Setting confidence intervals by ratio estimator

For the survey population total of a variable y when values of an auxiliary variable x are available a popular procedure is to employ the ratio estimator on drawing a simple random sample without replacement (SRSWOR) especially when the size of the sample is large. To set up a confidence interval for the total, various variance estimators are available to pair with the ratio estimator. We add a few more variance estimators studded with asymptotic design-cum-model properties. The ratio estimator is traditionally known to be appropriate when the regression of y on x is linear through the origin and the conditional variance of y given x is proportional to x. But through a numerical exercise by simulation we find the confidence intervals to fare better if the regression line deviates from the origin or if the conditional variance is disproportionate with x. Also, comparing the confidence intervals using alternative variance estimators we find our newly proposed variance estimators to yield favourably competitive results.     

Ratio estimator for the population mean using ranked set sampling

We adapt the ratio estimation using ranked set sampling, suggested by Samawi and Muttlak (Biometr J 38:753–764, 1996), to the ratio estimator for the population mean, based on Prasad (Commun Stat Theory Methods 18:379–392, 1989), in simple random sampling. Theoretically, we show that the proposed ratio estimator for the population mean is more efficient than the ratio estimator, in Prasad (1989), in all conditions. In addition, we support this theoretical result with the aid of a numerical example.      

On kernel density estimation based on different stratified sampling with optimal allocation

Kernel density estimation is probably the most widely used non parametric statistical method for estimating probability densities. In this paper, we investigate the performance of kernel density estimator based on stratified simple and ranked set sampling. Some asymptotic properties of kernel estimator are established under both sampling schemes. Simulation studies are designed to examine the performance of the proposed estimators under varying distributional assumptions. These findings are also illustrated with the help of a dataset on bilirubin levels in babies in a neonatal intensive care unit.     

Bootstrap confidence intervals for the coefficient of quartile variation

ABSTRACT

In non-normal populations, it is more convenient to use the coefficient of quartile variation rather than the coefficient of variation. This study compares the percentile and t-bootstrap confidence intervals with Bonett's confidence interval for the quartile variation. We show that empirical coverage of the bootstrap confidence intervals is closer to the nominal coverage (0.95) for small sample sizes (n = 5, 6, 7, 8, 9, 10 and 15) for most distributions studied. Bootstrap confidence intervals also have smaller average width. Thus, we propose using bootstrap confidence intervals for the coefficient of quartile variation when the sample size is small.     

Confidence interval estimation for the population coefficient of variation using ranked set sampling: a simulation study

In this paper, an evaluation of the performance of several confidence interval estimators of the population coefficient of variation (τ) using ranked set sampling compared to simple random sampling is performed. Two performance measures are used to assess the confidence intervals for τ, namely: width and coverage probabilities. Simulated data were generated from normal, log-normal, skew normal, Gamma, and Weibull distributions with specified population parameters so that the same values of τ are obtained for each distribution, with sample sizes n=15, 20, 25, 50, 100. A real data example representing birth weight of 189 newborns is used for illustration and performance comparison.     

Sample size needed to get given ratio of endpoints for confidence interval of standard deviation in a normal distribution

Confidence interval (CI) for a standard deviation in a normal distribution, based on pivotal quantity with a Chi-square distribution, is considered. As a measure of CI quality, the ratio of its endpoints is taken. There are given formulas for sample sizes so that this ratio does not exceed a fixed value. Both equally tailed and minimum ratio of endpoint CIs are considered.     

Comparing confidence intervals for the logarithmic series distribution

It is assumed that a small random sample of fixed size n is drawn from a logarithmic series distribution with parameter θ and that it is desired to estimate θ by means of a two-sided confidence interval. In this note Crow's system of confidence intervals is compared, in shortness of intervals, with Clopper and Pearson's, and the corresponding randomized counterparts.     

Estimation of the parameters of Downton's bivariate exponential distribution using ranked set sampling scheme

The parameters of Downton's bivariate exponential distribution are estimated based on a ranked set sample. Parametric and nonparametric methods are considered. The suggested estimators are compared to the corresponding ones based on simple random sampling. It turns out that some of the suggested estimators are significantly more efficient than the ones based on simple random sampling.     

Lifetime performance index based on ranked set sampling

In this article, we assume that the lifetimes of products follow a one-parameter exponential distribution and use both conjugate and Jeffreys prior distributions for finding a Bayes estimator based on the RSS scheme for lifetime performance index CL. We also obtained maximum likelihood and adhac estimators for CL. Monte Carlo simulation study is done for comparing the obtained estimators in two sampling schemes SRS and RSS. Fisher information has been utilized to obtain lower confidence bound of CL for both schemes in asymptotically state and use this bounds for hypothesis testing of CL.     

Monitoring the coefficient of variation using a variable sample size and sampling interval control chart

This article proposes a CV chart by using the variable sample size and sampling interval (VSSI) feature to improve the performance of the basic CV chart, for detecting small and moderate shifts in the CV. The proposed VSSI CV chart is designed by allowing the sample size and the sampling interval to vary. The VSSI CV chart's statistical performance is measured by using the average time to signal (ATS) and expected average time to signal (EATS) criteria and is compared with that of existing CV charts. The Markov chain approach is employed in the design of the chart.     

