollution load was not substantial enough to exceed the Danube self-purification potential. The City of Belgrade was also identified as serious pollution source. Assessment of assembled statistical procedure, which was based on the real environmental data, indicates that proposed tests are sufficiently robust to the observed level of nuisance factors with the exception of pronounced seasonality.

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In economic statistical design of a control chart, the economic-loss function is minimized subject to a constrained minimum value of power, maximum value of probability of false alarms and average time to signal an expected shift. This paper is concerned with the optimum economic statistical design of the X bar chart when quality characteristic has non-normal symmetric distribution. We considered three types of distributions: Student distribution, standard Laplace distribution and logistic distribution. For each of these distributions, we calculated theoretical distribution of standardized sample mean (or its best approximation) and approximated it with normal, Pearson VII and Johnson SU distributions. For considered example, constrained minimization of expected loss function was done using genetic algorithm in statistical software R. We compared results of economic statistical design of X-bar chart for theoretical distribution of standardized sample mean with the results for normal, Pearson and Johnson distributions. We found that, for all chosen distributions of quality characteristic, Pearson VII distribution and Johnson SU distribution give results very close to results based on theoretical distribution of standardized sample mean, while normal distribution gives much worse fit.

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