Joint distribution of simultaneous exposures to several carcinogens in a case–control study: sample size determination

Consider a population of individuals who are free of a disease under study, and who are exposed simultaneously at random exposure levels, say X,Y,Z,… to several risk factors which are suspected to cause the disease in the populationm. At any specified levels X=x, Y=y, Z=z, …, the incidence rate of the disease in the population ot risk is given by the exposure–response relationship r(x,y,z,…) = P(disease|x,y,z,…). The present paper examines the relationship between the joint distribution of the exposure variables X,Y,Z, … in the population at risk and the joint distribution of the exposure variables U,V,W,… among cases under the linear and the exponential risk models. It is proven that under the exponential risk model, these two joint distributions belong to the same family of multivariate probability distributions, possibly with different parameters values. For example, if the exposure variables in the population at risk have jointly a multivariate normal distribution, so do the exposure variables among cases; if the former variables have jointly a multinomial distribution, so do the latter. More generally, it is demonstrated that if the joint distribution of the exposure variables in the population at risk belongs to the exponential family of multivariate probability distributions, so does the joint distribution of exposure variables among cases. If the epidemiologist can specify the differnce among the mean exposure levels in the case and control groups which are considered to be clinically or etiologically important in the study, the results of the present paper may be used to make sample size determinations for the case–control study, corresponding to specified protection levels, i.e., size α and 1–β of a statistical test. The multivariate normal, the multinomial, the negative multinomial and Fisher's multivariate logarithmic series exposure distributions are used to illustrate our results.     

A generalised bivariate binomial distribution applicable in four-fold sampling

A bivariate generalisation of the Consul's(1974) quasi-binomial distributionCQBD) has been obtained with the help of an urn-model for eKplaining data obtained as a result of four-fold sampling, The distribution is expected to cover a very wide range of situations in four-fold sampling. The first and the second order moments of the distribution have been obtained,The distribution has been fitted to an observed set of data as an illustration and its limiting form has also been obtained.     

Semiparametric Likelihood Based Method for Goodness of Fit Tests and Estimation in Upgraded Mixture Models

We use Owen's (1988, 1990) empirical likelihood method in upgraded mixture models. Two groups of independent observations are available. One is z ₁, ..., z _n which is observed directly from a distribution F ( z ). The other one is x ₁, ..., x _m which is observed indirectly from F ( z ), where the x _i s have density ∫ p ( x | z ) dF ( z ) and p ( x | z ) is a conditional density function. We are interested in testing H ₀: p ( x | z ) = p ( x | z ; θ ), for some specified smooth density function. A semiparametric likelihood ratio based statistic is proposed and it is shown that it converges to a chi-squared distribution. This is a simple method for doing goodness of fit tests, especially when x is a discrete variable with finitely many values. In addition, we discuss estimation of θ and F ( z ) when H ₀ is true. The connection between upgraded mixture models and general estimating equations is pointed out.     

On using percentiles to fit data by a johnson distribution

The well-known Johnson system of distributions was developed by N. L. Johnson (1949). Slifker and Shapiro (1980) presented a criterion for choosing a member from the three distributional classes (S_B,S_L, and S_v) in the Johnson system to fit a set of data. The criterion is based on the value of a quantile ratio which depends on a specified positive z value and the parameters of the distribution. In this paper, we present some properties of the quantile ratio for various distributions and for some selected z values. Some comments are made on using the criterion for selecting a Johnson distribution to fit empirical data.     

Nonnull distribution of Wilks' statistic for Manova in the complex case

In this paper, the exact distribution of Wilks' likelihood ratio criterion, A, for MANOVA, in the complex case when the alternate hypothesis is of unit rank (i.e. the linear case) has been derived and the explicit expressions for the same for p = 2 and 3 (where p is the number of variates) and general f₁ (the error degrees of freedom) and f₂ (the hypothesis degrees of freedom), are given. For an unrestricted number of variables, a general form of the density and the distribution of A in this case, is also given. It has been shown that the total integral of the series obtained by taking a few terms only, rapidly approaches the theoretical value one as more terms are taken into account, and some percentage points have also been computed.     

