Pub Date : 2019-06-28DOI: 10.37920/sasj.2019.53.2.5
I. Visagie, F. Lombard
Chemical analyses of raw materials are often repeated in duplicate or triplicate. The assay values obtained are then combined using a predetermined formula to obtain an estimate of the true value of the material of interest. When duplicate observations are obtained, their average typically serves as an estimate of the true value. On the other hand, the "best of three" method involves taking three measurements and using the average of the two closest ones as estimate of the true value. In this paper, we consider another method which potentially involves three measurements. Initially two measurements are obtained and if their difference is sufficiently small, their average is taken as estimate of the true value. However, if the difference is too large then a third independent measurement is obtained. The estimator is then defined as the average between the third observation and the one among the first two which is closest to it. Our focus in the paper is the conditional distribution of the estimate in cases where the initial difference is too large. We find that the conditional distributions are markedly different under the assumption of a normal distribution and a Laplace distribution.
{"title":"On the conditional distribution of the mean of the two closest among a set of three observations","authors":"I. Visagie, F. Lombard","doi":"10.37920/sasj.2019.53.2.5","DOIUrl":"https://doi.org/10.37920/sasj.2019.53.2.5","url":null,"abstract":"Chemical analyses of raw materials are often repeated in duplicate or triplicate. The assay values obtained are then combined using a predetermined formula to obtain an estimate of the true value of the material of interest. When duplicate observations are obtained, their average typically serves as an estimate of the true value. On the other hand, the \"best of three\" method involves taking three measurements and using the average of the two closest ones as estimate of the true value. In this paper, we consider another method which potentially involves three measurements. Initially two measurements are obtained and if their difference is sufficiently small, their average is taken as estimate of the true value. However, if the difference is too large then a third independent measurement is obtained. The estimator is then defined as the average between the third observation and the one among the first two which is closest to it. Our focus in the paper is the conditional distribution of the estimate in cases where the initial difference is too large. We find that the conditional distributions are markedly different under the assumption of a normal distribution and a Laplace distribution.","PeriodicalId":53997,"journal":{"name":"SOUTH AFRICAN STATISTICAL JOURNAL","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48344256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-11-01DOI: 10.37920/sasj.2020.54.2.9
J. M. Zyl
Kurtosis is seen as a measure of the discrepancy between the observed data and a Gaussian distribution and is defined when the 4th moment is finite. In this work an empirical study is conducted to investigate the behaviour of the sample estimate of kurtosis with respect to sample size and the tail index when applied to heavy-tailed data where the 4th moment does not exist. The study will focus on samples from the symmetric stable distributions. It was found that the expected value of excess kurtosis divided by the sample size is finite for any value of the tail index and the sample estimate of kurtosis increases as a linear function of sample size and tail index. It is very sensitive to changes in the tail-index.
{"title":"An empirical study of the behaviour of the sample kurtosis in samples from symmetric stable distributions","authors":"J. M. Zyl","doi":"10.37920/sasj.2020.54.2.9","DOIUrl":"https://doi.org/10.37920/sasj.2020.54.2.9","url":null,"abstract":"Kurtosis is seen as a measure of the discrepancy between the observed data and a Gaussian distribution and is defined when the 4th moment is finite. In this work an empirical study is conducted to investigate the behaviour of the sample estimate of kurtosis with respect to sample size and the tail index when applied to heavy-tailed data where the 4th moment does not exist. The study will focus on samples from the symmetric stable distributions. It was found that the expected value of excess kurtosis divided by the sample size is finite for any value of the tail index and the sample estimate of kurtosis increases as a linear function of sample size and tail index. It is very sensitive to changes in the tail-index.","PeriodicalId":53997,"journal":{"name":"SOUTH AFRICAN STATISTICAL JOURNAL","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45702505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-04-18DOI: 10.37920/sasj.2017.51.1.3
C. Kumar, S. H. S. Dharmaja
In this paper, a new class of distribution namely “the inverse Kies distribution” is proposed and some of its important aspects are studied by deriving expressions for its percentile measures, raw moments, reliability measures etc. The maximum likelihood estimation of the parameters of the distribution is discussed and the distribution has been fitted to certain real life data sets. The asymptotic behaviour of maximum likelihood estimators of the parameters of the distribution are also studied by using simulated data sets. Keywords: Beta generalised inverse Weibull distribution, Failure rate, Generalised inverse Weibull distribution, Maximum likelihood estimation, Model selection, Percentile measures
