A study of generalized skew-normal distribution

Following the paper by Genton and Loperfido [Generalized skew-elliptical distributions and their quadratic forms, Ann. Inst. Statist. Math. 57 (2005), pp. 389–401], we say that Z has a generalized skew-normal distribution, if its probability density function (p.d.f.) is given by f(z)=2φ _p (z; ξ, Ω)π (z?ξ), z∈? ^p , where φ _p (·; ξ, Ω) is the p-dimensional normal p.d.f. with location vector ξ and scale matrix Ω, ξ∈? ^p , Ω>0, and π is a skewing function from ? ^p to ?, that is 0≤π (z)≤1 and π (?z)=1?π (z), ? z∈? ^p . First the distribution of linear transformations of Z are studied, and some moments of Z and its quadratic forms are derived. Next we obtain the joint moment-generating functions (m.g.f.’s) of linear and quadratic forms of Z and then investigate conditions for their independence. Finally explicit forms for the above distributions, m.g.f.’s and moments are derived when π (z)=κ (α′z), where α∈? ^p and κ is the normal, Laplace, logistic or uniform distribution function.     

A bayesian approach to prediction in analysis of variance settings

This paper takes the results of Lindley and smith ( 1972 ) one step further, by finding the predictive distribution of an observation y^* whose distribution is normal, and centred at A^* ₁θ₁ We then apply this distribution to the case of prediction based on data obtained in one and two wau ANOVA situations. For Example, it turns out that for two way ANOVA with interaction, the predictive mean, (which we would use as the predictor) is a weighted combination of sample main effects and interaction effects     

The Likelihood Ratio Test with the Box–Cox Transformation for the Normal Mixture Problem: Power and Sample Size Study

Abstract

Through simulation and regression, we study the alternative distribution of the likelihood ratio test in which the null hypothesis postulates that the data are from a normal distribution after a restricted Box–Cox transformation and the alternative hypothesis postulates that they are from a mixture of two normals after a restricted (possibly different) Box–Cox transformation. The number of observations in the sample is called N. The standardized distance between components (after transformation) is D = (μ₂ ? μ₁)/σ, where μ₁ and μ₂ are the component means and σ² is their common variance. One component contains the fraction π of observed, and the other 1 ? π. The simulation results demonstrate a dependence of power on the mixing proportion, with power decreasing as the mixing proportion differs from 0.5. The alternative distribution appears to be a non-central chi-squared with approximately 2.48 + 10N ^?0.75 degrees of freedom and non-centrality parameter 0.174N(D ? 1.4)² × [π(1 ? π)]. At least 900 observations are needed to have power 95% for a 5% test when D = 2. For fixed values of D, power, and significance level, substantially more observations are necessary when π ≥ 0.90 or π ≤ 0.10. We give the estimated powers for the alternatives studied and a table of sample sizes needed for 50%, 80%, 90%, and 95% power.     

Interval estimation for the Pareto distribution based on the progressive Type II censored sample

For the complete sample and the right Type II censored sample, Chen [Joint confidence region for the parameters of Pareto distribution. Metrika 44 (1996), pp. 191–197] proposed the interval estimation of the parameter θ and the joint confidence region of the two parameters of Pareto distribution. This paper proposed two methods to construct the confidence region of the two parameters of the Pareto distribution for the progressive Type II censored sample. A simulation study comparing the performance of the two methods is done and concludes that Method 1 is superior to Method 2 by obtaining a smaller confidence area. The interval estimation of parameter ν is also given under progressive Type II censoring. In addition, the predictive intervals of the future observation and the ratio of the two future consecutive failure times based on the progressive Type II censored sample are also proposed. Finally, one example is given to illustrate all interval estimations in this paper.     

Efficient sequential estimation in a markov branching process with immigration

Sequential estimation of parameters In a continuous time Markov branching process with Immigration with split rate λ₁ Immigration rate λ₂, offspring distribution {p₁j≥O) and Immigration distribution {p₂j≥l} is considered. A sequential version of the Cramér-Rao type information inequality is derived which gives a lower bound on the variances of unbiased estimators for any function of these parameters. Attaining the lower bounds depends on whether the sampling plan or stopping rule S, the estimator f, and the parametric function g = E(f) are efficient. All efficient triples (S,f,g) are characterized; It Is shown that for i = 1,2, only linear combinations of λ_ip_{ij j}'s or their ratios are efficiently estimable. Applications to a Yule process, a linear birth and death process with immigration and an M/M/∞ queue are also considered     

Information criteria for the predictive evaluation of bayesian models

As a natural successor of the information criteria AIC and ABIC, information criteria for the Bayes models were developed by evaluating the bias of the log likelihood of the predictive distribution as an estimate of its expected log-likelihood. Considering two specific situations for the true distribution, two information criteria, PIC₁ and PIC₂ are derived. Linear Gaussian cases are considered in details and the evaluation of the maximum a posteriori estimator is also considered. By a simple example of estimating the signal to noise ratio, it was shown that the PIC₂ is a good approximation to the expected log-likelihood in the entire region of the signal to noise ratio. On the other hand, PIC₁ performs good only for the smaller values of the variance ratio. For illustration, the problems of trend estimation and seasonal adjustment are considered. Examples show that the hyper-parameters estimated by the new criteria are usually closer to the best ones than those by the ABIC.     

