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In this article, a new approach which I call a “product moment generating function” (pmgf) is introduced and explained to find mean, variance, and higher moments. An advantage in this approach is illustrated using Gupta's [1] modified power series distributions (mpsd). Finding the higher moments of mpsd was not easy and Gupta employed recurrence relations among moments for obtaining higher moments.

This article examines the underlying concepts in making up an interval estimate of the normal population mean, and then suggests an alternative but "ad hoc" approach based on Kullback- Leibler information measure. In this new approach, both the sample information and sample precision are kept as separate entities, and this surprisingly shortens the interval estimate.

The relationship between fatal, injury, and property damage accident frequencies and traffic volume, segment length (SL), and vehicle miles traveled (VMT) is explored here. For this purpose, a generalized linear regression modeling framework was applied. Poisson, negative binomial, Gaussian, and log-normal distributions were evaluated in terms of their ability to model accident frequency for an interstate highway corridor in Colorado. Results indicate that, for injury, property damage, and total accidents, the Poisson regression with log-transformed predictors performed significantly better than the Poisson regression with linear predictors. For fatal accidents, the log-normal regression performed better than other regression models examined. However, the Poisson model performed better in terms of estimating accidents for the corridor. Models based on data disaggregated by median SL and annual average daily traffic (AADT) or VMT performed better than aggregated models. In addition, the SL-AADT–based models estimated a smaller difference between observed and expected total number of accidents for the corridor than the VMT-based model. SL and traffic volume were significant for injury and property damage accidents. For fatal accidents, an exposure-based predictor VMT was significant across all segment groups.

To quantify how worthy and reliable an interval estimate of a parameter, an index is defined. This index is derived for estimating the mean μ of a normal population. Using this index, a method is devised for determining the sample size formula. Advantages of this new sample size formula are pointed out.

An intriguing economic phenomenon in consumption studies is the law of diminishing return. In spite of its economic importance, this phenomenon has not received attention by stochastic modelers. In this article, an attempt is made to explain this phenomenon via a new probability model. Both the estimation and hypothesis testing procedures for the model parameters are derived, and illustrated using data on the number of viewers of a weekly television show in Denver, USA.

Due to reasons like absence of a proper sampling frame or inaccessibility to population units, data in statistical studies are sometimes contaminated by a phenomenon called length bias (LB). In this article, an asymptotic test statistic is derived to examine the homogeneity of a length biased sample from the Mean Exponential Family (MEF), a new class of distributions introduced by Shanmugam ( J. Statist. Plann. Inference 23 (1989) 227–291). Expressions for the test statistic are obtained for length biased binomial, Poisson, negative binomial, beta, gamma, normal, Pareto, Laplace, and Raleigh distributions as special cases. The results are illustrated. One among several advantages in our approach is the ability to quantify the specificity and sensitivity of LB and their instrinsic relations.

Air pollution is a serious threat to the healthy life in society. Nations across the continents in the 21st century need to evaluate the air pollution data to make up fair and implementable global policies, which support healthy life. In this process of examining the data, the following relevant question could be answered. Do developing or developed nations contribute more air pollutions? As this chapter illustrates, the pollutions related data in the public domain WebPages (see [1] through [21] for details) hide the answer. The developing nations aspire to improve their standard of life by advancing their industrial or agricultural productivities. The developed nations, on the contrary, tend to maintain their higher standard of living with the same or more industrial and agricultural productivities. When it comes to a discussion on who contribute the air pollutions, both the developed and developing nations argue and are willing to make a few policy changes but preach and expect the other side to do more to cut down the global air pollutions. How should this escalated conflict be resolved? What do the relevant data suggest? How a compromise between the developed and developing nations is plausible? To answer this and related questions, this chapter examines and assesses existing global air pollutions data using statistical techniques. The findings as they are stated in this chapter are interesting and pave ways to formulate equitable global policies to reduce air pollutions and thereby increase the healthy life at large.

In this robotic age of intelligent manufacturing, what is needed is a well-planned built-in automated statistical systems (AASS) which would by itself intelligently and timely figure out the need for a statistical study including statistical planning of either observational or experimental nature and also carry out a statistical expert systems analysis. This AASS is indispensable so far as saving time and reducing different complexities in a manufacturing process. This article examines impediments in establishing AASS from engineering and statistical viewpoints. The potential benefits of AASS are pointed out. It is time to think about AASS loud and clear.

