A simple approximation for the doubly noncentral t-distribution

A simple approximation for the doubly noncentral t-distribution, based upon the Fieller-Geary Theorem (1930) and approximate normality of the square root of the noncentral chi-square variable observed by Patnaik (1949), is developed, This approximation and an Edgeworth series expansion associated with it are evaluated. The simple approximation is seen to be reasonably accurate for most practical purposes.     

A partitioning of the power in the one sample Student t-test

The power of Student's t-test is partitioned to determine* the proportion of time the sample standard deviation is too small or the mean is misestimated given that rejection of the hypothesis under test has occurred     

The noncentral bivariate chisquared distribution and extensions

The distribution of certain correlated noncentral chisquared variates P, Q, is termed the noncentral bivariate chisquared distribution. Moment generating functions of the distributions of (P, Q), (P+Q) and other quadratic forms have been obtained. A relationship to the linear case of the noncentral Wishart distribution is indicated. Convolution properties and applications are presented.     

Probability-Based Process Capability Indices

This article deals with alternative process capability indices (PCIs) to traditional basic PCIs C _p , C _pk , and C _pm based on different fraction conforming type of probabilities. In view of various problems of constructing capability indices for univariate as well as multivariate set up, these alternative PCIs are very useful as compared to C _p , C _pk , and C _pm . Computing aspects of proposed PCIs are discussed for normal and non normal processes when process tolerance is symmetric as well as asymmetric. Generalization of these PCIs for multivariate set up is also discussed. Some simulation study results and real life problems are given for applications of proposed PCIs.     

A likelihood ratio test for the equality of proportions of two normal populations

In sampling inspection by variables, an item is considered defective if its quality characteristic Y falls below some specification limit L₀. We consider switching to a new supplier if we can be sure that the proportion of defective items for the new supplier is smaller than the proportion defective for the present supplier.

Assume that Y has a normal distribution. A test for comparing these proportions is developed. A simulation study of the performance of the test is presented.     

Distribution of the quotient of noncentral normal random variables

The Mellin convolution is used to derive in analytical form an exact 3-parameterprobabilitydensity function of the quotient of two noncentral normal random variables. In contrast with the 5-parameter probability density function previously derivedby Fieller (1932) and Hinkley (1969), this 3-parameter probability density function is feasible for computer evaluation of the mean and cumulative distribution function, which are needed, for example, when dealing with estimation and distribution problems in regression analysis and sampling theory. When the normal variables are independent, the probability density function reduces to a 2-parameter function, for which a computer program is operational. An illustrative example is given for one set of parameters when the normal variables are independent, in which themean and functional form of the probability density function are presented, together with a brief tabulation of the probability density function.     

An analysis of the type I error in the one sample Student t-test

In the one-sample Student t-test, the occurrence of a type-I error is dependent on the estimates of the mean and standard deviation for a fixed sample size, n. The statistic can achieve significance either by the sample mean being too different from the hypothesized mean or by the sample standard deviation being too small. The critical region is partitioned to determine the characteristics of samples in the critical region, assuming the null hypothesis is true. As might be conjectured from the use of the t-statistic, mis-estimation of the mean is shown to be the predominant characteristic of samples in the critical region for sample sizes larger than 20 and significance level greater than 0.01. Underestimation of the variance, unless accompanied by a misestimation of the mean, is a far less frequent characteristic of critical region samples.     

Tables of confidence limits on the tail area of the normal distribution

Tables are given of confidence limits on tail areas, γ, of the normal distribution, where γ = P{Y ≥ L}, and where L is a given number, and Y is normally distributed with unknown mean, μ, and unknown variance, σ².     

A multi sample test for the equality of coefficients of variation in normal populations

Two tests are derived for the hypothesis that the coefficients of variation of k normal populations are equal. The k samples may be of unequal size. The first test is the likelihood ratio test with the usual X²-approximation. A simulation study shows that the small sample behaviour under the null hypothesis is unsatisfactory. An alternative test, based on the sample coefficients of variation, appears to have somewhat better properties.     

A bivariate generalized geometric distribution with applications

This paper proposes a bivariate version of the univariate discrete generalized geometric distribution considered by Gómez–Déniz (2010 Gómez–Déniz, E. (2010). Another generalization of the geometric distribution. Test 19:399–415.[Crossref], [Web of Science ®] , [Google Scholar]). The proposed bivariate distribution can have a positive or negative correlation coefficient which can be used for modeling bivariate-dependent count data. After discussing some of its properties, maximum likelihood estimation is discussed. Two illustrative examples are given for fitting and demonstrating the usefulness of the new bivariate distribution proposed here.     

不同标准差估计方法下过程能力指数估计量的比较研究

估计量是统计推断的基础,通常无偏性是对一个好的估计量的基本要求。通过严格的数学推导,证明人们现在提出的过程能力指数（Cp）的估计量都是有偏的,且都有高估Cp的倾向;之后构造了Cp的两个无偏估计量;探讨了这两个无偏估计量的估计效率;最后通过试算和比较,认为当样本容量n较大,同时估计精度又不要求太高时,可直接使用p作为Cp的估计量;但在样本容量较小,或者对估计精度要求很高的场合,则必须选择p（D）作为Cp的估计量。     

A skew extension of the t-distribution, with applications

Summary. A tractable skew t -distribution on the real line is proposed. This includes as a special case the symmetric t -distribution, and otherwise provides skew extensions thereof. The distribution is potentially useful both for modelling data and in robustness studies. Properties of the new distribution are presented. Likelihood inference for the parameters of this skew t -distribution is developed. Application is made to two data modelling examples.     

A new generation of process capability indices

In quality control, we may confront imprecise concepts. One case is a situation in which upper and lower specification limits (SLs) are imprecise. If we introduce vagueness into SLs, we face quite new, reasonable and interesting processes, and the ordinary capability indices are not appropriate for measuring the capability of these processes. In this paper, similar to the traditional process capability indices (PCIs), we develop a fuzzy analogue by a distance defined on a fuzzy limit space and introduce PCIs, where instead of precise SLs we have two membership functions for upper and lower SLs. These indices are necessary when SLs are fuzzy, and they are helpful for comparing manufacturing process with fuzzy SLs. Some interesting relations among these introduced indices are proved. Numerical examples are given to clarify the method.     

A further note on the bivariate normal distribution

It is indicated to what extent the conditional normality of the distribution of one comnonent of a bivariate random vector given the value of the other component together with a restricted type of conditional normality or the marginal normality for the other component is equivalent to the bivariate normality of this random vector.     

A process capability index for discrete processes

Perakis and Xekalaki 2002, A process capability index that is based on the proportion of conformance. Journal of Statistical Computation and Simulation, 72(9), 707–718. introduced a process capability index that is based on the proportion of conformance of the process under study and has several appealing features. One of its advantages is that it can be used not only for continuous processes, as is the case with the majority of the indices considered in the literature, but also for discrete processes as well. In this article, the use of this index is investigated for discrete data under two alternative models, which are frequently considered in statistical process control. In particular, distributional properties and estimation of the index are considered for Poisson processes and for processes resulting in modeling attribute data. The performance of the suggested estimators and confidence limits is tested via simulation.