Visualizing Mean,Median, Mean Deviation,and Standard Deviation of a Set of Numbers

We review the existing visualizations of the mean and the median of a given set of numbers. Then we give an alternative visualization of the mean using the empirical cumulative distribution function of the given numbers. Next, we visualize the mean deviation (MD) and the mean square deviation (MSD) of the given numbers from any arbitrary value, including the variance. In light of these new visualizations, we revisit the well-known optimal properties of the MD from the median and the MSD from the mean. We also give a more elementary explanation of why the denominator of the sample variance of a set of numbers is one less than the sample size.     

Gender Disparities in Educational Trajectories in India: Do Females Become More Robust at Higher Levels?

Studies on educational attainments have tended to focus on attainments at a specific level of education. The change in disparities in attainment over the educational life cycle, however, has been neglected in literature—in India, for instance, the only exceptions are Vaid (2004) and Desai and Kulkarni (2008). This paper uses unit level National Sample Survey data (2004–2005) to examine gender disparities at different educational levels up to the high school level. Bi-variate analysis of gender disparities computed using the Sopher-Kundu index indicates that although there are substantial gender disparities at lower levels of education, this gets reversed at higher levels in several states. At the all-India level, too, in recent years, a reversal of disparity may be observed in urban areas. Econometric analysis using the variable parameter variant of the generalized ordered logit model reveals that robustness of females increase over the educational life cycle, though disparity remains, after controlling for socio-economic traits.     

A new graph parameter and a construction of larger graph without increasing radio <Emphasis Type="Italic">k</Emphasis>-chromatic number

For a positive integer \(k\ge 2\), the radio k-coloring problem is an assignment L of non-negative integers (colors) to the vertices of a finite simple graph G satisfying the condition \(|L(u)-L(v)| \ge k+1-d(u,v)\), for any two distinct vertices u, v of G and d(u, v) being distance between u, v. The span of L is the largest integer assigned by L, while 0 is taken as the smallest color. An \(rc_k\)-coloring on G is a radio k-coloring on G of minimum span which is referred as the radio k-chromatic number of G and denoted by \(rc_k(G)\). An integer h, \(0<h<rc_k(G)\), is a hole in a \(rc_k\)-coloring on G if h is not assigned by it. In this paper, we construct a larger graph from a graph of a certain class by using a combinatorial property associated with \((k-1)\) consecutive holes in any \(rc_k\)-coloring of a graph. Exploiting the same property, we introduce a new graph parameter, referred as \((k-1)\)-hole index of G and denoted by \(\rho _k(G)\). We also explore several properties of \(\rho _k(G)\) including its upper bound and relation with the path covering number of the complement \(G^c\).     

Effect of low back pain on social and professional life of drivers of Kolkata

Abstract Occupational driving has often been associated with a high prevalence of back pain. Buses constitute the backbone of the local transportation; so many people are engaged in the profession. The present study was to investigate the prevalence of low back pain (LBP) and due to that problem whether their social and professional life hampers or not. The present study was undertaken among 160 government undertaking male bus drivers. Modified Nordic musculoskeletal questionnaire was performed with physical assessment. Then Oswestry Low Back Pain Disability Questionnaire (QLBPDQ) was performed to evaluate the extent to which social and professional life is restricted by back pain. Afterwards, the sit-and-reach test and investigation of sitting postures was done. From the analysis of questionnaire it was revealed that drivers had to spend 8-10 hours daily for 6 days in a shift. It was further observed that all of them mainly suffering from LBP. From the QLBPDQ analysis it was found that LBP restricted their social and professional life. Sit and reach test and posture analysis also support this finding. This result is an absolute reflection of the alarming situations of the Kolkata bus drivers. It can be concluded that the bus drivers are highly stressed due to the hazardous working conditions, which in turn may additionally affects their health and overall work performance in the long run.     

Improving Revenue Management Decision Making for Airlines by Evaluating Analyst‐Adjusted Passenger Demand Forecasts*

To maximize revenue, airline revenue management analysts (RMAs) attempt to protect the right number of seats for late‐booking, high‐revenue‐generating passengers from low‐valued leisure passengers. Simulation results in the past showed that a major airline can generate approximately $500 million per year through efficient RM operations. Accurate passenger demand forecasts are required, because reduced forecast error significantly improves revenue. RMAs often adjust the system forecasts to improve revenue opportunity. Analysis of system forecast performance and analyst adjustment is complex, because one must account for all unseen demands throughout the life of a flight. This article proposes a method to account for unseen demand and evaluate forecast performance (adjusted or unadjusted) through a forecast monitoring system. Initial results from one major airline's origin‐destination market data justify the value of RMA forecasting adjustments.     

On adaptive procedures controlling the familywise error rate

The idea of modifying, and potentially improving, classical multiple testing methods controlling the familywise error rate (FWER) via an estimate of the unknown number of true null hypotheses has been around for a long time without a formal answer to the question whether or not such adaptive methods ultimately maintain the strong control of FWER, until Finner and Gontscharuk (2009) and Guo (2009) have offered some answers. A class of adaptive Bonferroni and S?idàk methods larger than considered in those papers is introduced, with the FWER control now proved under a weaker distributional setup. Numerical results show that there are versions of adaptive Bonferroni and S?idàk methods that can perform better under certain positive dependence situations than those previously considered. A different adaptive Holm method and its stepup analog, referred to as an adaptive Hochberg method, are also introduced, and their FWER control is proved asymptotically, as in those papers. These adaptive Holm and Hochberg methods are numerically seen to often outperform the previously considered adaptive Holm method.     

ROBUST INFERENCE IN PARAMETRIC MODELS USING THE FAMILY OF GENERALIZED NEGATIVE EXPONENTIAL DISPARITIES

We examine robust estimators and tests using the family of generalized negative exponential disparities, which contains the Pearson's chi‐square and the ordinary negative exponential disparity as special cases. The influence function and α‐influence function of the proposed estimators are discussed and their breakdown points derived. Under the model, the estimators are asymptotically efficient, and are shown to have an asymptotic breakdown point of 50%. The proposed tests are shown to be equivalent to the likelihood ratio test under the null hypothesis, and their breakdown points are obtained. The competitive performance of the proposed estimators and tests relative to those based on the Hellinger distance is illustrated through examples and simulation results. Unlike the Hellinger distance, several members of this family of generalized negative exponential disparities generate estimators which also possess excellent inlier‐controlling capability. The corresponding tests of hypothesis are shown to have better power breakdown than the Hellinger deviance test in the cases examined.