ON THE CONVERGENCE RESULT FOR THE SUPERCRITICAL BELLMAN-HARRIS PROCESS1

It was proved in Cohn (1982) that for any finite offspring mean, supercritical Bellman-Harris process {Z(t)} there exist some norming constants {C(t)} such that {Z(t)/C(t)} converges almost surely to a non-degenerate random variable W. {C(t)} were defined to be the μ-quantiles of {Z(t)}. Schuh (1982) has given an alternative proof of this result, identifying C(t) as “the Seneta constants” 1/(-log F_t^(-1)(γ)), where F₁(γ) = E(γ^Z(t)). Both proofs are long and complicated. It will be shown here that a much simpler proof can be devised from Cohn (1982), if use is made of an elementarily proved property given in Schuh (1982).     

A FINE-TUNED ESTIMATOR OF A GENERAL CONVERGENCE RATE

A general rate estimation method based on the in‐sample evolution of appropriately chosen diverging/converging statistics has recently been proposed by D.N. Politis [C. R. Acad. Sci. Paris, Ser. I, vol. 335, pp. 279–282, 2002] and T. McElroy & D.N. Politis [Ann. Statist., vol. 35, pp. 1827–1848, 2007]. In this paper, we show how a modification of the original estimators achieves a competitive rate of convergence. The modified estimators require the choice of a tuning parameter; an optimal such choice is generally a non‐trivial problem in practice. Some discussion to that effect is given, as well as a small simulation study in a heavy‐tailed setting.     

EUROPEAN OPTION PRICING WITH GENERAL TRANSACTION COSTS AND SHORT-SELLING CONSTRAINTS

In this paper, we study the problem of European Option Pricing in a market with short-selling constraints and transaction costs having a very general form. We consider two types of proportional costs and a strictly positive fixed cost. We study the problem within the framework of the theory of stochastic impulse control. We show that determining the price of a European option involves calculating the value functions of two stochastic impulse control problems. We obtain explicit expressions for the quasi-variational inequalities satisfied by the value functions and derive the solution in the case where the parameters of the price processes are constants and the investor's utility function is linear. We use this result to obtain a price for a call option on the stock and prove that this price is a nontrivial lower bound on the hedging price of the call option in the presence of general transaction costs and short-selling constraints. We then consider the situation where the investor's utility function has a general form and characterize the value function as the pointwise limit of an increasing sequence of solutions to associated optimal stopping problems. We thereby devise a numerical procedure to calculate the option price in this general setting and implement the procedure to calculate the option price for the class of exponential utility functions. Finally, we carry out a qualitative investigation of the option prices for exponential and linear-power utility functions.     

PHASE RANDOMIZATION: A CONVERGENCE DIAGNOSTIC TEST FOR MCMC

Most Markov chain Monte Carlo (MCMC) users address the convergence problem by applying diagnostic tools to the output produced by running their samplers. Potentially useful diagnostics can be borrowed from diverse areas such as time series. One such method is phase randomization. This paper describes this method in the context of MCMC, summarizes its characteristics, and contrasts its performance with those of the more common diagnostic tests for MCMC. It is observed that the new tool contributes information about third‐ and higher‐order cumulant behaviour which is important in characterizing certain forms of nonlinearity and non‐stationarity.     

ON CONVERGENCE RATES FOR NONPARAMETRIC POSTERIOR DISTRIBUTIONS

Rates of convergence of Bayesian nonparametric procedures are expressed as the maximum between two rates: one is determined via suitable measures of concentration of the prior around the “true” density f₀, and the other is related to the way the mass is spread outside a neighborhood of f₀. Here we provide a lower bound for the former in terms of the usual notion of prior concentration and in terms of an alternative definition of prior concentration. Moreover, we determine the latter for two important classes of priors: the infinite–dimensional exponential family, and the Pólya trees.     

ON A GENERAL DISTRIBUTION THEORY FOR A CLASS OF LIKELIHOOD RATIO CRITERIA

In this paper the exact distribution of a statistic with general moment function has been derived in a gamma series and also in a beta series. It is shown that the Box expansion can be obtained from the gamma series by collecting terms of the same order.     

A MORE GENERAL CRITERION FOR SUBSET SELECTION IN MULTIPLE LINEAR REGRESSION

ABSTRACT

In this article, we propose a more general criterion called S_p -criterion, for subset selection in the multiple linear regression Model. Many subset selection methods are based on the Least Squares (LS) estimator of β, but whenever the data contain an influential observation or the distribution of the error variable deviates from normality, the LS estimator performs ‘poorly’ and hence a method based on this estimator (for example, Mallows’ C_p -criterion) tends to select a ‘wrong’ subset. The proposed method overcomes this drawback and its main feature is that it can be used with any type of estimator (either the LS estimator or any robust estimator) of β without any need for modification of the proposed criterion. Moreover, this technique is operationally simple to implement as compared to other existing criteria. The method is illustrated with examples.     

ESTIMATION FOR THE GENERAL SAMPLE SELECTION MODELS

Consider a general regression model with an arbitrary and unknown link function and a stochastic selection variable that determines whether the outcome variable is observable or missing. The paper proposes U-statistics that are based on kernel functions as estimators for the directions of the parameter vectors in the link function and the selection equation, and shows that these estimators are consistent and asymptotically normal.     