Tables of cumulative distribution functions and percentiles of the standardized stable random variables

It is shown that Zolotarev's (1964) integral representation of the cumulative distribution function (c.d.f.) of stable random variables and the IMSL subroutine DCADRE (for numerical integration ) provide a natural and practically simple method for finding the values of c.d.f., the percentiles and the density function of such random variables. For symmetric stable random variables (r.v.'s ) Z_∝, values of P_∝(z) … P(0<Z_∝<z) for z … 0(.02)4.08 and ∝=.1(.2)1.9, as well as percentiles of these r.v.'s for ∝=.5(.1)2 and the percentage points .6, .7(.05).85(.025).9(.01).96(.005).995, are presented. For asymmetric stable r.v.'s we present values of their c.d.f.'s for z … 0(.1)4, ß= ?1(.25)1 and ∝=.1(.2)1.9. These result sare compared with related results of others which were obtained by using different procedure and standardization.     

On the distribution of Fisher's transformation of the correlation coefficient

As the sample size increases, the coefficient of skewness of the Fisher's transformation, z = (1/2) log ((l+r)/(l-r)), of the correlation coefficient decreases much more rapidly than the excess of its kurtosis. Hence, the usual normal approximation for its distribution can be improved by adjusting for the excess of its kurtosis. This is accomplished by mixing the approximating normal distribution with a logistic distribution. The resulting mixture approximation which can be used to estimate the probabilities, as well as the percentiles, compares favorably in both accuracy and simplicity, with the two best earlier approximations, namely, those due to Ruben (1966) and Kraemer (1973).     

Optimum stratification with two variates

For the 2×2 rectilinear stratification of a bivariate normal distribution with proportional and optimum allocation the dependence of the objective function z(x₁;y₁) on the coefficient of correlation ρ and the sampling fraction q=n/N is investigated. With proportional allocation for great values of ρ (but already for q=0) a so-called ρ-effect arises, which results in a saddle-point of z as “optimum” stratification point in the center of gravity of the distribution and two additional minima. With optimum allocation first for smaller values of q also the ρ-effect arises; for grater values of q a so-called q-effect is superposed, which results in a multitude of minima, saddle-points and maxima of z. All these points satisfy the generalized conditions of Dalenius, but for practical use only the global minimum is of interest.     

Conditionally gaussian distributions and an application to kalman filtering with stochastic regressors

The work reviews theory of conditionally Gaussian distributions, especially so called theorems on normal correlation. Three theorems are given: the basic, the recursive, and the conditional theorem on normal correlation. They assume that (a,y), (a,x,y), or (a,y,z) has a Gaussian distribution, ussert that (a,y), (a,x,y), and (a,y,z), respectively, are Gaussian, and give formulas for the corresponding conditional mean vectors and variance covariance matrices. A proof is presented for the recursive and the conditional theorem.     

The Discrete Normal Distribution

Abstract

The normal distribution has been playing a key role in stochastic modeling for a continuous setup. But its distribution function does not have an analytical form. Moreover, the distribution of a complex multicomponent system made of normal variates occasionally poses derivational difficulties. It may be worth exploring the possibility of developing a discrete version of the normal distribution so that the same can be used for modeling discrete data. Keeping in mind the above requirement we propose a discrete version of the continuous normal distribution. The Increasing Failure Rate property in the discrete setup has been ensured. Characterization results have also been made to establish a direct link between the discrete normal distribution and its continuous counterpart. The corresponding concept of a discrete approximator for the normal deviate has been suggested. An application of the discrete normal distributions for evaluating the reliability of complex systems has been elaborated as an alternative to simulation methods.     

