A general long-term aging model with different underlying activation mechanisms: Modeling,Bayesian estimation,and case influence diagnostics

In this paper we propose a general cure rate aging model. Our approach enables different underlying activation mechanisms which lead to the event of interest. The number of competing causes of the event of interest is assumed to follow a logarithmic distribution. The model is parameterized in terms of the cured fraction which is then linked to covariates. We explore the use of Markov chain Monte Carlo methods to develop a Bayesian analysis for the proposed model. Moreover, some discussions on the model selection to compare the fitted models are given, as well as case deletion influence diagnostics are developed for the joint posterior distribution based on the ψ-divergence, which has several divergence measures as particular cases, such as the Kullback–Leibler (K-L), J-distance, L₁ norm, and χ²-square divergence measures. Simulation studies are performed and experimental results are illustrated based on a real malignant melanoma data.     

An integral formula for the distribution of self-normalized Gaussian random samples

Given i.i.d. Gaussian random variables and after standardizing the sample by subtracting the sample mean and dividing it by the sample deviation, we obtain an integral formula for the distribution of these self-normalized variables. Using geometrical arguments, we obtain the distribution of each and the joint distribution of two of them. These formulas can be used to calculate the expected value of the particular type of Cramér von Mises statistic to test normality.     

Slashed generalized exponential distribution

In this paper, we introduce an extension of the generalized exponential (GE) distribution, making it more robust against possible influential observations. The new model is defined as the quotient between a GE random variable and a beta-distributed random variable with one unknown parameter. The resulting distribution is a distribution with greater kurtosis than the GE distribution. Probability properties of the distribution such as moments and asymmetry and kurtosis are studied. Likewise, statistical properties are investigated using the method of moments and the maximum likelihood approach. Two real data analyses are reported illustrating better performance of the new model over the GE model.     

Slashed generalized Rayleigh distribution

We introduce a new family of distributions suitable for fitting positive data sets with high kurtosis which is called the Slashed Generalized Rayleigh Distribution. This distribution arises as the quotient of two independent random variables, one being a generalized Rayleigh distribution in the numerator and the other a power of the uniform distribution in the denominator. We present properties and carry out estimation of the model parameters by moment and maximum likelihood (ML) methods. Finally, we conduct a small simulation study to evaluate the performance of ML estimators and analyze real data sets to illustrate the usefulness of the new model.     

Tests for compound periodicities and estimating a non linear function

We propose two tests for testing compound periodicities which are the uniformly most powerful invariant decision procedures against simple periodicities. The second test can provide an excellent estimation of a compound periodic non linear function from observed data. These tests were compared with the tests proposed by Fisher and Siegel by Monte Carlo studies and we found that all the tests showed high power and high probability of a correct decision when all the amplitudes of underlying periods were the same. However, if there are at least several different periods with unequal amplitudes, then the second test proposed always showed high power and high probability of a correct decision, whereas the tests proposed by Fisher and Siegel gave 0 for the power and 0 for the probability of a correct decision, whatever the standard deviation of pseudo normal random numbers. Overall, the second test proposed is the best of all in view of the probability of a correct decision and power.     

Optimal and lead-in adaptive allocation for binary outcomes: A comparison of Bayesian methods

We compare posterior and predictive estimators and probabilities in response-adaptive randomization designs for two- and three-group clinical trials with binary outcomes. Adaptation based upon posterior estimates are discussed, as are two predictive probability algorithms: one using the traditional definition, the other using a skeptical distribution. Optimal and natural lead-in designs are covered. Simulation studies show that efficacy comparisons lead to more adaptation than center comparisons, though at some power loss, skeptically predictive efficacy comparisons and natural lead-in approaches lead to less adaptation but offer reduced allocation variability. Though nuanced, these results help clarify the power-adaptation trade-off in adaptive randomization.     

M-estimation for functional linear regression

This paper studies M-estimation in functional linear regression in which the dependent variable is scalar while the covariate is a function. An estimator for the slope function is obtained based on the functional principal component basis. The global convergence rate of the M-estimator of unknown slope function is established. The convergence rate of the mean-squared prediction error for the proposed estimators is also established. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed procedure. Finally, the proposed method is applied to analyze the Berkeley growth data.     

Shrinkage estimation of proportion via logit penalty

By releasing the unbiasedness condition, we often obtain more accurate estimators due to the bias–variance trade-off. In this paper, we propose a class of shrinkage proportion estimators which show improved performance over the sample proportion. We provide the “optimal” amount of shrinkage. The advantage of the proposed estimators is given theoretically as well as explored empirically by simulation studies and real data analyses.