Morris (1982, 1983) shows that there exists only six distributions with variance as a quadratic function of the mean, and calls the group natural exponential-quadratic variance family (NEQVF). This note discovers that in NEQVF there are actually two factions which we call linear and nonlinear mean exponential families (MEF and NLMEF). A statistical utility of this characterization lies in determining a sample size based on Shannon's information criterion: As far as gathering information with respect to the population mean, there is a consensus formula in NEQVF for sample size determination, but with respect to the natural parameter, there are three opinions — one for LMEF and a different one for each member of NLMEF.

Among distributions that are used in survival analysis, of interest to us here is the Guaranteed exponential (for details about the distribution, see Gross and Clark 7). In this article, we present a new approach to testing guaranteed exponentiality. Using engineering data on the number of load cycles until a crack appears in the titanium and steel sheet in aircraft, medical data on the number of weeks tumor patients survived, data on modulus of rupture, a measurement on the breaking strength of lumber, and original I.Q. data on 112 children attending kindergarten classes in San Jose and San Mateo Counties, CA, we illustrate our method. The first two data sets are known to follow exponential distribution, while the last data set is known to follow normal distribution. Our method confirms these and nonexponentiality for the third data set. A discussion on the merits and disadvantages of our approach is included. Based on simulation, a comparison is made between the power of our test and EDF test statistic A2. Our test has excellent power for many nonexponential alternatives.

Operations research and management science handbook, edited by A. Ravi Ravindran, Boca Raton, CRC Press, 2008, -33 pp., $161.94, ISBN 978-0-8493-9721-9 Operations research: a practical introduction...

Encyclopedia of statistics in behavioural science, edited by Brian S. Everitt and David C. Howell, John Wiley and Sons Inc., 111 River Street, MS 8-01, Hoboken, NJ 07030 – 5774, USA, 2nd ed., Vols ...

Background: In times of an outbreak of a contagious deadly epidemic 1-4 such as severe acute respiratory syndrome (SARS), the patients are quarantined and rushed to an emergency department of a hospital for treatment. Paradoxically, the nurses who treat them to become healthy get infected in spite of the nurses’ precautionary defensive alertness. This is so unfortunate because the nurses might not have been in close contact with the virus otherwise in their life. The nurses’ sufficient immunity level is a key factor to avert hospital site infection. Is it possible to quantify informatics about the nurses’ immunity from the virus? Methods: The above question is answered with a development of an appropriate new model and methodology. This new frequency trend is named Bumped-up Binomial Distribution (BBD). Several useful properties of the BBD are derived, applied, and explained using SARS data 5 in the literature. Though SARS data are considered in the illustration, the contents of the article are versatile enough to analyze and interpret data from other contagious diseases. Results: With the help of BBD (3) and the Toronto data in Table 1, we have identified the informatics about the attending nurses’ sufficient immunity level. There were 32 nurses providing 16 patient care services. Though the nurses were precautionary to avoid infection, not all of them were immune to infection in those activities. Using the new methodology of this article, their sufficient immunity level is estimated to be only 0.25 in a scale of zero to one with a p-value of 0.001. It suggests that the nurses’ sufficient immunity level is statistically significant. The power of accepting the true alternative hypothesis of 0.50 immunity level, if it occurs, is calculated to be 0.948 in a scale of zero to one. It suggests that the methodology is powerful. Conclusions: The estimate of nurse’s sufficient immunity level is a helpful factor for the hospital administrators in the time of making work schedules and assignments of the nurses to highly contagious patients who come in to the emergency or regular wings of the hospital for treatment. When the approach and methodology of this article are applied, it would reduce if not a total elimination of the hospital site infections among the nurses and physicians.

Background: Due to severe pain, patients are impatient in several wings sporadically and more frequently in emergency wing of the hospitals. To efficiently administer in such environment and the hospital management seeks helpful strategies. The queuing concepts and related methodologies can help as this article has demonstrated by an analysis and interpretation of real data from a hospital in Malta. Methods: The queuing concepts are probabilistic and statistical ideas based approach. They require configuration of the rate and pattern of arriving patients, the rate and pattern of the service, the number of channels serving, the capacity of the waiting room, and the criterion for selecting patients for service etc. New ideas are presented in this article to manage in various scenarios of real life emergency operations. The pertinent queuing concepts and tools are made easier for the readers to comprehend and practice in their own situations in which they notice that the patients are impatient in their waiting. Results: Using the new ideas and formulas of this article, the data in the emergency wing of a hospital in Malta (a largest island of an archipelago situated in the center of the Mediterranean with a total population of a million) are analyzed and interpreted. The results clearly explain why there were a prolonged waiting times at the emergency department creating public dissatisfaction and patients were leaving without waiting to be seen. The total time spent by non-urgent patients with nurse and casualty officer is more in the second shift and lesser and lesser in the third and fourth shifts. The interactive time with a nurse by patient is statistically same in all three types: life-threatening, non-life threatening but urgent, and non-urgent. Very strikingly, the patients in all three groups wait longer to be seen by the nurse in shift three and lesser time in shifts two or four. Conclusion: In 21 st century with flourishing globalized medical tourism, a standardized approach to minimize efficiently the waiting time in emergency and other wings of the hospitals in developing as much as in developed nations is a necessity as this auricle has pointed out. The impediments and the remedies for an efficient standardization are overdue.

