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This paper deals with estimating the underlying parameter of a length-biased inverse Gaussian distribution, when the observations are prone to length-biased sampling. Length-biased sampling occurs when the observations of smaller lengths or dimensions are neglected from the sample. We focus on a particular type of sequential fixed-accuracy confidence interval for estimation purposes. This method proves to be both time and cost efficient as one is able to perform the estimation using an optimal number of observations according to some set criteria. We discuss the applicability of our proposed method with regards to estimating the cluster size of the "Eastern Tropical Pacific Dolphins", which are often vulnerable to length biasedness. [ABSTRACT FROM AUTHOR]

In this article, the length-biased power hazard rate distribution has introduced and investigated several statistical properties. This distribution reports an extension of several probability distributions, namely: exponential, Rayleigh, Weibull, and linear hazard rate. The procedure of maximum likelihood estimation is taken for parameters. Finally, the applicability of the model is explored by three real data sets. To examine, the performance of the technique, a simulation study is extracted. [ABSTRACT FROM AUTHOR]

The aims of this paper is to develop and establish the theoretical properties of a new survival distribution based on a preceding well-known distribution. The proposed distribution is called the length-biased Nakagami distribution. Also, we apply the proposed distribution to a set of actual data to show that the length-biased Nakagami distribution is appropriate for generating a survival model for data. [ABSTRACT FROM AUTHOR]

In this study, we introduce a new class of length–biased distribution, the length–biased Weibull–Rayleigh (LBWR) distribution, and provide its properties such as the limit behavior, survival function, hazard rate function, th moment, and moment generating function. Moreover, maximum likelihood estimation is used in the parameter estimation. The LBWR distribution was fitted to two hydrological datasets and its efficacy compared with Rayleigh, Weibull, Pareto, and Weibull–Rayleigh distributions, the results of which show that the novel distribution provides a better fit than the others. [ABSTRACT FROM AUTHOR]

Abstract Background In the presence of an intermediate clinical event, the analysis of time-to-event survival data by conventional approaches, such as the log-rank test, can result in biased results due to the length-biased characteristics. Methods In the present study, we extend the studies of Finkelstein and Nam & Zelen to propose new methods for handling interval-censored data with an intermediate clinical event using multiple imputation. The proposed methods consider two types of weights in multiple imputation: 1) uniform weight and 2) the weighted weight methods. Results Extensive simulation studies were performed to compare the proposed tests with existing methods regarding type I error and power. Our simulation results demonstrate that for all scenarios, our proposed methods exhibit a superior performance compared with the stratified log-rank and the log-rank tests. Data from a randomized clinical study to test the efficacy of sorafenib/sunitinib vs. sunitinib/sorafenib to treat metastatic renal cell carcinoma were analyzed under the proposed methods to illustrate their performance on real data. Conclusions In the absence of intensive iterations, our proposed methods show a superior performance compared with the stratified log-rank and the log-rank test regarding type I error and power.

This study examines regression analyses of length-biased and right-censored failure time data arising from the short-term and long-term hazard ratio model. Compared with some commonly used models, such as the proportional hazards models, the short-term and long-term hazard ratio model has the advantage of allowing crossing hazard functions and, thus, is more flexible. We propose two methods for estimating the regression parameters, the conditional likelihood approach, and the composite conditional likelihood approach. We establish the asymptotic and finite-sample properties of the proposed estimators, and the numerical results suggest that the methods work well in practical situations. In addition, the approaches are applied to a set of real data arising from a dementia study. [ABSTRACT FROM AUTHOR]

Weighted distributions, as an example of informative sampling, work appropriately under the missing at random mechanism since they neglect missing values and only completely observed subjects are used in the study plan. However, length-biased distributions, as a special case of weighted distributions, remove the subjects with short length deliberately, which surely meet the missing not at random mechanism. Accordingly, applying length-biased distributions jeopardizes the results by producing biased estimates. Hence, an alternate method has to be used such that the results are improved by means of valid inferences. We propose methods that are based on weighted distributions and joint modelling procedure and compare them in analysing longitudinal data. After introducing three methods in use, a set of simulation studies and analysis of two real longitudinal datasets affirm our claim. [ABSTRACT FROM AUTHOR]

In this paper, a ranked set sampling procedure with ranking based on a length-biased concomitant variable is proposed. The estimate for population mean based on this sample is given. It is proved that the estimate based on ranked set samples is asymptotically more efficient than the estimate based on simple random samples. Simulation studies are conducted to present the properties of the proposed estimate for finite sample size. Moreover, the consequence of ignoring length bias is also addressed by simulation studies and the real data analysis. [ABSTRACT FROM AUTHOR]

In this study, we reconsider weighted distribution from the perspective of missing mechanism since weighted distribution instead of being the distribution of the whole population of interest is only the distribution of respondents (sub-population). After defining some weighted distributions by different mechanisms for indicator of response, we show, by some simulation studies, that using weighted distributions may lead to biased estimates of parameters under the non-ignorable missing mechanism. On the other hand, joint modeling of the response and selection mechanism could result in more efficient and valid estimates of parameters. The lower root of mean squared errors of estimates from the joint modeling approach than those of the weighted distribution is a warranty to the statement that the joint modeling method is more efficient than weighted distribution; this is proved by diverse simulation studies along the article. However, these two methods of the weighted approach and joint modeling give similar results if the selection mechanism is at random. Finally, the methods are applied and compared in the analysis of one well-used real dataset. [ABSTRACT FROM AUTHOR]

