Inference about parameters in Binomial-Poisson distribution with additional incomplete data

ABSTRACT

This article considers the distribution of Binomial-Poisson random vector which has two components and includes two parameters: one is the rate of a Poisson distribution, the other is the proportion in a Binomial distribution. The inference about the two parameters is usually made based on only paired observations. However, the number of paired observations is, in general, not large enough because of either technical difficulty or budget limitation, and so one can not make efficient inferences with only paired data. Instead, it is often much easier and not too costly to have incomplete observation on only one component independently. In this article we will combine both the paired complete data and unpaired incomplete data for estimating the two parameters. The performances of various estimators are compared both analytically and numerically. It is observed that fully using the unpaired incomplete data can always improve the inference, and the improvement is very significant in the case when there are only a few paired complete observations.     

Estimation of p(y<x) in the gamma case

We consider estimation of P(Y<X) when X?Γr(M,λ) and Y?Γ(N,μ) are independent with M and N known. A concise representation of the UMVUE and several representations for the MLE are derived. Closed-form exact expressions of both MSE's and the bias of the MLE are obtained. Large-sample results are given and numerical comparison of the two point estimators is made. Confidence intervals are given.     

An identity for exponential distributions with the common location parameter and its applications

An identity for exponential distributions with an unknown common location parameter and unknown and possibly unequal scale parameters is established.Through use of the identity the maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (UMVUE) of a quantile of an exponential population are compared under the squared error loss.A class of estimators dominating both MLE and UMVUE is obtained by using the identity.     

Limiting distributions of MLE and UMVUE in the biparametric uniform distribution

In this paper, we study the asymptotic distributions of MLE and UMVUE of a parametric functionh(θ₁, θ₂) when sampling from a biparametric uniform distributionU(θ₁, θ₂). We obtain both limiting distributions as a convolution of exponential distributions, and we observe that the limiting distribution of UMVUE is a shift of the limiting distribution of MLE.     

Explicit expressions for gamma spacings and their moments

This work is concerned with the distributions of spacings from a two-parameter gamma distribution, when the shape parameter is a positive integer (or Erlang Distribution). We express the probability density functions of spacings and their moments in closed forms that are easy to implement using computer algebra systems like Mathematica.     

Generalized gamma parameter estimation and moment evaluation

The generalized gamma distribution includes the exponential distribution, the gamma distribution, and the Weibull distribution as special cases. It also includes the log-normal distribution in the limit as one of its parameters goes to infinity. Prentice (1974) developed an estimation method that is effective even when the underlying distribution is nearly log-normal. He reparameterized the density function so that it achieved the limiting case in a smooth fashion relative to the new parameters. He also gave formulas for the second partial derivatives of the log-density function to be used in the nearly log-normal case. His formulas included infinite summations, and he did not estimate the error in approximating these summations.

We derive approximations for the log-density function and moments of the generalized gamma distribution that are smooth in the nearly log-normal case and involve only finite summations. Absolute error bounds for these approximations are included. The approximation for the first moment is applied to the problem of estimating the parameters of a generalized gamma distribution under the constraint that the distribution have mean one. This enables the development of a correspondence between the parameters in a mean one generalized gamma distribution and certain parameters in acoustic scattering theory.     

Inference for the scale parameter of lifetime distribution of k-unit parallel system based on progressively censored data

In this paper, inference for the scale parameter of lifetime distribution of a k-unit parallel system is provided. Lifetime distribution of each unit of the system is assumed to be a member of a scale family of distributions. Maximum likelihood estimator (MLE) and confidence intervals for the scale parameter based on progressively Type-II censored sample are obtained. A β-expectation tolerance interval for the lifetime of the system is obtained. As a member of the scale family, half-logistic distribution is considered and the performance of the MLE, confidence intervals and tolerance intervals are studied using simulation.     

Estimation of Parameters of a Mixed Failure Time Distribution Which is a Mixture of Degenerate (Degenerate at Zero) and I. G. Distribution

The problem of estimation of parameters of a mixture of degenerate (at zero) and exponential distribution is considered by Jayade and Prasad (1990). The sampling scheme proposed in it is extended in this paper to a mixture of degenerate and Inverse Gaussian distribution. The Inverse Gaussian distribution is very relevant for studying reliability and life-testing problems. The inverse Gaussian being the first passage time distribution for Wiener process makes it particularly appropriate for failure or reaction time data analysis.     

Estimation of reliability with cumulative stress and strength degradation

In many real-life scenarios, system reliability depends on dynamic stress–strength interference where strength degrades and stress accumulates concurrently over time. In this paper, we consider the problem of estimating reliability of a system under deterministic strength degradation and cumulative damage due to shocks arriving according to a point process. Maximum likelihood estimation under two different sampling plans has been considered. Large sample properties in general are discussed. The method is illustrated through simulation and real-life data analysis.     