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.     

ON A SHARED ALLELE TEST OF RANDOM MATING

A simple random sample is observed from a population with a large number‘K’ of alleles, to test for random mating. Of n couples, n_ijkl have female genotype ij and male genotype kl (i, j, k, l{1,…, A‘}). The large contingency table is collapsed into three counts, n₀, n₁ and n₂ where n_p is the number of couples with s alleles in common (s = 0,1, 2). The counts are estimated by np?_o where n₀, is the estimated probability of a couple having s alleles in common under the hypothesis of random mating. The usual chi-square goodness of fit statistic X² compares observed (n_s) with expected (np?) over the three categories, s = 0,1,2. An empirical observation has suggested that X² is close to having a chi-square distribution with two degrees of freedom (X) despite a large number of parameters implicitly estimated in _e. This paper gives two theorems which show that x is indeed the approximate distribution of X² for large n and K₁“, provided that no allele type over-dominates the others.     

Estimation of the Burr type III distribution with application in unified hybrid censored sample of fracture toughness

In this paper, the statistical inference of the unknown parameters of a Burr Type III (BIII) distribution based on the unified hybrid censored sample is studied. The maximum likelihood estimators of the unknown parameters are obtained using the Expectation–Maximization algorithm. It is observed that the Bayes estimators cannot be obtained in explicit forms, hence Lindley's approximation and the Markov Chain Monte Carlo (MCMC) technique are used to compute the Bayes estimators. Further the highest posterior density credible intervals of the unknown parameters based on the MCMC samples are provided. The new model selection test is developed in discriminating between two competing models under unified hybrid censoring scheme. Finally, the potentiality of the BIII distribution to analyze the real data is illustrated by using the fracture toughness data of the three different materials namely silicon nitride (Si₃N₄), Zirconium dioxide (ZrO₂) and sialon (Si_6?xAl_xO_xN_8?x). It is observed that for the present data sets, the BIII distribution has the better fit than the Weibull distribution which is frequently used in the fracture toughness data analysis.     

Nonparametric location-scale models for censored successive survival times

Let (T₁,T₂) be gap times corresponding to two consecutive events, which are observed subject to (univariate) random right-censoring. The censoring variable corresponding to the second gap time T₂ will in general depend on this gap time. Suppose the vector (T₁,T₂) satisfies the nonparametric location-scale regression model T₂=m(T₁)+σ(T₁)?, where the functions m and σ are ‘smooth’, and ? is independent of T₁. The aim of this paper is twofold. First, we propose a nonparametric estimator of the distribution of the error variable under this model. This problem differs from others considered in the recent related literature in that the censoring acts not only on the response but also on the covariate, having no obvious solution. On the basis of the idea of transfer of tail information (Van Keilegom and Akritas, 1999), we then use the proposed estimator of the error distribution to introduce nonparametric estimators for important targets such as: (a) the conditional distribution of T₂ given T₁; (b) the bivariate distribution of the gap times; and (c) the so-called transition probabilities. The asymptotic properties of these estimators are obtained. We also illustrate through simulations, that the new estimators based on the location-scale model may behave much better than existing ones.     

Property of bivariate poisson distribution and its application to stochastic processes

Assuming that the random vectors X ₁ and X ₂ have independent bivariate Poisson distributions, the conditional distribution of X ₁ given X ₁?+?X ₂?=?n is obtained. The conditional distribution turns out to be a finite mixture of distributions involving univariate binomial distributions and the mixing proportions are based on a bivariate Poisson (BVP) distribution. The result is used to establish two properties of a bivariate Poisson stochastic process which are the bivariate extensions of the properties for a Poisson process given by Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, Academic Press, New York.     

The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation

In this paper, we consider the family of skew generalized t (SGT) distributions originally introduced by Theodossiou [P. Theodossiou, Financial data and the skewed generalized t distribution, Manage. Sci. Part 1 44 (12) ( 1998), pp. 1650–1661] as a skew extension of the generalized t (GT) distribution. The SGT distribution family warrants special attention, because it encompasses distributions having both heavy tails and skewness, and many of the widely used distributions such as Student's t, normal, Hansen's skew t, exponential power, and skew exponential power (SEP) distributions are included as limiting or special cases in the SGT family. We show that the SGT distribution can be obtained as the scale mixture of the SEP and generalized gamma distributions. We investigate several properties of the SGT distribution and consider the maximum likelihood estimation of the location, scale, and skewness parameters under the assumption that the shape parameters are known. We show that if the shape parameters are estimated along with the location, scale, and skewness parameters, the influence function for the maximum likelihood estimators becomes unbounded. We obtain the necessary conditions to ensure the uniqueness of the maximum likelihood estimators for the location, scale, and skewness parameters, with known shape parameters. We provide a simple iterative re-weighting algorithm to compute the maximum likelihood estimates for the location, scale, and skewness parameters and show that this simple algorithm can be identified as an EM-type algorithm. We finally present two applications of the SGT distributions in robust estimation.     

