Bayesian inference for Pareto distribution with prior conjugate and prior non conjugate

Comparison of Bayesian survival analysis and Cox regression analysis in simulated and breast cancer data sets

: A simple procedure is proposed to determine a sample size for estimating the mean weight of items in a problem of obtaining a batch of a large number of items. Suppose it is desired to obtain a large number N (sub s) of items for which individual counting is impractical, but one can demand a batch to weigh at least w units and hope that the number of items in the batch is close to the desired number N(sub s). If the items have mean weight theta, it is reasonable to have w equal to theta N sub s when theta is known. When theta is unknown, one can take a sample of size n, not bigger than N(sub s), estimate theta by a good estimator theta sub n and set w equal to theta sub n/sub s. The proposed procedure determines the sample size to be the integer closest to rhe CN sub s, where C is a function of the cost coefficients if the coefficient of variation rhe is known. It is shown to be optimal in some sense. If rho is unknown, a simple sequential procedure is proposed for which the average sample number is shown to be asymptotically equal to the optimal fixed sample size. When the weights are assumed to have a gamma distribution given theta and theta has a prior inverted gamma distribution, the optimal sample size in some sense can be found to be the nonnegative integer closest to rhe CN sub s + rho squared A(Rho C-1), where A is a known constant given in the prior distribution. Keywords: Nonparametric; Sequential procedure; Bayes procedure.

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.

This article compares maximum likelihood and Bayesian estimation of the correlated trait–correlated method (CT–CM) confirmatory factor model for multitrait–multimethod (MTMM) data. In particular, Bayesian estimation with minimally informative prior distributions—that is, prior distributions that prescribe equal probability across the known mathematical range of a parameter—are investigated as a source of information to aid convergence. Results from a simulation study indicate that Bayesian estimation with minimally informative priors produces admissible solutions more often maximum likelihood estimation (100.00% for Bayesian estimation, 49.82% for maximum likelihood). Extra convergence does not come at the cost of parameter accuracy; Bayesian parameter estimates showed comparable bias and better efficiency compared to maximum likelihood estimates. The results are echoed via 2 empirical examples. Hence, Bayesian estimation with minimally informative priors outperforms enables admissible solutions of the CT–CM model for MTMM data.

When Bayesian estimation is used to analyze Structural Equation Models (SEMs), prior distributions need to be specified for all parameters in the model. Many popular software programs offer default prior distributions, which is helpful for novel users and makes Bayesian SEM accessible for a broad audience. However, when the sample size is small, those prior distributions are not always suitable and can lead to untrustworthy results. In this tutorial, we provide a non-technical discussion of the risks associated with the use of default priors in small sample contexts. We discuss how default priors can unintentionally behave as highly informative priors when samples are small. Also, we demonstrate an online educational Shiny app, in which users can explore the impact of varying prior distributions and sample sizes on model results. We discuss how the Shiny app can be used in teaching; provide a reading list with literature on how to specify suitable prior distributions; and discuss guidelines on how to recognize (mis)behaving priors. It is our hope that this tutorial helps to spread awareness of the importance of specifying suitable priors when Bayesian SEM is used with small samples.

We presented approximate to Bayesian integrals of Gompertz distribution depending upon numerical integration and simulation study and showed how to study posterior distribution by means of simulation study. From the findings of above tables (1, 2, 3, 4) it has been found that the large sample distribution could be improved when prior is taken into account. In all cases (simulated data as well as real life data) normal approximation, T-K approximation, Bayesian estimates under informative priors are better than those under non-informative priors especially the Inverse levy distribution proves to be efficient with minimum posterior standard deviation. Further we accomplish that the posterior standard deviation based on different priors tends to decrease with the increase in sample size. It indicates that the estimators attained are consistent. It can also be detected that the performance of Bayes estimates under informative priors (inverse levy) is better than non-informative prior.

In this paper, Bayes estimators of the unknown shape and scale parameters of the Exponentiated Inverse Rayleigh Distribution (EIRD) have been derived using both the frequentist and bayesian methods. The Bayes theorem was adopted to obtain the posterior distribution of the shape and scale parameters of an Exponentiated Inverse Rayleigh Distribution (EIRD) using both conjugate and non-conjugate prior distribution under different loss functions (such as Entropy Loss Function, Linex Loss Function and Scale Invariant Squared Error Loss Function). The posterior distribution derived for both shape and scale parameters are intractable and a Lindley approximation was adopted to obtain the parameters of interest. The loss function were employed to obtain the estimates for both scale and shape parameters with an assumption that the both scale and shape parameters are unknown and independent. Also the Bayes estimate for the simulated datasets and real life datasets were obtained. The Bayes estimates obtained under dierent loss functions are close to the true parameter value of the shape and scale parameters. The estimators are then compared in terms of their Mean Square Error (MSE) using R programming language. We deduce that the MSE reduces as the sample size (n) increases.