Estimates of the quantiles of kendall's partial rank correlation coefficient

The sampling distribution of kendall's partial rank correlation coefficient, J_xy?z, is not known for N>4, where N is the number of subjectts. Moran (1951) used a direcr conbinatorial method to obtain the distribution of J_xy?z forN=4; however, ten minor computationa; errors in his Table 2apparently resulted in how erroneous entries for his frequency table. Since the parctial limits of the direct combinatorial approach have been reached once N>4, the first main objective of this paper was to obtain the exact distribution of J_xy?z for N=f, 6, and 7 using an electronic computer. The second was to use the Monte Carlo method to obtain reliable estimates of the quantiles of J_xy?z for N=8,9,...,30     

Estimation of parameters of the gped

Janardan (1973) introduced the generalized Polya-Eggenberger distribution as a limiting form of the generalized Markov-Polya distribution (GMPD), Ja¬nardan (1998) derived GPED formally by means of Lagrange's expansion and discussed its various properties systematically. Here, a new urn model is pro¬vided for the GPED. Moment estimators of the parameters are given in closed form. Maximum hkelihood estimators are also given. Some apphcations are provided.     

Tables of the cumulative non-central cm-square distribution-part 4

The cumulative non-central chi-square .distribution is tabulated for all combinations of values of λ = 0 (0.1) 1.0 (0.2) 3.0 (0.5) 5.0 (1.0) 34.0, y=l (I) 30 (2) 50 (5) 100 and y = 0.01 (0.01) 0.1 (0.1) 1.0 (0.2) 3.0 (0.5) 10.0 (1.0 30,0 (2.0) 50,0 (5.0) 165.0. The computations have been correctly rounded to five decimal places. Also, there is a discussion about the error involved in the computations. Furthermore, there is a discussion about possible interpolation in the table using the Lagrange's method     

Likelihood Ratio Tests Under Local and Fixed Alternatives in Monotone Function Problems

Abstract.  We focus on a class of non-standard problems involving non-parametric estimation of a monotone function that is characterized by n ^1/3 rate of convergence of the maximum likelihood estimator, non-Gaussian limit distributions and the non-existence of     -regular estimators. We have shown elsewhere that under a null hypothesis of the type ψ ( z ₀) =  θ ₀ ( ψ being the monotone function of interest) in non-standard problems of the above kind, the likelihood ratio statistic has a 'universal' limit distribution that is free of the underlying parameters in the model. In this paper, we illustrate its limiting behaviour under local alternatives of the form ψ _n ( z ), where ψ _n (·) and ψ (·) vary in O ( n ^−1/3) neighbourhoods around z ₀ and ψ _n converges to ψ at rate n ^1/3 in an appropriate metric. Apart from local alternatives, we also consider the behaviour of the likelihood ratio statistic under fixed alternatives and establish the convergence in probability of an appropriately scaled version of the same to a constant involving a Kullback–Leibler distance.     

Eliciting expert judgements about a set of proportions

Eliciting expert knowledge about several uncertain quantities is a complex task when those quantities exhibit associations. A well-known example of such a problem is eliciting knowledge about a set of uncertain proportions which must sum to 1. The usual approach is to assume that the expert's knowledge can be adequately represented by a Dirichlet distribution, since this is by far the simplest multivariate distribution that is appropriate for such a set of proportions. It is also the most convenient, particularly when the expert's prior knowledge is to be combined with a multinomial sample since then the Dirichlet is the conjugate prior family. Several methods have been described in the literature for eliciting beliefs in the form of a Dirichlet distribution, which typically involve eliciting from the expert enough judgements to identify uniquely the Dirichlet hyperparameters. We describe here a new method which employs the device of over-fitting, i.e. eliciting more than the minimal number of judgements, in order to (a) produce a more carefully considered Dirichlet distribution and (b) ensure that the Dirichlet distribution is indeed a reasonable fit to the expert's knowledge. The method has been implemented in a software extension of the Sheffield elicitation framework (SHELF) to facilitate the multivariate elicitation process.     

Rejoinder to 'Ahmed, M.S. (1998). A note on regression-type estimators using multiple auxiliary information.'