Dependent Right Censorship in the MOMW Distribution

In this paper, we consider paired survival data, in which pair members are subject to the same right censoring time, but they are dependent on each other. Assuming the Marshall–Olkin Multivariate Weibull distribution for the joint distribution of the lifetimes (X₁, X₂) and the censoring time X₃, we derive the joint density of the actual observed data and obtain maximum likelihood estimators, Bayes estimators and posterior regret Gamma minimax estimators of the unknown parameters under squared error loss and weighted squared error loss functions. We compare the performances of the maximum likelihood estimators and Bayes estimators numerically in terms of biases and estimated Mean Squared Error Loss.     

The inverse problem of multivariate and matrix-variate skew normal distributions

In this paper, we prove that the joint distribution of random vectors Z ₁ and Z ₂ and the distribution of Z ₂ are skew normal provided that Z ₁ is skew normally distributed and Z ₂ conditioning on Z ₁ is distributed as closed skew normal. Also, we extend the main results to the matrix variate case.     

A general class of capability indices in the case of asymmetric tolerances

Vannman has earlier studied a class of capability indices, containing the indices C _p , C _pk , C _pm and C _pmk , when the tolerances are symmetric. We study the properties of this class when the tolerances are asymmetric and suggest a new enlargened class of indices. Under the assumption of normality an explicit form of the distribution of the new class of the estimated indices is provided. Numerical investigations are made to explore the behavior of the estimators of the indices for different values of the parameters. Based on the estimator a decision rule that can be used to determine whether the process can be considered capable or not is provided and suitable criteria for choosing an index from the family are suggested.     

Determination of Single Sampling Plans by Attributes Under the Conditions of Zero-Inflated Poisson Distribution

When the manufacturing process is well monitored, occurrence of nondefects would be a frequent event in sampling inspection. The appropriate probability distribution of the number of defects is a zero-inflated Poisson (ZIP) distribution. In this article, determination of single sampling plans (SSPs) by attributes using unity values is considered, when the number of defects follows a ZIP distribution. The operating characteristic (OC) function of the sampling plan is derived. Plan parameters are obtained for some sets of values of (p₁, α, p₂, β). Numerical illustrations are given to describe the determination of SSP under ZIP distribution and to study its performance in comparison with Poisson SSP.     

Estimation of Pr{yp > max(y1, …, yp-1)}: Predictive approach in exponential case

The structural-inference approach to predictive distributions is used to derive the estimator of P = Pr{Y_p > max(Y₁, …, Y_p-₁)} when the independent random variables Y₁, …, Y_p follow exponential distributions with unequal location parameters and equal scale parameters. The result is Equation (4.6).     

The information for the direction of dependence in l1 regression

Fitting a linear regression for a response variable by minimising the sum of absolute deviations, L₁ regression, may be viewed as a maximum likelihood procedure applied to the Laplace distribution. An interesting bivariate case is where the conditional distribution of the response X₂ given X₁ and the marginal distribution of the explanatory variable X₁ are both Laplace. In this context we show there is information to distinguish the direction of dependence between X₁ and X₂ from observations. That is we may distinguish the model in which X₁ is dependent on X₂ from that in which X₂ is dependent on X₁ This is not true for L₂ regression based on the Normal distribution.     

Exact inference for a simple step-stress model with Type-II hybrid censored data from the exponential distribution

In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress factors such as temperature, voltage and load on the lifetimes of experimental units. Step-stress test, which is a special class of accelerated life-tests, allows the experimenter to increase the stress levels at fixed times during the experiment in order to obtain information on the parameters of the life distributions more quickly than under normal operating conditions. In this paper, we consider a new step-stress model in which the life-testing experiment gets terminated either at a pre-fixed time (say, _Tm+1$T_{m+1}$) or at a random time ensuring at least a specified number of failures (say, r out of n). Under this model in which the data obtained are Type-II hybrid censored, we consider the case of exponential distribution for the underlying lifetimes. We then derive the maximum likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model with lifetimes being exponentially distributed. The exact distributions of the MLEs of parameters are obtained through the use of conditional moment generating functions. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs and the parametric bootstrap methods, and assess their performance through a Monte Carlo simulation study. Finally, we present two examples to illustrate all the methods of inference discussed here.     

Estimation After Selection Under Reflected Normal Loss Function

Let X ₁ and X ₂ be two independent random variables from normal populations Π₁, Π₂ with different unknown location parameters θ₁ and θ₂, respectively and common known scale parameter σ. Let X ₍₂₎ = max (X ₁, X ₂) and X ₍₁₎ = min (X ₁, X ₂). We consider the problem of estimating the location parameter θ _M (or θ _J ) of the selected population under the reflected normal loss function. We obtain minimax estimators of θ _M and θ _J . Also, we provide sufficient conditions for the inadmissibility of invariant estimators of θ _M and θ _J .