Background: In 1985, the center for disease control coined the name: “Acquired Immune Deficiency Syndrome (AIDS)” to refer a deadly illness. The World Health Organization (WHO) estimated that about 33.4 million people were suffering with AIDS and two million people (including 330,000 children) died in 2009 alone in many parts of the world. A scary fact is that the public worry about situations which might spread AIDS according to reported survey result in Meulders et al. (2001). This article develops and illustrates an appropriate statistical methodology to understand the meanings of the data. Methods: While the binomial model is a suitable underlying model for their responses, the data mean and dispersion violates the model’s required functional balance between them. This violation is called over-under dispersion. This article creates an innovative approach to assess whether the functional imbalance is too strong to reject the binomial model for the data. In a case of rejecting the model, what is a correct way of warning the public about the spreads of AIDS in a specified situation? This question is answered. Results: In the survey data about how AIDS/HIV might spread according to fifty respondents in thirteen nations, the functional balance exists only in three cases: “needle”, “blood” and “sex” justifying using the usual binomial model (1). In all other seven cases: “glass”, “eating”, “object”, “toilet”, “hands”, “kissing”, and “care” of an AIDS or HIV patient, there is a significant imbalance between the dispersion and its functional equivalence in terms of the mean suggesting that the new binomial called imbalanced binomial distribution (6) of this article should be used. The statistical power of this methodology is indeed excellent and hence the practitioners should make use of it. Conclusion: The new model called imbalanced binomial distribution (6) of this article is versatile enough to be useful in other research topics in the disciplines such as medicine, drug assessment, clinical trial outcomes, business, marketing, finance, economics, engineering and public health.

1. Environmental Statistics: Methods and Applications, by Vic Barnett, 2005, Blackwell Publishing Professional, 2121 State Avenue, Ames, IA 50014, USA, ISBN: 0 8138 1112 0, pp. 175+xi, $99.95. 2. S...

[1] Encyclopedia of Biostatistics (Second Edition): Volumes 1 to 8, edited by Peter Armitage and Theodore Colton, 2005, John Wiley and Sons Inc., 111 River Street, MS 8-01, Hoboken, NJ 07030-5774, ...

Who discovered zero? The historians and archeologists have been debating about zero. The mathematical conception of zero was present in a spiritual form about 12,000 B.C. in India. The zero is a tr...

Background: Smoking is generally known to be carcinogenic and health hazardous. What is not clear is whether the smoking impacts on the woman’s reproductive process. There have been medical debates on whether a woman in the child bearing age may delay her pregnancy due to smoking. A definitive conclusion on this issue has not been reached perhaps due to a lack of appropriate data evidence. The missing link to answer the question might be exercising a suitable model to extract the pertinent data information on the number of missed menstrual cycles by smoking women versus non-smoking women . This article develops and demonstrates a statistical methodology to answer the question. Methods: To construct such a needed methodology, a new statistical distribution is introduced as an underlying model for the data on the number of missed menstrual cycles by women who smoke. This new distribution is named Tweaked Geometric Distribution (TGD). Several useful properties of the TGD are derived and explained using a historical data in the literature. Results: In the data of 100 smokers and 486 non-smokers, on the average, smoking women missed 3.22 menstrual cycles and non-smoking women missed only 1.96 menstrual cycles before becoming pregnant. The smoking women exhibited more variation than the non-smoking women and it suggests that the non-smoking women are more homogeneous while the smoking women are more heterogeneous . Furthermore, the impairment level to pregnancy due to smoking among the 486 women is estimated to be 5% in a possible scale of zero to one. The 5% impairment level appears like a small amount , but its impact can be felt once it is cast in terms of fecundity . What is fecundity? The terminology fecundity refers the chance for a woman to become pregnant. The fecundity is 0.24 for smoking woman while it is 0.34 for non-smoking woman. The fecundity of a non-smoking woman is more than twice the fecundity of a smoking woman. Conclusion: The smoking is really disadvantageous to any one in general and particularly to a woman who wants to become pregnant.