Tests for One or Two Outliers in Normal Samples with Unknown Variance: A Correction

In a paper in this Journal, McMillan (1971) compared three procedures for the detection of outliers in samples from a Normal population. One of these was a sequential application of a maximum residual test. One step in McMillan's derivation of the probability of detecting two outliers using this test is incorrect. A correct derivation is presented in this note and some numerical results in McMillan's paper are appropriately amended. McMillan's qualitative statements on the performance of the test are essentially unaffected.     

Investigation on the Mean Field Assumptions for Switching Field Distribution Measurements

Since the transition of the research focus from longitudual to perpendicular recording system in early 2000, there has been significant progress in understanding and developing metrologies to characterize the switching field distribution (SFD). However, most of the metrologies developed for characterizing SFD have relied on mean field assumptions. In this paper, we investigate the mean field assumptions commonly used in the SFD metrologies. For micromagnetic simulation data without intergranular exchange interactions, we find that the interaction fields are independent of the local environment of the spins, which suggests that the mean field approximation is applicable. Moreover, the dependency between the interaction field and the magnetization is rather linear with and without thermal fluctuations. For the experimental data obtained from a sample with moderate intergranular exchange interactions, we find that the mean field approximation is reasonable in situations with thermal fluctuation. But for the case without thermal fluctuation, such approximation might be called into question.     

Simplified Estimators for the Normal Distribution When Samples Are Singly Censored or Truncated

Easily computed estimates of the mean and variance of a Normal distribution obtained from samples which are truncated or censored are described. The estimates require only a single auxiliary function which is conveniently tabulated. Worked examples are given for sample data subject to left and right truncation. Additional examples are given for left and right censoring for the two cases in which either the point of censoring is fixed or the number of censored items is known.     

Testing the Mean and Standard Deviation of a Normal Distribution Using Quantiles

Assuming a normally distributed population and a large sample size, three tests of hypotheses are presented using sample quantiles. First, we test whether μ = μ₁ and σ = σ₁ or μ = μ₂ and σ = σ₂, using one optimum quantile. Then test statistics and acceptance regions are given for testing the mean and standard deviation independently. The efficiency of each test is given relative to the best test using all the sample values.     

Estimation of Variance Components in Two-Stage Nested Designs With Composite Samples

If the cost of measurement is high, it is often expedient to composite several samples before taking measurements. Optimal designs for estimating (individually and jointly) the mean and variance components in two-stage nested designs are discussed, using the so-called analysis of variance estimation procedure. This traditional procedure is then compared numerically with two other unbiased estimation procedures; specific conditions are given whereby a particular procedure is superior to the other procedures.     

The Folded Normal Distribution: Accuracy of Estimation By Maximum Likelihood

This paper is a sequel to two previous papers on the subject of the Folded Normal Distribution appearing in Technometrics, S, pp. 543–550 and 551–562, (1961). In these earlier papers methods for estimating the parameters of the Folded Normal Distribution were proposed. Thii paper gives approximations for the standard errors for the maximum liklihood estimates of these parameters.     

Maximum Likelihood Estimation in the Weibull Distribution Based On Complete and On Censored Samples

This paper is concerned with the two-parameter Weibull distribution which is widely employed as a model in life testing. Maximum likelihood equations are derived for estimating the distribution parameters from (i) complete samples, (ii) singly censored samples and (iii) progressively (multiple) censored samples. Asymptotic variance-covariance matrices are given for each of these sample types. An illustrative example is included.     

Estimation of the Shape Parameter of the Weibull Distribution Using Linear Regression Methods: Non‐Censored Samples

The two‐parameter Weibull distribution is one of the most widely applied probability distributions, particularly in reliability and lifetime modelings. Correct estimation of the shape parameter of the Weibull distribution plays a central role in these areas of statistical analysis. Many different methods can be used to estimate this parameter, most of which utilize regression methods. In this paper, we presented various regression methods for estimating the Weibull shape parameter and an experimental study using classical regression methods to compare the results of the methods. A complete list of the parameter estimators considered in this study is as follows: ordinary least squares (OLS), weighted least squares (WLS, Bergman, F&T, Lu), non‐parametric robust Theil's (Theil) and weighted Theil's (WeTheil), robust Winsorized least squares (WinLS), and M‐estimators (Huber, Andrew, Tukey, Cauchy, Welsch, Hampel and Logistic). Estimator performances were compared based on bias and mean square error criteria using Monte‐Carlo simulations. The simulation results demonstrated that for small, complete, and non‐outlier data sets, the Bergman, F&T, and Lu estimators are more efficient than the others. When the data set contains one or two outliers in the X direction, Theil is the most efficient estimator. Copyright © 2012 John Wiley & Sons, Ltd.     

The Robust Regression Methods for Estimating of Finite Population Mean Based on SRSWOR in Case of Outliers

The ordinary least square (OLS) method is commonly used in regression analysis. But in the presence of outlier in the data, its results are unreliable. Hence, the robust regression methods have been suggested for a long time as alternatives to the OLS to solve the outliers problem. In the present study, new ratio type estimators of finite population mean are suggested using simple random sampling without replacement (SRSWOR) utilizing the supplementary information in Bowley’s coefficient of skewness with quartiles. For these proposed estimators, we have used the OLS, Huber-M, Mallows GM-estimate, Schweppe GM-estimate, and SIS GM-estimate methods for estimating the population parameters. Theoretically, the mean square error (MSE) equations of various estimators are obtained and compared with the OLS competitor. Simulations for skewed distributions as the Gamma distribution support the results, and an application of real data set containing outliers is considered for illustration.     