In the estimators t ₃ , t ₄ , t ₅ of Mukerjee, Rao & Vijayan (1987), b _{y x} and b _{y z} are partial regression coefficients of y on x and z , respectively, based on the smaller sample. With the above interpretation of b _{y x} and b _{y z} in t ₃ , t ₄ , t ₅ , all the calculations in Mukerjee at al. (1987) are correct. In this connection, we also wish to make it explicit that b _{x z} in t ₅ is an ordinary and not a partial regression coefficient. The 'corrected' MSEs of t ₃ , t ₄ , t ₅ , as given in Ahmed (1998 Section 3) are computed assuming that our b _{y x} and b _{y z} are ordinary and not partial regression coefficients. Indeed, we had no intention of giving estimators using the corresponding ordinary regression coefficients which would lead to estimators inferior to those given by Kiregyera (1984). We accept responsibility for any notational confusion created by us and express regret to readers who have been confused by our notation. Finally, in consideration of the above, it may be noted that Tripathi & Ahmed's (1995) estimator t ₀ , quoted also in Ahmed (1998), is no better than t ₅ of Mukerjee at al. (1987).     

The flexible skew Laplace distribution

ABSTRACT

Skew-symmetric distributions have been discussed by several research-ers. In this article we construct a skew-symmetric Laplace distribution, which is the generalization of distribution given by Ali et al. (2009 Ali, M., Pal, M., Woo, J. (2009). Skewed reflected distributions generated by the Laplace kernel. Aust. J. Statist. 38:45–58. [Google Scholar]) and Nekoukhou and Alamatsaz (2012 Nekoukhou, V., Alamatsaz, M.H. (2012). A family of skew-symmetric-Laplace distributions. Statist. Papers. 53(3):685–696.[Crossref], [Web of Science ®] , [Google Scholar]). This new distribution contains more parameters, and this induces flexibility properties, such as unimodality or bimodality. We study on some properties of this distribution. In the last section we also provide an application with a real data. Concerning example has recently been discussed by Nekoukhou et al. (2013 Nekoukhou, V., Alamatsaz, M.H., Aghajani, A.H. (2013). A flexible skew-generalized normal distribution. Commun. Statist. Theory Methods. 42(13):2324–2334.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to apply to their model. We compare the behavior of our distribution to their distribution on this example.     

The Kumaraswamy skew-t distribution and its related properties

Skew normal distribution is an alternative distribution to the normal distribution to accommodate asymmetry. Since then extensive studies have been done on applying Azzalini’s skewness mechanism to other well-known distributions, such as skew-t distribution, which is more flexible and can better accommodate long tailed data than the skew normal one. The Kumaraswamy generalized distribution (Kw ? F) is another new class of distribution which is capable of fitting skewed data that can not be fitted well by existing distributions. Such a distribution has been widely studied and various versions of generalization of this distribution family have been introduced. In this article, we introduce a new generalization of the skew-t distribution based on the Kumaraswamy generalized distribution. The new class of distribution, which we call the Kumaraswamy skew-t (KwST) has the ability of fitting skewed, long, and heavy-tailed data and is more flexible than the skew-t distribution as it contains the skew-t distribution as a special case. Related properties of this distribution family such as mathematical properties, moments, and order statistics are discussed. The proposed distribution is applied to a real dataset to illustrate the estimation procedure.     

Bayesian Statistical Inference for Weighted Exponential Distribution

Exponential distribution has an extensive application in reliability. Introducing shape parameter to this distribution have produced various distribution functions. In their study in 2009, Gupta and Kundu brought another distribution function using Azzalini's method, which is applicable in reliability and named as weighted exponential (WE) distribution. The parameters of this distribution function have been recently estimated by the above two authors in classical statistics. In this paper, Bayesian estimates of the parameters are derived. To achieve this purpose we use Lindley's approximation method for the integrals that cannot be solved in closed form. Furthermore, a Gibbs sampling procedure is used to draw Markov chain Monte Carlo samples from the posterior distribution indirectly and then the Bayes estimates of parameters are derived. The estimation of reliability and hazard functions are also discussed. At the end of the paper, some comparisons between classical and Bayesian estimation methods are studied by using Monte Carlo simulation study. The simulation study incorporates complete and Type-II censored samples.     

Stochastic Monotonicity and Conditioning in the Limit

Suppose that {( X _n , Y _n )} is a sequence of pairs of cector-valued stochastic variables which converges weakly to ( X , Y ), and that { y _n } converges to y . Sufficient conditions for the conditional distribution of X _n given Y = y are given in terms of stochastic monotonicity. Conditions, which guarantee that also moments of the conditional distributions converge to the moments of the ones of the limit, are also derived.