This article discusses regression analysis of length-biased and right-censored failure time data arising from the accelerated failure time model. A key feature of such data is the informative censoring induced by the length-biased sampling, and several methods have been proposed in the literature for their analysis. However, these may be less efficient or apply only to limited situations. We propose a kernel-smoothed composite likelihood method for estimation of covariate effects. The proposed estimators are proved to be consistent and asymptotically normal. Simulation studies conducted to assess the finite sample performance of the method suggest that it works well for practical situations. An illustrative example is provided. [ABSTRACT FROM AUTHOR]

Length-Biased distributions are a special case of the more general form known as weighted distributions. We can exploit the conceptuality of Length-Biased distribution in the development of appropriate models for lifetime data. Its method is adjusting the original probability density function from real data and the expectation of those data. This modification can lead to correct conclusions of the models. Therefore, we introduced the Length-Biased version of the weighted Exponentiated inverted Weibull distribution in this paper. Various properties and the expressions for moments, coefficient of skewness, coefficient of kurtosis, moment generating function, hazard rate function, etc. are derived. The maximum likelihood estimates of the parameters of the proposed distribution are determined. The study results suggest that this distribution is an efficacious model in life time data analysis and other related fields. [ABSTRACT FROM AUTHOR]

In this article, we introduce a mixture inverse Gaussian (MIG) model based on the Student-t distribution and apply it to bacterium-based protein production for food industry. This model is mainly useful to describe data that follow positively skewed distributions and accommodate atypical observations in a better way than its classical version. Specifically, we present a characterization of the MIG-t distribution. In addition, we carry out a hazard analysis of this distribution centered mainly on its hazard rate. Furthermore, we discuss the maximum likelihood method, which produces--in this case--robust parameter estimates. Moreover, to evaluate the potential influence of atypical observations, we produce a diagnostic analysis for the model. Finally, we apply the obtained results to novel bacterium-based protein production data and statistically compare two types of protein producers using the likelihood ratio test based on the MIG-t model as an alternative methodology to the procedures available until now. This fact is very important, since the evaluation of protein production using both constructions allows practitioners to choose the most productive one before the bacterial culture is scaled to an industrial level.En este articulo, introducimos un modelo Gaussiano inverso (MIG) mezclado basado en la distribucion t-Student y lo aplicamos a la produccion de proteinas basada en bacterias para la industria de alimentos. Este modelo es especialmente util para describir datos que siguen una distribucion con sesgo positivo ya que permite acomodar observaciones atipicas de mejor forma que su versión clasica. Espec{i}ficamente, presentamos una caracterizacion de la distribución MIG-t y realizamos un analisis de confiabilidad de esta distribucion centrado principalmente en la tasa de fallas. También, discutimos el metodo de verosimilitud maxima, el cual proporciona en este caso estimaciones robustas de los parametros del modelo. Con el fin de evaluar la influencia potencial de observaciones atipicas, proponemos un analisis de diagnostico para la distribucion. Finalmente, aplicamos los resultados obtenidos al análisis de datos nuevos de produccion de proteina basada en bacterias utilizada en la industria de alimentos y comparamos estadísticamente dos tipos de bacterias productoras usando la prueba de razon de verosimilitudes basada en el modelo MIG-t como una metodologia alternativa a los procedimientos disponibles a la fecha. Este punto es muy importante, ya que la evaluacion de produccion de proteinas usando dos construcciones distintas permite a los investigadores escoger el tipo mas productivo antes de proceder al cultivo industrial a gran escala.

In some statistical applications, data may not be considered as a random sample of the whole population and some subjects have less probability of belonging to the sample. Consequently, statistical inferences for such data sets, usually yields biased estimation. In such situations, thelength-biasedversion of the original random variable as a special weighted distribution often produces better inferences. An alternative weighted distribution based on the mean residual life is suggested to treat the biasedness. The Rayleigh distribution is applied in many real applications, hence the proposed method of weighting is performed to produce a new lifetime distribution based on the Rayleigh model. In addition, statistical properties of the proposed distribution is investigated. A simulation study and a real data set are prepared to illustrate that the mean residual weighted Rayleigh distribution gives a better fit than the original and also thelength-biasedRayleigh distribution. [ABSTRACT FROM AUTHOR]

In this article, we discuss nonparametric estimation of a mean residual life function from length-biased data. Precisely, we prove strong uniform consistency and weak converge of the nonparametric mean residual life estimator in length-biased setting. [ABSTRACT FROM AUTHOR]

Left-truncation often arises when patient information, such as time of diagnosis, is gathered retrospectively. In some cases, the distribution function, sayG(x), of left-truncated variables can be parameterized asG(x; θ), where θ∈Θ⊂Rqand θ is aq-dimensional vector. Under semiparametric transformation models, we demonstrated that the approach of Chenet al.(Semiparametric analysis of transformation models with censored data. Biometrika. 2002;89:659–668) can be used to analyse this type of data. The asymptotic properties of the proposed estimators are derived. A simulation study is conducted to investigate the performance of the proposed estimators. [ABSTRACT FROM AUTHOR]

The complication in analyzing tumor data is that the tumors detected in a screening program tend to be slowly progressive tumors, which is the so-called length-biased sampling that is inherent in screening studies. Under the assumption that all subjects have the same tumor growth function, Ghosh (2008) developed estimation procedures for proportional hazards model. In this article, by modeling growth function as a function of covariates, we demonstrate that Ghosh (2008)'s approach can be extended to the case when each subject has a specific growth function. A simulation study is conducted to demonstrate the potential usefulness of the proposed estimators for the regression parameters in the proportional and additive hazards model. [ABSTRACT FROM AUTHOR]