A Similarity Measure Based on Species Proportions

ABSTRACT

There are several indices for measuring the similarity of two populations, including the ratio of the number of shared species to the number of distinct species (Jaccard's index) and the conditional probability of observing a shared species (Smith et al., 1996 Smith , W. , Solow , A. R. , Preston , P. E. ( 1996 ). An estimator of species overlap using a modified beta-binomial model. Biometrics 52 : 1472 – 1477 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). However, these indices only take into account the number of species and species proportions of shared species. In this article, we propose a new similarity index which includes the species proportions of both the shared and non shared species in each population, and also propose a Nonparametric Maximum Likelihood Estimator (NPMLE) for this index. Bootstrap and delta methods are used to evaluate the standard errors of the NPMLE. Based on a loss function, we also compare a class of nonparametric estimators for the proposed index in various situations.     

Estimation in the Three-Parameter Gamma Distribution Based on the Profile Log-Likelihood Function

In this article, we express the profile log-likelihood function for the three-parameter gamma distribution in terms of the location parameter only and we study its properties. The behavior of the profile function is examined as the location parameter tends to the boundary values, i.e., to ? ∞ and to the minimum value of the sample. As a result, we obtain that if the log-likelihood function has a local maximum then it has another stationary value which is a saddle point. The results are supported with the use of simulation results.     

Estimation of Parameters of Mixed Failure Time Distribution Based on an Extended Modified Sampling Scheme

Abstract

The problem of estimation of parameters of a mixture of degenerate (at zero) and exponential distribution is considered by Dixit and Prasad [Dixit, V. U. (Nee: Jayade, V. D.), Prasad, M. S. (1990 Dixit, V. U. and Prasad, M. S. 1990. Estimation of parameters of mixed failure time distribution. Commun.in Statist.-Theory Meth, 19(12): 4667–4678. (Nee: Jayade, V. D.) [Google Scholar]). Estimation of parameters of mixed failure time distribution. Commun.in Statist.-Theory Meth., 19(12):4667–4678]. The sampling scheme proposed in it is extended to k positive observations in Dixit [Dixit, V. U. (1993 Dixit, V. U. 1993. “Statistical Inference for AR (1) Process with Mixed Errors”. Kolhapur, , India: Shivaji University. Unpublished Ph.D. thesis [Google Scholar]). Statistical Inference for AR (1) Process with Mixed Errors. Unpublished Ph.D. thesis, Shivaji University Kolhapur, India] and moment estimator, MLE and UMVUE based on it are obtained and their finite sample and asymptotic properties are studied. These results are presented in this paper. It is interesting to mention that the sampling scheme proposed by Shinde and Shanubhogue [Shinde, R. L., Shanubhogue, A. (2000 Shinde, R. L. and Shanubhogue, A. 2000. Estimation of parameters and the mean life of a mixed failure time distribution. Commun. Statist.-Theory Meth, 29(11): 2621–2642. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Estimation of parameters and the mean life of a mixed failure time distribution. Commun. Statist.-Theory Meth. 29(11):2621–2642] is a particular case of the sampling scheme proposed in Dixit [Dixit, V. U. (1993 Dixit, V. U. 1993. “Statistical Inference for AR (1) Process with Mixed Errors”. Kolhapur, , India: Shivaji University. Unpublished Ph.D. thesis [Google Scholar]). Statistical Inference for AR (1) Process with Mixed Errors. Unpublished Ph.D. thesis, Shivaji University Kolhapur, India] for n = k.     

Stress-Strength Reliability of a Two-Parameter Bathtub-shaped Lifetime Distribution Based on Progressively Censored Samples

Based on progressively Type-II censored samples, this article deals with inference for the stress-strength reliability R = P(Y < X) when X and Y are two independent two-parameter bathtub-shape lifetime distributions with different scale parameters, but having the same shape parameter. Different methods for estimating the reliability are applied. The maximum likelihood estimate of R is derived. Also, its asymptotic distribution is used to construct an asymptotic confidence interval for R. Assuming that the shape parameter is known, the maximum likelihood estimator of R is obtained. Based on the exact distribution of the maximum likelihood estimator of R an exact confidence interval of that has been obtained. The uniformly minimum variance unbiased estimator are calculated for R. Bayes estimate of R and the associated credible interval are also got under the assumption of independent gamma priors. Monte Carlo simulations are performed to compare the performances of the proposed estimators. One data analysis has been performed for illustrative purpose. Finally, we will generalize this distribution to the proportional hazard family with two parameters and derive various estimators in this family.     