In this paper, we propose Bayesian estimation of unknown parameters for a multinomial logit model. Since the multinomial logit model does not have conjugate prior distribution, we apply a Metropolis-Hastings (MH) algorithm. The MH algorithm requires prior and proposal distributions, which are frequently chosen according to the researcher’s belief or subjectivity. Unless we give appropriate distributions, the result is an inefficient or unbalanced estimation. Hence, we propose a combined empirical prior distribution that assumes the independence of the prior distribution for each of the parameter groups with an estimation procedure to obtain the appropriate estimation. We confirm the validity of the proposal method using multiple convergence diagnostics.

The objective of this study is to examine the properties of Bayes estimators of the shape parameter of the Power Function Distribution (PFD-I), by using two different prior distributions for the parameter θ and different loss functions that were compared with the maximum likelihood estimators. In many practical applications, we may have two different prior information about the prior distribution for the shape parameter of the Power Function Distribution, which influences the parameter estimation. So, we used two different kinds of conjugate priors of shape parameter θ of the Power Function Distribution (PFD-I) to estimate it. The conjugate prior function of the shape parameter θ was considered as a combination of two different prior distributions such as gamma distribution with Erlang distribution and Erlang distribution with exponential distribution and Erlang distribution with non-informative distribution and exponential distribution with the non-informative distribution. We derived Bayes estimators for shape parameter θ of the Power Function Distribution (PFD-I) according to different loss functions such as the squared error loss function (SELF), the weighted error loss function (WSELF) and modified linear exponential (MLINEX) loss function (MLF), with two different double priors. In addition to the classical estimation (maximum likelihood estimation). We used simulation to get the results of this study, for different cases of the shape parameter of the Power Function Distribution used to generate data for different samples sizes.

BackgroundAnalyses of multicenter studies often need to account for center clustering to ensure valid inference. For binary outcomes, it is particularly challenging to properly adjust for center when the number of centers or total sample size is small, or when there are few events per center. Our objective was to evaluate the performance of generalized estimating equation (GEE) log-binomial and Poisson models, generalized linear mixed models (GLMMs) assuming binomial and Poisson distributions, and a Bayesian binomial GLMM to account for center effect in these scenarios.MethodsWe conducted a simulation study with few centers (≤30) and 50 or fewer subjects per center, using both a randomized controlled trial and an observational study design to estimate relative risk. We compared the GEE and GLMM models with a log-binomial model without adjustment for clustering in terms of bias, root mean square error (RMSE), and coverage. For the Bayesian GLMM, we used informative neutral priors that are skeptical of large treatment effects that are almost never observed in studies of medical interventions.ResultsAll frequentist methods exhibited little bias, and the RMSE was very similar across the models. The binomial GLMM had poor convergence rates, ranging from 27% to 85%, but performed well otherwise. The results show that both GEE models need to use small sample corrections for robust SEs to achieve proper coverage of 95% CIs. The Bayesian GLMM had similar convergence rates but resulted in slightly more biased estimates for the smallest sample sizes. However, it had the smallest RMSE and good coverage across all scenarios. These results were very similar for both study designs.ConclusionsFor the analyses of multicenter studies with a binary outcome and few centers, we recommend adjustment for center with either a GEE log-binomial or Poisson model with appropriate small sample corrections or a Bayesian binomial GLMM with informative priors.

Selection of the best set of scales is problematic when developing signal‐driven approaches for pixel‐based image segmentation. Often, different possibly conflicting criteria need to be fulfilled in order to obtain the best trade‐off between uncertainty (variance) and location accuracy. The optimal set of scales depends on several factors: the noise level present in the image material, the prior distribution of the different types of segments, the class‐conditional distributions associated with each type of segment as well as the actual size of the (connected) segments. We analyse, theoretically and through experiments, the possibility of using the overall and class‐conditional error rates as criteria for selecting the optimal sampling of the linear and morphological scale spaces. It is shown that the overall error rate is optimized by taking the prior class distribution in the image material into account. However, a uniform (ignorant) prior distribution ensures constant class‐conditional error rates. Consequently, we advocate for a uniform prior class distribution when an uncommitted, scale‐invariant segmentation approach is desired. Experiments with a neural net classifier developed for segmentation of dynamic magnetic resonance (MR) images, acquired with a paramagnetic tracer, support the theoretical results. Furthermore, the experiments show that the addition of spatial features to the classifier, extracted from the linear or morphological scale spaces, improves the segmentation result compared to a signal‐driven approach based solely on the dynamic MR signal. The segmentation results obtained from the two types of features are compared using two novel quality measures that characterize spatial properties of labelled images. Copyright © 2002 John Wiley & Sons, Ltd.