A gamma goodness-of-fit test based on characteristic independence of the mean and coefficient of variation

Characterization theorems in probability and statistics are widely appreciated for their role in clarifying the structure of the families of probability distributions. Less well known is the role characterization theorems have as a natural, logical and effective starting point for constructing goodness-of-fit tests. The characteristic independence of the mean and variance and of the mean and the third central moment of a normal sample were used, respectively, by Lin and Mudholkar [1980. A simple test for normality against asymmetric alternatives. Biometrika 67, 455–461] and by Mudholkar et al. [2002a. Independence characterizations and testing normality against skewness-kurtosis alternatives. J. Statist. Plann. Inference 104, 485–501] for developing tests of normality. The characteristic independence of the maximum likelihood estimates of the population parameters was similarly used by Mudholkar et al. [2002b. Independence characterization and inverse Gaussian goodness-of-fit. Sankhya A 63, 362–374] to develop a test of the composite inverse Gaussian hypothesis. The gamma models are extensively used for applied research in the areas of econometrics, engineering and biomedical sciences; but there are few goodness-of-fit tests available to test if the data indeed come from a gamma population. In this paper we employ Hwang and Hu's [1999. On a characterization of the gamma distribution: the independence of the sample mean and the sample coefficient of variation. Ann. Inst. Statist. Math. 51, 749–753] characterization of the gamma population in terms of the independence of sample mean and coefficient of variation for developing such a test. The asymptotic null distribution of the proposed test statistic is obtained and empirically refined for use with samples of moderate size.     

Inference procedures for the variance gamma model and applications

Goodness-of-fit tests for the family of the four-parameter normal–variance gamma distribution are constructed. The tests are based on a weighted integral incorporating the empirical characteristic function of suitably standardized data. Non-standard algorithms are employed for the computation of the maximum-likelihood estimators of the parameters involved in the test statistic, while Monte Carlo results are used in order to compare the new test with some classical goodness-of-fit methods. A real-data application is also included.     

A new generalization of the gamma distribution with application to negatively skewed survival data

We propose a three-parameter distribution referred to as the reflected- shifted-truncated gamma (RSTG) distribution to model negatively skewed data. Various properties of the proposed distribution are derived. The estimation of the model parameters is approached by maximum likelihood methods and the observed information matrix is derived. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. Using information theoretic criteria, we compare the RSTG distribution to the exponential, generalized F, generalized gamma, Gompertz, log-logistic, lognormal, Rayleigh, and Weibull distributions in three negatively skewed real datasets.     

The empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior under Stein's loss function

For the hierarchical Poisson and gamma model, we calculate the Bayes posterior estimator of the parameter of the Poisson distribution under Stein's loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein's Loss (PESL). We also obtain the Bayes posterior estimator of the parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior by two methods. In numerical simulations, we have illustrated: The two inequalities of the Bayes posterior estimators and the PESLs; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we exploit the attendance data on 314 high school juniors from two urban high schools to illustrate our theoretical studies.     

An algorithm for the evaluation of the incomplete gamma function and the first two partial derivatives with respect to the parameter

A simple to use, straight coded, Fortran 4 algorithm, is presented. This algorithm has the ability to:

1) evaluate the Incomplete Gamma function, Y(r,λx), for parameter values in the range 0 < r < 20.0 and upper limit of integration values in the range 0 75.0;

2) evaluate both the first and second partial derivatives of y with respect to the parameter

3) evaluate both the Euler Di and Trigamma functions, ψ(r) and if ψ′(r)for 0<rf.20.0.

In all cases the accuracy is nine or more significant figures. The user has several choices of data output format     

Empirical bayes estimation of a distribution function with a gamma process prior

A sequence of empirical Bayes estimators is given for estimating a distribution function. It is shown that ‘i’ this sequence is asymptotically optimum relative to a Gamma process prior, ‘ii’ the overall expected loss approaches the minimum Bayes risk at a rate of n , and ‘iii’ the estimators form a sequence of proper distribution functions. Finally, the numerical example presented by Susarla and Van Ryzin ‘Ann. Statist., 6, 1978’ reworked by Phadia ‘Ann. Statist., 1, 1980, to appear’ has been analyzed and the results are compared to the numerical results by Phadia     

Estimation of dynamic panel data models with both individual and time-specific effects

This paper proposes a generalized least squares and a generalized method of moment estimators for dynamic panel data models with both individual-specific and time-specific effects. We also demonstrate that the common estimators ignoring the presence of time-specific effects are inconsistent when N→∞$N\to \infty$ but T is finite if the time-specific effects are indeed present. Monte Carlo studies are also conducted to investigate the finite sample properties of various estimators. It is found that the generalized least squares estimator has the smallest bias and root mean square error, and also has nominal size close to the empirical size. It is also found that even when there is no presence of time-specific effects, there is hardly any efficiency loss of the generalized least squares estimator assuming its presence compared to the generalized least squares estimator allowing only the presence of individual-specific effects.     