A Note on the Estimate of Treatment Effect from a Cox Regression Model When the Proportionality Assumption Is Violated

ABSTRACT

Cox proportional hazards regression model has been widely used to estimate the effect of a prognostic factor on a time-to-event outcome. In a survey of survival analyses in cancer journals, it was found that only 5% of studies using Cox proportional hazards model attempted to verify the underlying assumption. Usually an estimate of the treatment effect from fitting a Cox model was reported without validation of the proportionality assumption. It is not clear how such an estimate should be interpreted if the proportionality assumption is violated. In this article, we show that the estimate of treatment effect from a Cox regression model can be interpreted as a weighted average of the log-scaled hazard ratio over the duration of study. A hypothetic example is used to explain the weights.     

On a Selection Weibull Distribution

In this article, a selection Weibull distribution is investigated. First, some properties and representations of the model with some plots of the density and hazard rate functions are illustrated. Second, some simple relations of this model with some distributions discussed. In addition, maximum likelihood estimators obtained with numerical methods, and compared by three sub-models with a data set that shows the performance of our model. Finally, a simulation study presented for all parameters.     

A note on bias reduction of maximum likelihood estimates for the scalar skew t distribution

The skew t distribution is a flexible parametric family to fit data, because it includes parameters that let us regulate skewness and kurtosis. A problem with this distribution is that, for moderate sample sizes, the maximum likelihood estimator of the shape parameter is infinite with positive probability. In order to try to solve this problem, Sartori (2006) has proposed using a modified score function as an estimating equation for the shape parameter. In this note we prove that the resulting modified maximum likelihood estimator is always finite, considering the degrees of freedom as known and greater than or equal to 2.     

A note on asymptotic distributions in directed exponential random graph models with bi-degree sequences

The asymptotic normality of a fixed number of the maximum likelihood estimators (MLEs) in the directed exponential random graph models with an increasing bi-degree sequence has been established recently. In this article, we further derive a central limit theorem for a linear combination of all the MLEs with an increasing dimension. Simulation studies are provided to illustrate the asymptotic results.     

Asymptotic Normality of Extreme Quantile Estimators Based on the Peaks-Over-Threshold Approach

The POT (Peaks-Over-Threshold) approach consists of using the generalized Pareto distribution (GPD) to approximate the distribution of excesses over thresholds. In this article, we establish the asymptotic normality of the well-known extreme quantile estimators based on this POT method, under very general assumptions. As an illustration, from this result, we deduce the asymptotic normality of the POT extreme quantile estimators in the case where the maximum likelihood (ML) or the generalized probability-weighted moments (GPWM) methods are used. Simulations are provided in order to compare the efficiency of these estimators based on ML or GPWM methods with classical ones proposed in the literature.     

A Novel Collinearity-Influential Observation Diagnostic Measure Based on a Group Deletion Approach

High leverage points can induce or disrupt multicollinearity patterns in data. Observations responsible for this problem are generally known as collinearity-influential observations. A significant amount of published work on the identification of collinearity-influential observations exists; however, we show in this article that all commonly used detection techniques display greatly reduced sensitivity in the presence of multiple high leverage collinearity-influential observations. We propose a new measure based on a diagnostic robust group deletion approach. Some practical cutoff points for existing and developed diagnostics measures are also introduced. Numerical examples and simulation results show that the proposed measure provides significant improvement over the existing measures.     

A Note on Wald's Type Chi-squared Tests of Fit

The Wald's method for constructing chi-squared tests of fit has been formulated more accurately. It is shown that Wald's type statistics will follow the central chi-squared distribution if and only if the limit covariance matrix of standardized frequencies will not depend on unknown parameters. Several examples that illustrate this important fact are presented. In particular, it is shown that the goodness-of-fit statistic developed by Moore and Stubblebine does not follow the chi-squared limit distribution, and, hence, cannot be used for testing multivariate normality.     

Saddlepoint condition on a predictor to reconfirm the need for the assumption of a prior distribution

Saddlepoint conditions on a predictor are introduced and developed to reconfirm the need for the assumption of a prior distribution in constructing a useful inferential procedure. A condition yields that the predictor induced from the maximum likelihood estimator is the worst under a loss, while the predictor induced from a suitable posterior mean is the best. This result indicates the promising role of Bayesian criteria, such as the deviance information criterion (DIC). As an implication, we critique the conventional empirical Bayes method because of its partial assumption of a prior distribution.     