{"title":"Inverse Kies distribution: Properties and applications","authors":"C. Kumar, S. H. S. Dharmaja","doi":"10.37920/sasj.2017.51.1.3","DOIUrl":"https://doi.org/10.37920/sasj.2017.51.1.3","url":null,"abstract":"In this paper, a new class of distribution namely “the inverse Kies distribution” is proposed and some of its important aspects are studied by deriving expressions for its percentile measures, raw moments, reliability measures etc. The maximum likelihood estimation of the parameters of the distribution is discussed and the distribution has been fitted to certain real life data sets. The asymptotic behaviour of maximum likelihood estimators of the parameters of the distribution are also studied by using simulated data sets. Keywords: Beta generalised inverse Weibull distribution, Failure rate, Generalised inverse Weibull distribution, Maximum likelihood estimation, Model selection, Percentile measures","PeriodicalId":53997,"journal":{"name":"SOUTH AFRICAN STATISTICAL JOURNAL","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46669627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-04-18DOI: 10.37920/sasj.2017.51.1.5
N. Nair, P. Sankaran, Nidhi P. Ramesh
Discrete distributions with bathtub shaped hazard rates have recently become of interest in reliability modelling and analysis. In the present work, we address the problem of obtaining distributions having such hazard rates when the lifetime is discrete. The methods considered here include discretising continuous bathtub models, construction using the score function, construction from decreasing hazard rate distributions and some other methods currently available in the continuous case. We discuss properties and applications of the discretised quadratic hazard model which has a bathtub shaped hazard rate. Keywords: Bathtub hazard rate, Discrete distribution, Quadratic hazard model, Score function, Upside down bathtub hazard rate
{"title":"Discrete distributions with bathtub-shaped hazard rates","authors":"N. Nair, P. Sankaran, Nidhi P. Ramesh","doi":"10.37920/sasj.2017.51.1.5","DOIUrl":"https://doi.org/10.37920/sasj.2017.51.1.5","url":null,"abstract":"Discrete distributions with bathtub shaped hazard rates have recently become of interest in reliability modelling and analysis. In the present work, we address the problem of obtaining distributions having such hazard rates when the lifetime is discrete. The methods considered here include discretising continuous bathtub models, construction using the score function, construction from decreasing hazard rate distributions and some other methods currently available in the continuous case. We discuss properties and applications of the discretised quadratic hazard model which has a bathtub shaped hazard rate. Keywords: Bathtub hazard rate, Discrete distribution, Quadratic hazard model, Score function, Upside down bathtub hazard rate","PeriodicalId":53997,"journal":{"name":"SOUTH AFRICAN STATISTICAL JOURNAL","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43159821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-04-18DOI: 10.37920/sasj.2017.51.1.1
G. Frank, A. Dörre
Non-parametric estimation for a linear regression model under random double-truncation is investigated, i.e. the variables are observed if and only if the dependent variable lies in a random interval. The method requires only weak distribution assumptions to ensure identifiability, but does not require any specific distribution family for any variable, neither for the truncation variables nor for the error term. By using non-parametric estimators of several distribution functions, consistent and asymptotically normal estimators are established. A simulation study shows the tendency that the lower the probability of observation, the higher the mean squared error of the estimators, even for the same number of observations. Finally, the method is applied to a doubly truncated data set of German companies, where the age-at-insolvency is of interest. Keywords: Insolvency risk, Linear regression, Non-parametric, Random double-truncation
{"title":"Linear regression with randomly double-truncated data","authors":"G. Frank, A. Dörre","doi":"10.37920/sasj.2017.51.1.1","DOIUrl":"https://doi.org/10.37920/sasj.2017.51.1.1","url":null,"abstract":"Non-parametric estimation for a linear regression model under random double-truncation is investigated, i.e. the variables are observed if and only if the dependent variable lies in a random interval. The method requires only weak distribution assumptions to ensure identifiability, but does not require any specific distribution family for any variable, neither for the truncation variables nor for the error term. By using non-parametric estimators of several distribution functions, consistent and asymptotically normal estimators are established. A simulation study shows the tendency that the lower the probability of observation, the higher the mean squared error of the estimators, even for the same number of observations. Finally, the method is applied to a doubly truncated data set of German companies, where the age-at-insolvency is of interest. Keywords: Insolvency risk, Linear regression, Non-parametric, Random double-truncation","PeriodicalId":53997,"journal":{"name":"SOUTH AFRICAN STATISTICAL JOURNAL","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47852007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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