An Equivalence of Conditional and Unconditional Maximum Likelihood Estimators via Infinite Replication of Observations

Motivated by an application with complex survey data, we show that for logistic regression with a simple matched-pairs design, infinitely replicating observations and maximizing the conditional likelihood results in an estimator exactly identical to the unconditional maximum likelihood estimator based on the original sample, which is inconsistent. Therefore, applying conditional likelihood methods to a pseudosample with observations replicated a large number of times can lead to an inconsistent estimator; this casts doubt on one possible approach to conditional logistic regression with complex survey data. We speculate that for more general designs, an asymptotic equivalence holds.     

Combinatorially composing Chebyshev polynomials

We present a combinatorial proof of two fundamental composition identities associated with Chebyshev polynomials. Namely, for all m,n?0$m,n?0$, _Tm(_Tn(x))=_Tmn(x)$T_m(T_n(x))=T_{\mathit{mn}}(x)$ and _Um−1(_Tn(x))_Un−1(x)=_Umn−1(x).$U_{m-1}(T_n(x))U_{n-1}(x)=U_{\mathit{mn}-1}(x).$     

The q-Bernstein basis as a q-binomial distribution

The q-Bernstein basis, used in the definition of the q-Bernstein polynomials, is shown to be the probability mass function of a q-binomial distribution. This distribution is defined on a sequence of zero–one Bernoulli trials with probability of failure at any trial increasing geometrically with the number of previous failures. A modification of this model, with the probability of failure at any trial decreasing geometrically with the number of previous failures, leads to a second q-binomial distribution that is also connected to the q-Bernstein polynomials. The q-factorial moments as well as the usual factorial moments of these distributions are derived. Further, the q-Bernstein polynomial B_n(f(t),q;x) is expressed as the expected value of the function f([X_n]_q/[n]_q) of the random variable X_n obeying the q-binomial distribution. Also, using the expression of the q-moments of X_n, an explicit expression of the q-Bernstein polynomial B_n(f_r(t),q;x), for f_r(t) a polynomial, is obtained.     

Approximations for a fixed sample size selection procedure for negative binomial populations

A large sample approximation of the least favorable configuration for a fixed sample size selection procedure for negative binomial populations is proposed. A normal approximation of the selection procedure is also presented. Optimal sample sizes required to be drawn from each population and the bounds for the sample sizes are tabulated. Sample sizes obtained using the approximate least favorable configuration are compared with those obtained using the exact least favorable configuration. Alternate form of the normal approximation to the probability of correct selection is also presented. The relation between the required sample size and the number of populations involved is studied.     

Exact power comparison of three criteria to assess the independence between two sets of variates

Powers of the three criteria are evaluated for testing the hypothesis of the independence between a -set and a q-set of variates in a (p + q) -variate normal population. They are: (1) the likelihood ratio type criterion, Wt (2) the largest root criterion, r1, and (3) criterion of the sum of roots, V. For p= 2, Pillai and Jayachandran, and others have studied for the restricted range of the alternative hypothesis. Recently the power of the largest root was investigated in detail by Sugiyama and %%. In this paper, their power functions are compared in a wide range of the alternative hypotheses. The powers of rl and V are locally optimum, but the W shows a large power in a wide range.     

Abel series distributions with applications to fluctuations of sample functions of stochastic processes

A two-parameter class of discrete distributions, Abel series distributions, generated by expanding a suitable pa,rametric function into a series of Abel polynomials is discussed. An Abel series distribution occurs in fluctuations of sample functions of stochastic processes and has applications in insurance risk, queueing, dam and storage processes. The probability generating function and the factorial moments of the Abel series distributions are obtained in closed forms. It is pointed out that the name of the generalized Poisson distribution of Consul and Jain is justified by the form of its generating function. Finally it is shown that this generalized Poisson distribution is the only member of the Abel series distributions which is closed under convolution.     

Splines as a Useful and Convenient Statistical Tool

The framework for a unified statistical theory of spline regression assuming fixed knots using the truncated polynomial or “+” function representation is presented. In particular, a partial ordering of some spline models is introduced to clarify their relationship and to indicate the hypotheses that can be tested by using either standard multiple regression procedures or a little-used conditional test developed by Hotelling (1940). The construction of spline models with polynomial pieces of different degrees is illustrated. A numerical example from a chemical experiment is given by using the GLM procedure of the statistical software package SAS (Barr et al. 1976).     

Automatic Bayesian quantile regression curve fitting

Quantile regression, including median regression, as a more completed statistical model than mean regression, is now well known with its wide spread applications. Bayesian inference on quantile regression or Bayesian quantile regression has attracted much interest recently. Most of the existing researches in Bayesian quantile regression focus on parametric quantile regression, though there are discussions on different ways of modeling the model error by a parametric distribution named asymmetric Laplace distribution or by a nonparametric alternative named scale mixture asymmetric Laplace distribution. This paper discusses Bayesian inference for nonparametric quantile regression. This general approach fits quantile regression curves using piecewise polynomial functions with an unknown number of knots at unknown locations, all treated as parameters to be inferred through reversible jump Markov chain Monte Carlo (RJMCMC) of Green (Biometrika 82:711–732, 1995). Instead of drawing samples from the posterior, we use regression quantiles to create Markov chains for the estimation of the quantile curves. We also use approximate Bayesian factor in the inference. This method extends the work in automatic Bayesian mean curve fitting to quantile regression. Numerical results show that this Bayesian quantile smoothing technique is competitive with quantile regression/smoothing splines of He and Ng (Comput. Stat. 14:315–337, 1999) and P-splines (penalized splines) of Eilers and de Menezes (Bioinformatics 21(7):1146–1153, 2005).     

Detecting the interaction of quantity and quality: A study of Fisher's analysis

In Sections 49 and 50 of the Design of Experiments, Fisher discusses an experiment designed to compare the effects of several types of manure on yield. Each type of manure is applied at three dosage levels: zero, single, and double doses. Fisher points out that the usual contrasts constructed for a factorial experiment are unsatisfactory in this setting. In particular, since the response curves necessarily meet at the zero dose, the usual notion of interaction as a lack of parallelism cannot apply. Fisher then gives an appropriate definition for interaction in this setting. This paper is concerned with a class of orthogonal polynomials that can be used as an aid in the detection of this modified definition of interaction.