A Study On Q Chart For Short Runs

Suatu peralatan yang penting dalam kawalan mutu ialah carta kawalan Shewhart. Kelemahan carta kawalan Shewhart adalah ia hanya dikhaskan untuk pengeluaran bervolum tinggi. Walau bagaimanapun, sejak kebelakangan m1, wujud kecenderungan di kalangan pengeluar untuk menghasilkan lot bersaiz kecil atau pengeluaran bervolum rendah. Kecenderungan ini disebabkan teknik tepat pada masa, pengeluaran serentak, penyediaan kerja kedaian, pengurangan inventori dalam proses dan nilai kos yang semakin dititikberatkan. An important tool in quality control is the Shewhart control chart. The disadvantage of a Shewhart control chart is that it is only used for high volume manufacturing. However, in recent years, there exist a trend among manufacturers to produce smaller lot sizes or low volume manufacturing. This trend is due to just-in-time techniques (JIT), synchronous productions, job-shop settings and the reduction of inprocess inventories and costs

A comparison of asymptotic and bootstrapping approach in constructing confidence interval of the concentration parameter in von mises distribution

Bootstrap is a resampling procedure for estimating the distributions of statistics based on independent observations. Basically, bootstrapping has been established for the use of parameter estimation of linear data. Thus, the used of bootstrap in confidence interval of the concentration parameter, κ in von Mises distribution which fitted the circular data is discussed in this paper. The von Mises distribution is the ’natural’ analogue on the circle of the Normal distribution on the real line and widely used to describe circular variables. The distribution has two parameters, namely mean direction, μ and concentration parameter, κ, respectively. The confidence interval based on the calibration bootstrap method will be compared with the existing method, confidence interval based on the asymptotic to the distribution of . Simulation studies were conducted to examine the empirical performance of the confidence intervals. Numerical results suggest the superiority of the proposed method based on measures of coverage probability and expected length. The confidence intervals were illustrated using daily wind direction data recorded at maximum wind speed for seven stations in Malaysia. From point estimates of the concentration parameter and the respective confidence interval, we note that the method works well for a wide range of κ values. This study suggests that the method of obtaining the confidence intervals can be applied with ease and provides good estimates

Modelling Rainfall Amounts using Mixed-Gamma Model for Kuantan District

An efficient design of flood mitigation and construction of crop growth models depend upon good understanding of the rainfall process and characteristics. Gamma distribution is usually used to model nonzero rainfall amounts. In this study, the mixed-gamma model is applied to accommodate both zero and nonzero rainfall amounts. The mixed-gamma model presented is for the independent case. The formulae of mean and variance are derived for the sum of two and three independent mixed-gamma variables, respectively. Firstly, the gamma distribution is used to model the nonzero rainfall amounts and the parameters of the distribution (shape and scale) are estimated using the maximum likelihood estimation method. Then, the mixed-gamma model is defined for both zero and nonzero rainfall amounts simultaneously. The formulae of mean and variance for the sum of two and three independent mixed-gamma variables derived are tested using the monthly rainfall amounts from rainfall stations within Kuantan district in Pahang Malaysia. Based on the Kolmogorov-Smirnov goodness of fit test, the results demonstrate that the descriptive statistics of the observed sum of rainfall amounts is not significantly different at 5% significance level from the generated sum of independent mixed-gamma variables. The methodology and formulae demonstrated can be applied to find the sum of more than three independent mixed-gamma variables

Independent Mixed-gamma Variables for Modelling Rainfall

Understanding the rainfall process and characteristics are crucial to the efficient design of flood mitigation and construction of crop growth models. Modelling rainfall is not limited to fit the historical data to a suitable distribution but the model should be able to generate synthetic rainfall data. In this study, we derive sets of formulae of mean and variance for the sum of two and three independent mixed-gamma variables, respectively. Firstly, the positive data is fitted to gamma model marginally and the shape and scale parameters are estimated using the maximum likelihood estimation method. Then, the mixed-gamma model is defined to include zero and positive data. The formulae of mean and variance for the sum of two and three independent mixed-gamma variables are derived and tested using the daily rainfall totals from Pooraka station in South Australia for the period of 1901-1990. The results demonstrate that the values of generated mean and using formula are close to the observed mean. However, the values of the variance are sometimes over-estimated or under-estimated of the observed values. The observed variance is lower possibly due to correlation between the experimental data, that have not been included in the mixed-gamma models. The Kolmogorov–Smirnov and Anderson–Darling goodness of fit tests are used to assess the fit between the observed sum and the generated sum of independent mixed-gamma variables. In both cases, the observed sum is not significantly different from the generated sum of independent mixed-gamma model at 5% significance level. This methodology and formulae derived can be applied to find the sum of more than three independent mixed-gamma variables and the general form of the formulae can be derived

