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The accurate values of population parameters such as population proportion, population mean, and population standard deviation (or variance) are usually unknown. These are fixed values that can only be estimated from the data collected from the samples. The estimates of each of these parameters are sample proportion, the sample mean, and sample standard deviation (or variance). To obtain the values of these sample statistics, data are required that have particular distribution and central...
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The normal, a continuous distribution, is the most important of all the distributions. Its graph is a bell-shaped symmetrical curve, which is observed in almost all disciplines. Some of these include psychology, business, economics, the sciences, nursing, and, of course, mathematics. Some instructors may use the normal distribution to help determine students’ grades. Most IQ scores are normally distributed. Often real-estate prices fit a normal distribution. The normal distribution is...
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An unbiased point estimate is often insufficient to predict a population estimate, such as population mean or population proportion. In this scenario, a confidence interval is used. A confidence interval is an estimate similar to a  sample proportion. However, unlike the point estimate which is a single value, the confidence interval  contains a range of values. These values have lower and upper limits, known as confidence limits, and can be designated as L1 and L2, respectively.
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The interval estimate of any variable is known as the prediction interval. It helps decide if a point estimate is dependable.
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The z and the Student t distribution estimate the population mean using the sample mean and standard deviation. However, to decide which distribution to use for a calculation, one needs to determine the sample size, the nature of the distribution, and whether the population standard deviation is known. If the population standard deviation is known and the population is normally distributed, or if the sample size is greater than 30, the z distribution is preferred. The Student t distribution is...
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The population standard deviation is rarely known in many day-to-day examples of statistics. When the sample sizes are large, it is easy to estimate the population standard deviation using a confidence interval, which provides results close enough to the original value. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
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The Zero-Adjusted Log-Symmetric Distributions: Point and Intervalar Estimation.

Diego Risco-Cosavalente1, Francisco José A Cysneiros1

  • 1Universidade Federal de Pernambuco, Departamento de Estatística, Av. Prof Luiz Freire, s/n, Cidade Universitária, 50670-901 Recife, PE, Brazil.

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Summary
This summary is machine-generated.

A new zero-adjusted log-symmetric distribution is introduced for semi-continuous data. This flexible model, estimated using maximum likelihood, shows promise for real-world applications with varying data tails.

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Area of Science:

  • Statistics
  • Probability Theory
  • Data Analysis

Background:

  • Semi-continuous data, common in fields like biostatistics and econometrics, often requires specialized modeling approaches.
  • Existing distributions may not adequately capture the characteristics of data with a high proportion of zero values and a continuous positive component.

Purpose of the Study:

  • To introduce and investigate a novel class of semi-continuous probability distributions: the zero-adjusted log-symmetric (ZALS) distributions.
  • To derive key properties and parameter estimation methods for the ZALS family.
  • To assess the performance of the proposed estimators through simulation and demonstrate practical utility with real data.

Main Methods:

  • Theoretical derivation of ZALS distribution properties.
  • Application of the maximum likelihood estimation (MLE) method for parameter estimation.
  • Development of confidence intervals (CIs) for the estimated parameters.
  • Conducting a simulation study to evaluate MLE performance across different tail behaviors (lighter/heavier).
  • Illustrative application using a real-world dataset.

Main Results:

  • The ZALS distribution is formally defined and its fundamental properties are established.
  • Maximum likelihood estimators and confidence intervals for ZALS parameters are derived.
  • Simulation results indicate that the MLEs perform adequately in both lighter and heavier-tailed scenarios.
  • The real data application demonstrates the practical applicability and flexibility of the ZALS distribution.

Conclusions:

  • The proposed zero-adjusted log-symmetric distribution offers a flexible and effective tool for modeling semi-continuous data.
  • The maximum likelihood method provides reliable parameter estimation for this new class of distributions.
  • The ZALS family is a valuable addition to the statistical toolkit for analyzing complex datasets with zero-inflated and continuous components.