Multi - sample test of equal gamma distribution scale parameters in presence of unknown common shape parameter

Shiue and Bain proposed an approximate F statistic for testing equality of two gamma distribution scale parameters in presence of a common and unknown shape parameter. By generalizing Shiue and Bain's statistic we develop a new statistic for testing equality of L >= 2 gamma distribution scale parameters. We derive the distribution of the new statistic ESP for L = 2 and equal sample size situation. For other situations distribution of ESP is not known and test based on the ESP statistic has to be performed by using simulated critical values. We also derive a C(α) statistic CML and develop a likelihood ratio statistic, LR, two modified likelihood ratio statistics M and MLB and a quadratic statistic Q. The distribution of each of the statistics CML, LR, M, MLB and Q is asymptotically chi-square with L - 1 degrees of freedom. We then conducted a monte-carlo simulation study to compare the perfor- mance of the statistics ESP, LR, M, MLB, CML and Q in terms of size and power. The statistics LR, M, MLB and Q are in general liberal and do not show power advantage over other statistics. The statistic CML, based on its asymptotic chi-square distribution, in general, holds nominal level well. It is most powerful or nearly most powerful in most situations and is simple to use. Hence, we recommend the statistic CML for use in general. For better power the statistic ESP, based on its empirical distribution, is recommended for the special situation for which there is evidence in the data that λ₁ < … < λ_L and n₁ < … < n_L, where λ₁ …, λ_L are the scale parameters and n₁,…, n_L are the sample sizes.     

The average likelihood and a fiducial approximation: one parameter members of the generalized gamma distributions

The average likelihood, defined as the integral of the like-lihood function over the parameter space, has been used as a criterion for model selection The form of the average likelihood considered uses a uniform prior. An approximation is presented based on fiducial distributions. The sampling distributions of the average likelihood and its fiducial approximation are derived for cases of sampling from one parameter members of the general-ized gamma distributions.     

Reliability modeling of systems with two dependent degrading components based on gamma processes

Abstract

Many engineering systems have multiple components with more than one degradation measure which is dependent on each other due to their complex failure mechanisms, which results in some insurmountable difficulties for reliability work in engineering. To overcome these difficulties, the system reliability prediction approaches based on performance degradation theory develop rapidly in recent years, and show their superiority over the traditional approaches in many applications. This paper proposes reliability models of systems with two dependent degrading components. It is assumed that the degradation paths of the components are governed by gamma processes. For a parallel system, its failure probability function can be approximated by the bivariate Birnbaum–Saunders distribution. According to the relationship of parallel and series systems, it is easy to find that the failure probability function of a series system can be expressed by the bivariate Birnbaum–Saunders distribution and its marginal distributions. The model in such a situation is very complicated and analytically intractable, and becomes cumbersome from a computational viewpoint. For this reason, the Bayesian Markov chain Monte Carlo method is developed for this problem that allows the maximum likelihood estimates of the parameters to be determined in an efficient manner. After that, the confidence intervals of the failure probability of systems are given. For an illustration of the proposed model, a numerical example about railway track is presented.     

A Multi-Stage Two-Machines Replacement Strategy Using Mixture Models,Bayesian Inference,and Stochastic Dynamic Programming

If at least one out of two serial machines that produce a specific product in manufacturing environments malfunctions, there will be non conforming items produced. Determining the optimal time of the machines' maintenance is the one of major concerns. While a convenient common practice for this kind of problem is to fit a single probability distribution to the combined defect data, it does not adequately capture the fact that there are two different underlying causes of failures. A better approach is to view the defects as arising from a mixture population: one due to the first machine failures and the other due to the second one. In this article, a mixture model along with both Bayesian inference and stochastic dynamic programming approaches are used to find the multi-stage optimal replacement strategy. Using the posterior probability of the machines to be in state λ₁, λ₂ (the failure rates of defective items produced by machine 1 and 2, respectively), we first formulate the problem as a stochastic dynamic programming model. Then, we derive some properties for the optimal value of the objective function and propose a solution algorithm. At the end, the application of the proposed methodology is demonstrated by a numerical example and an error analysis is performed to evaluate the performances of the proposed procedure. The results of this analysis show that the proposed method performs satisfactorily when a different number of observations on the times between productions of defective products is available.