Independent Gamma Variables of McKay Distribution for Rainfall Model

The application of mathematical models of rainfall is crucial in order to have a better understanding in terms of rainfall characteristics. The rainfall models has been used widely to improve water management, to construct hydrological structures and as an input in climatological studies. In this study, the gamma distribution is used to fit the marginals of monthly rainfall data. The parameters of gamma distributions, shape and scale, are estimated using the maximum likelihood estimation method. Then the sum of monthly rainfall amount is modelled using the Mckay ditribution for independent gamma variables

Comparison of Sum of Two Correlated Gamma Variables for Alouini's Model and McKay Distribution

A statistical model for rainfall is useful to describe the relationship between rainfall at a given location and other weather-related variables. It is also able to provide a principled way to quantify the uncertainty that associates rainfall processes, which is crucial to the efficient design of environmental projects and to improve crop production. Various statistical models have been used for rainfall such as a right-skewed distribution including the exponential, gamma or mixed-exponential to model the rainfall intensities and Markov chain model to model rainfall occurrences

Modelling Catchment Rainfall Using Sum of Correlated Gamma Variables

One of the major difficulties in simulating rainfall is the need to accurately represent rainfall accumulations. An accurate simulation of monthly rainfall should also provide an accurate simulation of yearly rainfall by summing the monthly totals. A major problem in this regard is that rainfall distributions for successive months may not be independent. Thus the rainfall accumulation problem must be represented as the summation of dependent random variables. This study is aimed to show if the statistical parameters for several stations within a particular catchment is known, then a weighted sum is used to determine a rainfall model for the entire local catchment. A spatial analysis for the sum of rainfall volumes from four selected meteorological stations within the same region using the monthly rainfall data is conducted. The sum of n correlated gamma variables is used to model the sum of monthly rainfall totals from four stations when there is significant correlation between the stations

A comparison of asymptotic and bootstrapping approach in constructing confidence interval of the concentration parameter in von mises distribution

Bootstrap is a resampling procedure for estimating the distributions of statistics based on independent observations. Basically, bootstrapping has been established for the use of parameter estimation of linear data. Thus, the used of bootstrap in confidence interval of the concentration parameter, κ in von Mises distribution which fitted the circular data is discussed in this paper. The von Mises distribution is the ’natural’ analogue on the circle of the Normal distribution on the real line and widely used to describe circular variables. The distribution has two parameters, namely mean direction, µ and concentration parameter, κ, respectively. The confidence interval based on the calibration bootstrap method will be compared with the existing method, confidence interval based on the asymptotic to the distribution of . Simulation studies were conducted to examine the empirical performance of the confidence intervals. Numerical results suggest the superiority of the proposed method based on measures of coverage probability and expected length. The confidence intervals were illustrated using daily wind direction data recorded at maximum wind speed for seven stations in Malaysia. From point estimates of the concentration parameter and the respective confidence interval, we note that the method works well for a wide range of κ values. This study suggests that the method of obtaining the confidence intervals can be applied with ease and provides good estimates

Applied Statistics Module, Version 2: Science, Engineering, Technology

This module is intended to facilitate teaching and learning of Statistics courses in colleges and universities at degree level. It is specially tailored for students of science, engineering and technology. This book provides examples and exercises that present important ideas of statistics in a realistic setting to show connections between theory and application in industry and scientific research. The materials in this module also integrate well with computer software packages especially in the chapters on descriptive statistics, hypothesis testing, analysis of variance and regression models. The use of Microsoft Excel is emphasised in this module