Two new confidence intervals for the coefficient of variation in a normal distribution

In this article we introduce an approximately unbiased estimator for the population coefficient of variation, τ, in a normal distribution. The accuracy of this estimator is examined by several criteria. Using this estimator and its variance, two approximate confidence intervals for τ are introduced. The performance of the new confidence intervals is compared to those obtained by current methods.     

Change of origin and scale in ratio and difference methods of estimation in sampling

This paper discusses two estimators of the mean of a finite population based on a simple random sample from it, when supplementary information on a variable positively correlated with the variable of interest is available. Simultaneous reductions in absolute bias and mean square error of the estimator are seen as compared with those of the traditional estimator in the ratio method of estimation. The suggested estimators are simple for computation and there is no appreciable increase in the cost as well.     

Inference of the derivative of nonparametric curve based on confidence distribution

Abstract

This paper focuses on inference based on the confidence distributions of the nonparametric regression function and its derivatives, in which dependent inferences are combined by obtaining information about their dependency structure. We first give a motivating example in production operation system to illustrate the necessity of the problems studied in this paper in practical applications. A goodness-of-fit test for polynomial regression model is proposed on the basis of the idea of combined confidence distribution inference, which is the Fisher’s combination statistic in some cases. On the basis of this testing results, a combined estimator for the p-order derivative of nonparametric regression function is provided as well as its large sample size properties. Consequently, the performances of the proposed test and estimation method are illustrated by three specific examples. Finally, the motivating example is analyzed in detail. The simulated and real data examples illustrate the good performance and practicability of the proposed methods based on confidence distribution.     

Generalized confidence interval estimation for the mean of delta-lognormal distribution: an application to New Zealand trawl survey data

Highly skewed and non-negative data can often be modeled by the delta-lognormal distribution in fisheries research. However, the coverage probabilities of extant interval estimation procedures are less satisfactory in small sample sizes and highly skewed data. We propose a heuristic method of estimating confidence intervals for the mean of the delta-lognormal distribution. This heuristic method is an estimation based on asymptotic generalized pivotal quantity to construct generalized confidence interval for the mean of the delta-lognormal distribution. Simulation results show that the proposed interval estimation procedure yields satisfactory coverage probabilities, expected interval lengths and reasonable relative biases. Finally, the proposed method is employed in red cod densities data for a demonstration.     

On an asymptotic relative efficiency concept based on expected volumes of confidence regions

ABSTRACT

The paper deals with an asymptotic relative efficiency concept for confidence regions of multidimensional parameters that is based on the expected volumes of the confidence regions. Under standard conditions the asymptotic relative efficiencies of confidence regions are seen to be certain powers of the ratio of the limits of the expected volumes. These limits are explicitly derived for confidence regions associated with certain plugin estimators, likelihood ratio tests and Wald tests. Under regularity conditions, the asymptotic relative efficiency of each of these procedures with respect to each one of its competitors is equal to 1. The results are applied to multivariate normal distributions and multinomial distributions in a fairly general setting.     

Influence Functions for the Coefficient of Variation,Its Inverse,and CV Comparisons

Expressions are found for the influence function of the coefficient of variation, CV, and its reciprocal, the signal to noise ratio. These functions are free of units, which permits the comparison of the values of the CVs of continuous positive distributions to a perturbation by a small amount of probability at x. For a CV ≤0.5, the influence function response will be negative, of modest size, for values of x near E(X). For such values of a CV and of x, the influence function for 1/CV will be positive and its values will be substantial. These results imply similar behavior by the sample coefficient of variation or its reciprocal, which is supported by simulation studies in the literature. Values of the CV ≥1 are associated with large negative responses of their influence functions. The distributions producing such responses often have densities that decrease from positive infinite to zero on the positive axis with a long tail to the right. An influence function for the difference of two coefficients of variation is also obtained.     

Modified inference about the mean of the exponential distribution using moving extreme ranked set sampling

The maximum likelihood estimator (MLE) and the likelihood ratio test (LRT) will be considered for making inference about the scale parameter of the exponential distribution in case of moving extreme ranked set sampling (MERSS). The MLE and LRT can not be written in closed form. Therefore, a modification of the MLE using the technique suggested by Maharota and Nanda (Biometrika 61:601–606, 1974) will be considered and this modified estimator will be used to modify the LRT to get a test in closed form for testing a simple hypothesis against one sided alternatives. The same idea will be used to modify the most powerful test (MPT) for testing a simple hypothesis versus a simple hypothesis to get a test in closed form for testing a simple hypothesis against one sided alternatives. Then it appears that the modified estimator is a good competitor of the MLE and the modified tests are good competitors of the LRT using MERSS and simple random sampling (SRS).     

A note on the satterthwaite confidence interval for a variance

When using a Satterthwaite chi-squared approximation, it is generally thought that the approximation is satisfactory when it is applied to a positive linear combination of mean squares. In this note, we describe how the Williams - Tukey idea for getting a confidence interval for the among groups variance in a random one-way model can be incorporated into Satterthwaite’s procedure for getting a confidence interval for a variance. This adjusted Satterthwaite procedure insures that his chi-squared approximation is always applied to positive linear combinations of mean squares. A small simulation is included which suggests that the adjustment to the Satterthwaite procedure is effective.     

On asymptotic distribution of the test statistic for the mean of the non-isotropic principal component

In this paper, the authors derived the large sample distribution of the t statistic based upon the observations on the first principal component instead of the original variables. It is shown that the above statistic is distributed asymptotically as Student's t distribution.