A fast robust method for fitting gamma distributions

The art of fitting gamma distributions robustly is described. In particular we compare methods of fitting via minimizing a Cramér Von Mises distance, an L ₂ minimum distance estimator, and fitting a B-optimal M-estimator. After a brief prelude on robust estimation explaining the merits in terms of weak continuity and Fréchet differentiability of all the aforesaid estimators from an asymptotic point of view, a comparison is drawn with classical estimation and fitting. In summary, we give a practical example where minimizing a Cramér Von Mises distance is both efficacious in terms of efficiency and robustness as well as being easily implemented. Here gamma distributions arise naturally for “in control” representation indicators from measurements of spectra when using fourier transform infrared (FTIR) spectroscopy. However, estimating the in-control parameters for these distributions is often difficult, due to the occasional occurrence of outliers.     

SMALL SAMPLE BIAS CORRECTION FOR HUBER'S PROPOSAL-2 SCALE M-ESTIMATOR

The most popular and perhaps universal estimator of location and scale in robust estimation, where the population is normal with possible small departures, is Huber's Proposal‐2 M‐estimator. This paper gives the first‐order small sample bias correction for the scale estimator, verifying the calculation through theory and simulation. Other ways of reducing small sample bias, say by jackknifing or bootstrapping, can be computationally intensive, and would not be routinely used with this iteratively derived estimator. It is suggested that bias‐reduced estimates of scale are most useful when forming confidence intervals for location and/or scale based on the asymptotic distribution.     

CALCULATING THE TRUE COST OF PRIVATE SCHOOL EDUCATION: BASED ON A SURVEY OF SOUTH AUSTRALIAN SCHOOL FEES

The cost of a child's education should not be a mystery to consumers, yet little has been published about the true costs, and it is no simple task for parents to calculate. This paper seeks to fill a gap in the research literature that exists in relation to informing parents and educationalists alike as to the true cost of private school education. The term cost is used in the economic sense as encompassing not only the price paid for the product (school fees) but also the opportunity cost to the family in terms of what they forego in order to finance private schooling. Identifying opportunity cost is a useful means of assessing the impact of factors, other than financial, that are influential in determining whether or not private education is purchased. From a survey of South Australian private schools, we have been able to provide a detailed account of the price variations between the fees charged, as well as identifying a variety of additional ancillary charges. Based on the survey figures, we have projected the amount of money that needs to be taken out of the family budget in order to pay for a child's private schooling. At the time the survey was conducted the cost of private school education across three strata, on average, ranged from $13,400 to $42,246. We calculated that incremental family pretax incomes of between $20,303 and $64,009 would be required to fund this education. The amounts represent an opportunity cost to a family whereby money could be put to alternative uses such as home renovation, an overseas holiday, or a buffer against unexpected life events. We conclude that the outlay is substantial for many families and that greater transparency of all costs is needed for families to make informed financial plans to accommodate the overall cost of education for their children.     

Investigation of the performance of trimmed estimators of life time distributions with censoring

For the lifetime (or negative) exponential distribution, the trimmed likelihood estimator has been shown to be explicit in the form of a β‐trimmed mean which is representable as an estimating functional that is both weakly continuous and Fréchet differentiable and hence qualitatively robust at the parametric model. It also has high efficiency at the model. The robustness is in contrast to the maximum likelihood estimator (MLE) involving the usual mean which is not robust to contamination in the upper tail of the distribution. When there is known right censoring, it may be perceived that the MLE which is the most asymptotically efficient estimator may be protected from the effects of ‘outliers’ due to censoring. We demonstrate that this is not the case generally, and in fact, based on the functional form of the estimators, suggest a hybrid defined estimator that incorporates the best features of both the MLE and the β‐trimmed mean. Additionally, we study the pure trimmed likelihood estimator for censored data and show that it can be easily calculated and that the censored observations are not always trimmed. The different trimmed estimators are compared by a modest simulation study.     

Using multivariate time series methods to estimate location and climate change effects on temperature readings employed in electricity demand simulation

Long‐term historical daily temperatures are used in electricity forecasting to simulate the probability distribution of future demand but can be affected by changes in recording site and climate. This paper presents a method of adjusting for the effect of these changes on daily maximum and minimum temperatures. The adjustment technique accommodates the autocorrelated and bivariate nature of the temperature data which has not previously been taken into account. The data are from Perth, Western Australia, the main electricity demand centre for the South‐West of Western Australia. The statistical modelling involves a multivariate extension of the univariate time series ‘interleaving method’, which allows fully efficient simultaneous estimation of the parameters of replicated Vector Autoregressive Moving Average processes. Temperatures at the most recent weather recording location in Perth are shown to be significantly lower compared to previous sites. There is also evidence of long‐term heating due to climate change especially for minimum temperatures.     

Quality indicator survey versus traditional survey in New York State: a comparison of results from annual nursing home surveys

Despite the passage of OBRA’87 for nursing home reform, concerns about care in facilities continue. The Centers for Medicare and Medicaid developed new regulations and the Traditional Survey (TS) process for annual nursing home survey. The survey is conducted by state regional offices to determine facility compliance with federal regulations. Despite the regulations and new survey process, the TS inconsistently identified problems. A computerized process called the Quality Indicator Survey (QIS) was subsequently developed. This study was designed to compare results from TS and QIS on overall deficiencies, select quality indicators, high-severity deficiencies, and severity differences of seven quality indicators in New York State over a 6-year period from 2010 through 2015. Results of t-tests determined a significant difference in the overall mean number of deficiencies (p < .001), and on four indicators: choices (p < .001), nursing staff (p < .001), dental (p < .001), and dignity (p < .05). Facilities using the TS showed a higher mean number of harm level or higher deficiencies (p < .001). Chi-square tests for severity levels showed significantly more higher severity deficiencies on two quality indicators: nutrition (p < 0.001) and hydration (p < 0.05). Thus, the QIS produced a greater mean number of deficiencies, while TS produced more higher severity deficiencies in New York State.     

Carbon Labelling and Low‐income Country Exports: A Review of the Development Issues

This article discusses the carbon accounting and carbon‐labelling schemes being developed to address growing concerns over climate change. Its particular concern is their impact on small stakeholders, especially low‐income countries. The popular belief that trade is by definition problematic is not true; carbon efficiencies elsewhere in the supply chain may more than offset emissions from transportation. Indeed, low‐income countries may offer important opportunities for carbon emission reductions because of their favourable climatic conditions and use of low energy‐intensive production techniques. However, their effective inclusion in labelling schemes will require innovative solutions to provide low‐cost data collection and certification.     

An outlier problem in the determination of ore grade

Data from recordings of ore assays from the Western Australian goldfields provide motivation to devise new tests for outliers when observations are distributed with the same mean but diff ering variances. In the case of equal variances, tests for a single outlier reduce to well-known tests of discordancy. A block discordancy test for k outliers is also described. The question of whether or not one should omit any observation(s) in the calculation of the mean recoverable gold content is addressed in the context of whether or not the data contain outliers, as judged by a normal model for the 'logged' ore assay values. The given data suggest that models with 'logged' values that follow long-tailed approximately normal distributions may be appropriate.