A crucial decision in designing a spatial sample for soil survey is the number of sampling locations required to answer, with sufficient accuracy and precision, the questions posed by decision makers at different levels of geographic aggregation. In the Indian Soil Health Card (SHC) scheme, many thousands of locations are sampled per district. In this paper the SHC data are used to estimate the mean of a soil property within a defined study area, e.g., a district, or the areal fraction of the study area where some condition is satisfied, e.g., exceedence of a critical level. The central question is whether this large sample size is needed for this aim. The sample size required for a given maximum length of a confidence interval can be computed with formulas from classical sampling theory, using a prior estimate of the variance of the property of interest within the study area. Similarly, for the areal fraction a prior estimate of this fraction is required. In practice we are uncertain about these prior estimates, and our uncertainty is not accounted for in classical sample size determination (SSD). This deficiency can be overcome with a Bayesian approach, in which the prior estimate of the variance or areal fraction is replaced by a prior distribution. Once new data from the sample are available, this prior distribution is updated to a posterior distribution using Bayes’ rule. The apparent problem with a Bayesian approach prior to a sampling campaign is that the data are not yet available. This dilemma can be solved by computing, for a given sample size, the predictive distribution of the data, given a prior distribution on the population and design parameter. Thus we do not have a single vector with data values, but a finite or infinite set of possible data vectors. As a consequence, we have as many posterior distribution functions as we have data vectors. This leads to a probability distribution of lengths or coverages of Bayesian credible intervals, from which various criteria for SSD can be derived. Besides the fully Bayesian approach, a mixed Bayesian-likelihood approach for SSD is available. This is of interest when, after the data have been collected, we prefer to estimate the mean from these data only, using the frequentist approach, ignoring the prior distribution. The fully Bayesian and mixed Bayesian-likelihood approach are illustrated for estimating the mean of log-transformed Zn and the areal fraction with Zn-deficiency, defined as Zn concentration <0.9 mg kg −1, in the thirteen districts of Andhra Pradesh state. The SHC data from 2015–2017 are used to derive prior distributions. For all districts the Bayesian and mixed Bayesian-likelihood sample sizes are much smaller than the current sample sizes. The hyperparameters of the prior distributions have a strong effect on the sample sizes. We discuss methods to deal with this. Even at the mandal (sub-district) level the sample size can almost always be reduced substantially. Clearly SHC over-sampled, and here we show how to reduce the effort while still providing information required for decision-making. R scripts for SSD are provided as supplementary material.

ABSTRACTA uniform shrinkage prior (USP) distribution on the unknown variance component of a random-effects model is known to produce good frequency properties. The USP has a parameter that determines the shape of its density function, but it has been neglected whether the USP can maintain such good frequency properties regardless of the choice for the shape parameter. We investigate which choice for the shape parameter of the USP produces Bayesian interval estimates of random effects that meet their nominal confidence levels better than several existent choices in the literature. Using univariate and multivariate Gaussian hierarchical models, we show that the USP can achieve its best frequency properties when its shape parameter makes the USP behave similarly to an improper flat prior distribution on the unknown variance component.

This paper considers properties of half-normal distribution using informative priors under the Bayesian criterion. It employs the squared root inverted gamma, Chi-square and Rayleigh distributions as the prior distribution to construct the Posterior distributions of the respective distributional parameters. Hyperparameters are elicited via prior predictive distribution. The properties of posterior distribution are studied, and their graphs are presented using a real data set. A comprehensive simulation scheme is conducted using informative priors. Bayes estimates are obtained using the loss functions (squared error loss function, modified loss function, quadratic loss function and Degroot loss function). Statistical inferences interval estimates and Bayesian hypothesis testing are presented to demonstrate the usefulness of the study.

The exponential-logarithmic is a new lifetime distribution with decreasing failure rate and interesting applications in the biological and engineering sciences. Thus, a Bayesian analysis of the parameters would be desirable. Bayesian estimation requires the selection of prior distributions for all parameters of the model. In this case, researchers usually seek to choose a prior that has little information on the parameters, allowing the data to be very informative relative to the prior information. Assuming some noninformative prior distributions, we present a Bayesian analysis using Markov Chain Monte Carlo (MCMC) methods. Jeffreys prior is derived for the parameters of exponential-logarithmic distribution and compared with other common priors such as beta, gamma, and uniform distributions. In this article, we show through a simulation study that the maximum likelihood estimate may not exist except under restrictive conditions. In addition, the posterior density is sometimes bimodal when an improper prior density is used.