Estimation of the Parameters of Two Parameter Exponential Distributions from Censored Samples

In life testing a two parameter exponential distribution is often required to provide an adequate representation of the observed results. Sufficient statistics and interval estimates for the parameters are given for the case of censored data. Numerical examples are provided.     

Maximum likelihood Estimation of Parameters in the Inverse Gaussian Distribution,With Unknown Origin

Maximum likelihood estimation is applied to the three-parameter Inverse Gaussian distribution, which includes an unknown shifted origin parameter. It is well known that for similar distributions in which the origin is unknown, such as the lognormal, gamma, and Weibull distributions, maximum likelihood estimation can break down. In these latter cases, the likelihood function is unbounded and this leads to inconsistent estimators or estimators not asymptotically normal. It is shown that in the case of the Inverse Gaussian distribution this difticulty does not arise. The likelihood remains bounded and maximum likelihood estimation yields a consistent estimator with the usual asymptotic normality properties. A simple iterative method is suggested for the estimation procedure. Numerical examples are given in which the estimates in the Inverse Gaussian model are compared with those of the lognormal and Weibull distributions.     

Generating a Variable from the Tail of the Normal Distribution

The performance of three rules for dealing with outliers in small samples of size n from the normal distribution N(μ, σ²) is investigated when the primary objective of sampling is to obtain an accurate estimate of σ². It is assumed that at most one observation in the sample may be biased, arising from either N(μ + aσ, σ²) or N(μ, (1 + b) σ²). Performance of each rule is measured in terms of “Protection”, the fractional decrease in MSE obtained by using the rule when a biased observation actually is present in the sample. Numerical results have been obtained for n 5 ≤ 11 when μ is known and n = 3 when μ is unknown.     

Optimization of Liquid Scintillation Counter for Tritium Estimation in Water Samples

Radiochemistry - The liquid scintillation counting procedure for tritium quantification in environmental samples was optimized with respect to sample to cocktail volume ratio, fixing of region of...     

Distribution of Badial Error in the Bivariate Elliptical Normal Distribution

This note is concerned with the distribution of radial error in the general bivariate normal distribution with correlated and heteroscedastic errors. Measures of location and of dispersion of the radial error are obtained.     

Maximum-Likelihood Estimation of the Parameters of Gamma and Weibull Populations from Complete and from Censored Samples

Iterative procedures are given for joint maximum-likelihood estimation, from complete and censored samples, of the three parameters of Gamma and of Weibull populations. For each of these populations, the likelihood function is written down, and the three maximum-likelihood equations are obtained. In each case, simultaneous solution of these three equations would yield joint maximum-likelihood estimators for the three parameters. The iterative procedures proposed to solve the equations are applicable to the most general case, in which all three parameters are unknown, and also to special cases in which any one or any two of the parameters are known. Numerical examples are worked out in which the parameters are estimated from the first m failure times in simulated life tests of n items (m ≤ n), using data drawn from Gamma and Weibull populations, each with two different values of the shape parameter.     

The Folded Normal Distribution: Two Methods of Estimating Parameters from Moments

The general formula for the rth moment of the folded normal distribution is obtained, and formulae for the first four non-central and central moments are calculated explicitly.

To illustrate the mode of convergence of the folded normal to the normal distribution, as μ/σ = θ increases, the shape factors β _f1 and β _f2 were calculated and the relationship between them represented graphically.

Two methods, one using first and second moments (Method I) and the other using second and fourth moments (Method II) of estimating the parameters μ and σ of the parent normal distribution are presented and their standard errors calculated. The accuracy of both methods, for various values of θ, are discussed.     

Linear Order Statistic Estimation for the Two-Parameter Weibull and Extreme-Value Distributions from Type II Progressively Censored Samples

Point estimation for the scale and location parameters of the extreme-value (Type I) distribution by linear functions of order statistics from Type II progressively censored samples is investigated. Four types of linear estimators are considered: the best linear unbiased (BLU), an approximation to the BLU, unweighted regression, and a linearized maximum likelihood. Linear transformations of the estimators are also considered for reducing mean square errors. Exact bias, variance, and mean square error comparisons of the estimators are made for several censoring patterns. Since the natural logarithms of Weibull variates have extreme-value distributions, the investigation is applicable to estimation for Weibull distributions.     

Estimation of Parameters in the Weibdl Distribution

A new method of estimation is used to obtain two simple estimators of the parameters in a Weibull distribution. These estimators are similar to the estimators given by Gumbel, Miller and Freund, and Menon. Some useful properties of these estimators are developed which make it practical to use Monte Carlo methods to determine the variances and biases of the estimators for various sample sizes. Comparisons of the estimators can be made and unbiasing factors calculated in some cases. The variances of the estimators are also compared to the Cramer-Rao lower bounds for regular unbiased estimators.     

The Correct Use of the Sample Mean Absolute Deviation in Confidence Intervals for a Normal Variate

In short routine experiments involving small samples from a normal distribution, it is often found to be more convenient to use the sample mean absolute deviation (m) rather than the sample standard deviation (S) in forming confidence intervals. Approximate and exact methods for correctly using m in confidence intervals are given. The expected lengths of the intervals using m are then compared with the expected lengths of the intervals using S. The results indicate that the increase in expected length using m rather than S is insignificant, whereas there is a decrease in expected length using m when one can take an additional observation.