A hypothesis testing procedure for the evaluation on the lifetime performance index of products with Burr XII distribution under progressive type I interval censoring

ABSTRACT

It is a very important topic these days to assessing the lifetime performance of products in manufacturing or service industries. Lifetime performance indices C_L is used to measure the larger-the-better type quality characteristics to evaluate the process performance for the improvement of quality and productivity. The lifetimes of products are assumed to have Burr XII distribution. The maximum likelihood estimator is used to estimate the lifetime performance index based on the progressive type I interval censored sample. The asymptotic distribution of this estimator is also developed. We use this estimator to build the new hypothesis testing algorithmic procedure with respect to a lower specification limit. Finally, two practical examples are given to illustrate the use of this testing algorithmic procedure to determine whether the process is capable.     

Variance Estimation in Spatial Regression Using a Non-parametric Semivariogram Based on Residuals

Abstract.  The empirical semivariogram of residuals from a regression model with stationary errors may be used to estimate the covariance structure of the underlying process. For prediction (kriging) the bias of the semivariogram estimate induced by using residuals instead of errors has only a minor effect because the bias is small for small lags. However, for estimating the variance of estimated regression coefficients and of predictions, the bias due to using residuals can be quite substantial. Thus we propose a method for reducing this bias. The adjusted empirical semivariogram is then isotonized and made conditionally negative-definite and used to estimate the variance of estimated regression coefficients in a general estimating equations setup. Simulation results for least squares and robust regression show that the proposed method works well in linear models with stationary correlated errors.     

Multiple-Steps Step-Stress Accelerated Degradation Modeling Based on Wiener and Gamma Processes

During the step-stress accelerated degradation test (SSADT) experiment, the operator usually elevates the stress level at a predetermined time-point for all tested products that had not failed. This time-point is determined by the experience of the operator and the test is carried on until the performance degradation value of the product crosses the threshold value. In fact, this mode only works when a lot of products have been used in the experiment. But in the SSADT experiment, the number of products is relatively few, and so the test control to the products should be done more carefully. Considering the differences in products, we think the time-point of elevating stress level varies randomly from product-to-product. We consider a situation in which when the degradation value crosses a pre-specified value, the stress level is elevated. Under the circumstances, the time at which the degradation path crosses the pre-specified value is uncertain, and so we may regard it as a random variable. We discuss multiple-steps step-stress accelerated degradation models based on Wiener and gamma processes, respectively, and we apply the Bayesian Markov chain Monte Carlo (MCMC) method for such analytically intractable models to obtain the maximum likelihood estimates (MLEs) efficiently and present some computational results obtained from our implementation.     

A Note on a Specification Test for Time Series Models Based on Spectral Density Estimation

In a recent paper, Paparoditis [Scand. J. Statist. 27 (2000) 143] proposed a new goodness‐of‐fit test for time series models based on spectral density estimation. The test statistic is based on the distance between a kernel estimator of the ratio of the true and the hypothesized spectral density and the expected value of the estimator under the null and provides a quantification of how well the parametric density fits the sample spectral density. In this paper, we give a detailed asymptotic analysis of the corresponding procedure under fixed alternatives.     

The sampling distribution of the W estimator of the number of valid signatures on a petition

Abstract

A new non linear estimator, W, for the number of valid, unique signatures on a petition has been shown better, for the cases enumerated and with certain restrictions, than a popular Goodman-type statistic, G. This article extends those results with relaxed conditions by developing the exact probability mass function and mean of W and a close approximation of the variance (Var(W)). If the proportion of valid signatures among unique and duplicated signatures is the same, then Var(W) is approximately a function of the means and variances of the two sample statistics. Using the delta method, we estimate Var(W), with the resulting approximation shown to be good, even when the condition of equal proportions does not hold. We compare W to G and establish which estimator is preferred for different intervals of the design parameters. Data from a Washington State petition illustrate the findings.     

Bayesian,MLE, and GMM Estimation of a Spot Rate Model

ABSTRACT

We develop Markov chain Monte Carlo algorithms for estimating the parameters of the short-term interest rate model. Using Monte Carlo experiments we compare the Bayes estimators with the maximum likelihood and generalized method of moments estimators. We estimate the model using the Japanese overnight call rate data.     

A Bivariate Distribution for the Reliability Analysis of Failure Data in Presence of a Forewarning or Primer Event

This article presents a bivariate distribution for analyzing the failure data of mechanical and electrical components in presence of a forewarning or primer event whose occurrence denotes the inception of the failure mechanism that will cause the component failure after an additional random time. The characteristics of the proposed distribution are discussed and several point estimators of parameters are illustrated and compared, in case of complete sampling, via a large Monte Carlo simulation study. Confidence intervals based on asymptotic results are derived, as well as procedures are given for testing the independence between the occurrence time of the forewarning event and the additional time to failure. Numerical applications based on failure data of cable insulation specimens and of two-component parallel systems are illustrated.