One of the pillars of experimental science is sampling. Based on the analysis of samples, estimations for populations are made. There is an entire science based on sampling. Distribution of the population, of the sample, and the connection among those two (including sampling distribution) provides rich information for any estimation to be made. Distributions are split into two main groups: continuous and discrete. The present study applies to continuous distributions. One of the challenges of sampling is its accuracy, or, in other words, how representative the sample is of the population from which it was drawn. To answer this question, a series of statistics have been developed to measure the agreement between the theoretical (the population) and observed (the sample) distributions. Another challenge, connected to this, is the presence of outliers - regarded here as observations wrongly collected, that is, not belonging to the population subjected to study. To detect outliers, a series of tests have been proposed, but mainly for normal (Gauss) distributions—the most frequently encountered distribution. The present study proposes a statistic (and a test) intended to be used for any continuous distribution to detect outliers by constructing the confidence interval for the extreme value in the sample, at a certain (preselected) risk of being in error, and depending on the sample size. The proposed statistic is operational for known distributions (with a known probability density function) and is also dependent on the statistical parameters of the population—here it is discussed in connection with estimating those parameters by the maximum likelihood estimation method operating on a uniform U(0,1) continuous symmetrical distribution.

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A Test Detecting the Outliers for Continuous Distributions Based on the Cumulative Distribution Function of the Data Being Tested

Background The purpose of this study was to evaluate hospital admissions for revision surgeries associated with robotic arm–assisted unicompartmental knee arthroplasty (rUKA) vs manually instrumented UKA (mUKA) procedures. Methods Patients ≥18 years of age who received either a mUKA or a rUKA procedure were candidates for inclusion and were identified by the presence of appropriate billing codes. Procedures performed between March 1, 2013 and July 31, 2015 were used to calculate the rate of surgical revisions occurring within 24-months of the index procedure. Following propensity matching, 246 rUKA and 492 mUKA patients were included. Revision rates and the associated costs were compared between the two cohorts. The Mann-Whitney U test was used to compare continuous variables, and Fisher’s exact tests was used to analyze discrete categorical variables. Results At 24 months after the primary UKA procedure, patients who underwent rUKA had fewer revision procedures (0.81% [2/246] vs 5.28% [26/492]; P = .002), shorter mean length of stay (2.00 vs 2.33 days; P > .05), and incurred lower mean costs for the index stay plus revisions ($26,001 vs $27,915; P > .05) than mUKA patients. Length of stay at index and index costs were also lower for rUKA patients (1.77 vs 2.02 days; P = .0047) and ($25,786 vs $26,307; P > .05). Conclusions The study results demonstrate that patients who underwent rUKA had fewer revision procedures, shorter length of stay, and incurred lower mean costs (although not statistically different) during the index admission and at 24 months postoperatively. These results could be important for payers as the prevalence of end-stage knee osteoarthritis increases alongside the demand for cost-efficient treatments.

Revision Analysis of Robotic Arm-Assisted and Manual Unicompartmental Knee Arthroplasty.

Hearing loss is common among older adults. Thus, it was of interest to explore differences in health care utilization and costs associated with hearing loss and hearing aid use. Hearing loss and hearing aid use were assessed through self-reports and included 5 categories: no hearing loss, aided mild, unaided mild, aided severe, and unaided severe hearing loss. Health care utilization and costs were obtained from medical claims. Those with aided mild or severe hearing loss were significantly more likely to have an emergency department visit. Conversely, those with aided severe hearing loss were about 15% less likely to be hospitalized. Individuals with unaided severe hearing loss had the highest annual medical costs ($14349) compared with those with no hearing loss ($12118, P < .001). In this study, those with unaided severe hearing loss had the highest medical costs. Further studies should attempt to better understand the relationship between hearing loss, hearing aid use, and medical costs.

Self-Reported Hearing Loss in Older Adults Is Associated With Higher Emergency Department Visits and Medical Costs:

The moment(MM) and least squares(LS) estimations of the parameters are derived for the Pareto distribution in the presence of outliers. Further, we have derived a mixture method(MIX) of estimations with MM and LS that shows that the MIX is more efficient. In the final section we have given an example of actual data from a medical insurance company.

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Efficient Estimation of the Parameters of the Pareto Distribution in the Presence of Outliers

The problem of multiple upper outlier detection in gamma sample is considered. Balasooriya and Gadag (1994) proposed a location and scale invariant test based on the test statistic Z k for testing the k upper outliers in two-parameter exponential sample. In this article, the test statistic is extended for gamma sample and the null distribution of the statistic is obtained. A simulation study is carried out to compare the performance of the tests and is found that the test based on Z k is more powerful than the test based on the test statistic proposed by Jabbari Nooghabi et al. (2010).

Testing for Upper Outliers in Gamma Sample

Zerbet and Nikulin presented the new statistic Z k for detecting outliers in exponential distribution. They also compared this statistic with Dixon's statistic D k . In this article, we extend this approach to gamma distribution and compare the result with Dixon's statistic. The results show that the test based on statistic Z k is more powerful than the test based on the Dixon's statistic.

Detecting Outliers in Gamma Distribution

The maximum likelihood, moment and mixture of the estimators are derived for samples from the gamma distribution in the presence of outliers generated from uniform distribution. These estimators are compared empirically when all the parameters are unknown; their bias and determinants are investigated with the help of numerical technique. We have shown that these estimators are asymptotically unbiased. At the end, we conclude that mixture estimators are better than the maximum likelihood and moment estimators.

Estimation of Parameters of the Gamma Distribution in the Presence of Outliers Generated From Uniform Distribution

In this article, we have estimated the scale parameter of exponential distribution with a prior information. A shrinkage estimator is derived for parameter of exponential distribution contaminated with outliers and in the presence of LINEX loss function. An admissible estimator based on the LINEX loss function are compared with different methods of estimations. Numerical study are used to compare the estimators. AMS (MOS) Subject Classification Codes: 62Fxx; 62F10; 62F15

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On Bayesian Shrinkage Estimator of Parameter of Exponential Distribution with Outliers

Gupta and Kundu (1999) defined the cumulative distribution function of the generalized exponential (GE). It has many properties that are quite similar to those of the gamma distribution. This paper deals with the estimation of the parameters of the generalized exponential distribution in the presence of one outlier generated from uniform distribution. The maximum likelihood, moment and mixture of the estimators are derived. These estimators are compared empirically when all of the parameters are unknown. Their bias and mean square error (MSE) are investigated with help of numerical techniques.

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Estimation of the parameters of the generalized exponential distribution in the presence of outliers generated from uniform distribution

The maximum likelihood, moment and mixture of the estimators are derived for samples from the gamma distribution in the presence of outliers generated from gamma distribution. These estimators are compared empirically when all parameters are unknown; their bias and mean squares error are investigated with the help of numerical technique. We have shown that these estimators are asymptotically unbiased. At the end, we conclude that mixture estimators are better than the maximum likelihood and moment estimators.

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Parviz Nasiri

Papers

Detecting Outliers in Gamma Distribution

Estimation of Parameters of the Gamma Distribution in the Presence of Outliers Generated From Uniform Distribution

On Bayesian Shrinkage Estimator of Parameter of Exponential Distribution with Outliers

Estimation of the parameters of the generalized exponential distribution in the presence of outliers generated from uniform distribution

Parameters Estimation of the Gamma Distribution in the Presence of Outliers Generated From Gamma Distribution