In his seminal book, Shewhart (1931) makes no demand on the distribution of the characteristic to be plotted on a control chart. How then can we explain the idea that normality is, if not required, at least highly desirable? I believe that it has come about through the many statistical studies of control-chart behavior. If one is to study how a control chart behaves, it is necessary to relate it to some distribution. The obvious choice is the normal distribution because of its ubiquity as a satisfactory model. This is bolstered by the existence of the Central Limit Theorem.

Sampling: Design and Analysis

This introductory text on the design and analysis of sample surveys emphasizes the practical aspects of survey problems. It begins with brief chapters on the role of sample surveys in the modern world. Thereafter, each chapter introduces a sample survey design or estimation procedure by describing the pertinent practical problem. The authors describe the methodology proposed for solving the problem and provide the details of the estimation procedure, including a compact presentation of the formulas needed to complete the analysis. Then, a practical example is worked out in complete detail. At the end of each chapter, a wealth of exercises gives students ample opportunity to practice the techniques and stretch their grasp of ideas.

Elementary Survey Sampling

A new estimator of population total has been developed following the method of Murthy (1957) by using the Shahbaz and Hanif (2003) General selection procedure. Two special cases have been obtained of the general estimator. Empirical study has been carried out to obtain the most suitable value of the constant involved.

Sampling with Unequal Probabilities

The present paper deals with a modified ratio type variance estimator for estimation of population variance of the study variable, when the population median of the auxiliary variable is known. The bias and the mean squared error of the proposed estimator are obtained and also derived the conditions for which the proposed estimator performs better than the traditional ratio type variance estimator suggested by Isaki[10] and the modified ratio type variance estimators suggested by Kadilar and Cingi[11]. Further we have compared the efficiencies of the proposed estimator with that of traditional ratio type variance estimator and existing modified ratio type variance estimators for certain known populations. From the numerical study it is observed that the proposed estimator performs better than the traditional ratio type variance estimator and existing modified ratio type variance estimators.

/pdf/variance-estimation-using-median-of-the-auxiliary-variable-1j884nvmdd.pdf

Variance Estimation Using Median of the Auxiliary Variable

In this paper we have proposed a class of modified ratio type variance estimators for estimation of population variance of the study variable using Quartiles and their functions of the auxiliary variable are known. The biases and mean squared errors of the proposed estimators are obtained and also derived the conditions for which the proposed estimators perform better than the traditional ratio type variance estimator and existing modified ratio type variance estimators. Further we have compared the proposed estimators with that of traditional ratio type variance estimator and existing modified ratio type variance estimators for certain known populations. From the numerical study it is observed that the proposed estimators perform better than the traditional ratio type variance estimator and existing modified ratio type variance estimators.

/pdf/variance-estimation-using-quartiles-and-their-functions-of-38phm8uyo1.pdf

Variance Estimation Using Quartiles and their Functions of an Auxiliary Variable

We study an M/G/1 queue with second optional service. Poisson arrivals with mean arrival rate A (> 0), all demand the first 'essential' service, whereas only some of them demand the second 'optional' service. The service times of the first essential service are assumed to follow a general (arbitrary) distribution with distribution function B 1 (v) and that of the second optional service with general (arbitrary) distribution with distribution function B 2 (v). The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly.

An M/G/1 queue with second optional service with general service time distribution

Recently, many authors introduced ratio-type estimators for estimating the mean, or the ratio, for a finite populations. Most of the articles are discussing this problem under simple random sampling design, with more assumptions on the auxiliary variable such as the coefficient of variation, and kurtosis are assumed to be known. Gupta and Shabbir (2008) have suggested an alternative form of ratio-type estimators and they assumed the coefficient of variation of the auxiliary variable must be known; this assumption is crucialfor this estimator.An estimator of the population ratio, under general sampling design, is proposed.Further, exact and an unbiased variance estimator of this estimator are obtained, and the Godambe-Joshi lower bound is asymptotically attainable for this estimator. The assumption on the coefficient of variation of the auxiliary variable is not needed for the proposed estimator. Simulation results from real data set and simulations from artificial population, show that the performance of the proposed estimator is better than Gupta and Shabbir (2008) and Hartley and Ross (1954) estimators.

http://www.stat.tugraz.at/AJS/ausg122/122Al-Jararha.pdf

A Ratio Estimator Under General Sampling Design

A wider class of chain based estimators for the variance of a finite population has been proposed by using two auxiliary variables.

The class of chain estimators for a finite population variance using double sampling

Singh (2013) considered the dual problem to the calibration of design weights to obtain a new generalized linear regression estimator (GREG) for the finite population total In this work, we have made an attempt to suggest a way to use the dual calibration of the design weights in case of multi-auxiliary variables; in other words, we have made an attempt to give an answer to the concern in Remark 2 of Singh (2013) work The same idea is also used to generalize the GREG estimator proposed by Deville and Sarndal (1992) It is not an easy task to find the optimum values of the parameters appear in our approach; therefore, few suggestions are mentioned to select values for such parameters based on a random sample Based on real data set and under simple random sampling without replacement design, our approach is compared with other approaches mentioned in this paper and for different sample sizes Simulation results show that all estimators have negligible relative bias, and the multivariate case of Singh (2013) estimator is more efficient than other estimators

A Dual Problem of Calibration of Design Weights Based on Multi-Auxiliary Variables

Consider the standard multiple linear regression model $y=x\beta+\varepsilon.$ If the correlation matrix $x^tx$  is ill-conditioned, the ordinary least squared estimate (ols) $\hat{\beta}$ of $\beta$ is not the best choice. In this paper, multiple regularization  parameters for different coefficients in ridge regression are proposed. The Mean Squared Error (MSE) of a ridge estimate  based on the multiple regularization parameters is less than or equal to the MSE of the ridge estimate based on Hoerl and Kennard, 1970. The proposed approach, depending on the condition numbers, leave's zero for the largest eigenvalue of $x^tx$ and gives the largest value  for the smallest eigenvalue of $x^tx.$ Furthermore, if $x^tx$ is nearly a unit matrix, $x^tx$  is not an ill-conditioned one. The proposed approach gives approximately the same results as the ols estimates. The proposed approach can also be modified to give other new ridge parameters. The modified approach depends on the eigenvalues of  $x^tx$ and  differ from the ridge parameter proposed by Khalaf and Shukur by a factor. The body fat data set has severe multicollinearity and is used to compare different approaches.

/pdf/new-approaches-for-choosing-the-ridge-parameters-1yvauvybb5.pdf

Jehad Al-Jararha

Papers

An M/G/1 queue with second optional service with general service time distribution

A Ratio Estimator Under General Sampling Design

The class of chain estimators for a finite population variance using double sampling

A Dual Problem of Calibration of Design Weights Based on Multi-Auxiliary Variables

New approaches for choosing the ridge parameters