Interval estimation of the mean and quantiles of a lognormal distribution is addressed based on a Type I singly censored sample. A special case of interest is that of a sample containing values below a single detection limit. Generalized inferential procedures which use maximum likelihood estimation based approximate pivotal quantities, and some likelihood based methods, are proposed. The latter include methodology based on the signed log-likelihood ratio test (SLRT) statistic and the modified signed log-likelihood ratio test (MSLRT) statistic. The merits of the methods are evaluated for a left-censored sample using Monte Carlo simulation. For inference concerning the lognormal mean, the SLRT is to be preferred for left-tailed testing, generalized inference for right-tailed testing, and all three approaches provide nearly the same performance for two-tailed testing. These conclusions hold even when the proportion of censored values is as large as 0.70. For inference concerning quantiles, both the generalized inference approach and the MSLRT approach are satisfactory. In view of its simplicity and ease of understanding and implementation, the generalized inference procedure is to be preferred. The results are illustrated with two examples. Technical derivations are given on the Technometrics website as supplementary material.
In this article, additional properties of the Gumbel-Burr XII distribution, denoted by (GBXII(L)), defined in (Osatohanmwen et al., 2017), are studied. We consider some useful characterizations for the GBXII(L) distribution and some of its properties. A simulation study is conducted to assess the performance of the MLEs and the usefulness of the GBXII(L) distribution is illustrated by means of three real data sets. The simulation study suggests that the maximum likelihood method can be used to estimate the distribution parameters, and the three examples show that the GBXII(L) is very flexible in fitting different shapes of data. A log-GBXII(L) regression model is proposed and a survival data is used in an application of the proposed regression model. The log-GBXII(L) regression model is adequate and can be used in comparison to other models.
A count data that have excess number of zeros, ones, twos or threes are commonplace in experimental studies. But these inflated frequencies at particular counts may lead to overdispersion and thus may cause difficulty in data analysis. So to get appropriate results from them and to overcome the possible anomalies in parameter estimation, we may need to consider suitable inflated distribution. Generally, Inflated Poisson or Inflated Negative Binomial distribution are the most commonly used for modeling and analyzing such data. Geometric distribution is a special case of Negative Binomial distribution. This work deals with parameter estimation of a Geometric distribution inflated at certain counts, which we called Generalized Inflated Geometric (GIG) distribution. Parameter estimation is done using method of moments, empirical probability generating function based method and maximum likelihood estimation approach. The three types of estimators are then compared using simulation studies and finally a Swedish fertility dataset was modeled using a GIG distribution.
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