Kontrola kvaliteta praćenjem centralne tendencije negausovih slučajnih veličina

Abstract

Razmatran je dizajn kontrolnih grafika za pracenje centralne tendencije negausovih slucajnih velicina sa simetricnim ili pozitivno asimetricnim raspodelama. U slucaju negausovih simetricnih raspodela, u ovoj disertaciji je predložen modifikovani kontrolni grafik uzoracke sredine. Za odabranu Studentovu, Laplasovu, logisticku i uniformnu raspodelu izracunata je teorijska raspodela standardizovane uzoracke sredine i aproksimirana je Pirsonovom raspodelom II tipa ili Pirsonovom raspodelom VII tipa. Izracunate su širina kontrolnih granica i moc kontrolnog grafika uzoracke sredine, za datu verovatnocu greške prve vrste. Dobijeni rezultati ukazuju da odgovarajuca Pirsonova raspodela može poslužiti kao veoma dobra aproksimacija raspodele standardizovane uzoracke sredine. Za primenu kontrolnog grafika uzoracke sredine u praksi, pore ene su mere koeficijenta spljoštenosti i upotreba predloženog grafika ilustrovana je na podacima. U slucaju pozitivno asimetricnih raspodela, u ovoj disertaciji je predložen jednostrani kontrolni grafik medijane za pracenje centralne tendencije karakteristike kvaliteta. Za odabranu eksponencijalnu, gama i Vejbulovu raspodelu izvedena je teorijska raspodela uzoracke medijane i aproksimirana je Pirsonovom raspodelom I tipa ili Pirsonovom raspodelom VI tipa. Dobijene vrednosti za gornju kontrolnu granicu i moc kontrolnog grafika medijane za teorijsku raspodelu medijane i odgovarajucu Pirsonovu raspodelu su vrlo bliske. Za primenu predloženog kontrolnog grafika medijane, upore ene su mere koeficijenata asimetrije i spljoštenosti i zatim je konstruisan kontrolni grafik medijane za date podatke. Pored statistickog dizajna kontrolnih grafika za pracenje centralne tendencije negausovih slucajnih velicina, razmatran je i njihov optimalni ekonomsko-statisticki dizajn. U ovoj disertaciji je predložena upotreba genetskog algoritma za minimizaciju funkcije ocekivanog gubitka pod ogranicenjima. Odabrane su iste simetricne raspodele kao u slucaju statistickog dizajna kontrolnog grafika uzoracke sredine, odnosno iste pozitivno asimetricne raspodele kao u slucaju statistickog dizajna kontrolnog grafika medijane. Za sve odabrane raspodele karakteristike kvaliteta, odgovarajuca Pirsonova raspodela daje rezultate vrlo bliske rezultatima za teorijsku raspodelu standardizovane uzoracke sredine (uzoracke medijane).

Design of control chart for monitoring central tendency of nongaussian random variables with symmetric or positively skewed distributions is considered. In the case of nongaussian symmetric distributions, modified X bar control chart is proposed in this dissertation. For chosen Student, Laplace, logistic and uniform distributions, theoretical distribution of the standardized sample mean is calculated and approximated with Pearson type II or Pearson type VII distributions. Width of control limits and power of X bar control chart are established, for a given probability of type I error. The results imply that the corresponding Pearson distribution represents very good approximation of the distribution of the standardized sample mean. For implementation of X bar control chart in practice, measures of sample kurtosis are compared and the usage of proposed chart is illustrated on given data. In the case of positively skewed distributions, one sided median control chart for monitoring central tendency of quality characteristics is proposed in this dissertation. For chosen exponential, gamma and Weibull distributions, theoretical distribution of sample median is calculated and approximated with Pearson type I or Pearson type VI distributions. Calculated values of upper control limits and power of median control chart for theoretical distribution of sample median and corresponding Pearson distribution are very close. For implementation of median control chart in practice, measures of sample skewness and sample kurtosis are compared and then proposed median chart is constructed for given data. Besides the statistical design of control charts for monitoring central tendency of nongaussian random variables, their optimal economic statistical design is also considered. Use of genetic algorithms for constrained minimization of expected loss function is proposed in this dissertation. Same symmetric distributions as in the case of statistical design of the X bar control chart and positively skewed distributions as in the case of statistical design of median control chart are chosen. For all chosen distributions of quality characteristic, a corresponding Pearson distribution gives results very close to results based on the theoretical distribution of the standardized sample mean (sample median).