LARGE-SAMPLE INFERENCE FOR THE GENERAL HALF-NORMAL DISTRIBUTION

ABSTRACT

The paper considers the development of inferential techniques for the general half-normal distribution based on the maximum likelihood method. Point estimates and large-sample confidence sets are derived for the distribution's parameters, and the use of the confidence sets as a basis for hypothesis testing is discussed. Data on the percentage body fat of elite male athletes are used to illustrate the application of the new methods.     

THE CONVERGENCE RATE OF SEQUENTIAL FIXED-WIDTH CONFIDENCE INTERVALS FOR REGULAR FUNCTIONALS

Let X₁X₂,.be i.i.d. random variables and let U_n= (n r)^-1S?^(n,r) h (X_i1,., X_ir,) be a U-statistic with EU_n= v, v unknown. Assume that g(X₁) =E[h(X₁,.,X_r) - v |X₁]has a strictly positive variance s?². Further, let a be such that φ(a) - φ(-a) =α for fixed α, 0 < α < 1, where φ is the standard normal d.f., and let S²_n be the Jackknife estimator of n Var U_n. Consider the stopping times N(d)= min {n: S²_n: + n^-1≤²a^-2},d > 0, and a confidence interval for v of length 2d,of the form I_n,d= [U_n,-d, Un + d]. We assume that Var U_n is unknown, and hence, no fixed sample size method is available for finding a confidence interval for v of prescribed width 2d and prescribed coverage probability α Turning to a sequential procedure, let I_N(d),d be a sequence of sequential confidence intervals for v. The asymptotic consistency of this procedure, i.e. lim_{d → 0}P(v ∈ I_N(d),d)=α follows from Sproule (1969). In this paper, the rate at which |P(v ∈ I_N(d),d) converges to α is investigated. We obtain that |P(v ∈ I_N(d),d) - α| = 0 (d^{1/2-(1+k)/2(1+m)}), d → 0, where K = max {0,4 - m}, under the condition that E|h(X₁, X_r)|^m < ∞m > 2. This improves and extends recent results of Ghosh & DasGupta (1980) and Mukhopadhyay (1981).     

JACKKNIFE TESTS FOR HETEROSCEDASTICITY IN THE GENERAL LINEAR MODEL

In this paper, tests based on the Jackknife technique are proposed to test for heteroscedasticity in the linear regression model when the errors are non-normal. These are the Jackknifed Goldfeld-Quandt (GQ), and jack-knife related variations of White (H), Lagrange multiplier (LM), Glejser (GL) and Bickel (B) tests. The power of the proposed tests is compared with that of GQ, H, LM, GL and B tests; and the robustness to the error distribution is analyzed under several heteroscedastic assumptions. The GQ test is by far the best test if the error distribution is close to normal, however, GQ test is not robust against non-normal errors. By applying the jackknife technique to the regression a more robust statistic (GQJRG) is produced but the cost is a loss in power. The GQJRG statistic generally is not M powerful as the Bickel (BlOLS) and Glejser (GLlOLS) statistics.     

THE RATE OF CONVERGENCE IN THE NORMAL APPROXIMATION FOR THE WILCOXON R-ESTIMATOR

We exploit the fact that the Wilcoxon score R-estimator of the slope in a linear regression model minimises Gini's mean difference of the residuals to obtain a Berry-Esseen rate of convergence result for the Wilcoxon R-estimator.     

RANDOMIZED SEARCH STRATEGIES FOR FINDING OPTIMAL OR NEAR OPTIMAL BLOCK AND ROW-COLUMN DESIGNS

We consider a family of effective and efficient strategies for generating experimental designs of several types with high efficiency. These strategies employ randomized search directions and at some stages allow the possibility of taking steps in a direction of decreasing efficiency in an effort to avoid local optima. Hence our strategies have some affinity with the simulated annealing algorithm of combinatorial optimization. The methods work well and compare favourably with other search strategies. We have implemented them for incomplete block designs, optionally resolvable, and for row-column designs.     

CANONICAL VARIATE ANALYSIS—A GENERAL MODEL FORMULATION

A general model, specifying the population means as a function of the population canonical vectors, provides a natural basis for considering many aspects of canonical variate analysis. These aspects include: estimation for elliptical densities, and robust estimation; estimation of missing values; adequacy of hypothetical variables; regression in dummy variable space; and comparison of solutions.     

ON UMP INVARIANT F-TEST PROCEDURES IN A GENERAL LINEAR MODEL

A simple method of setting linear hypotheses for a split mean vector testable by F-tests in a general linear model, when the covariance matrix has a general form and is completely unknown, is provided by extending the method discussed in Ukita et al. The critical functions in these F-tests are constructed as UMP invariants, when the covariance matrix has a known structure. Further critical functions in F-tests of linear hypotheses for the other split mean vector in the model are shown to be UMP invariant if the same known structure of the covariance matrix is assumed.     

A ROLE FOR VARIOGRAMS

In time series texts and journals, variograms are mentioned seldom, if at all. The autocovariance function is preferred. However there are situations where the variogram can be estimated with moderate precision but the autocovariance function cannot, because the variance of the process is not well known. If the problem to be solved does not require the process variance for its solution then it is generally more straightforward to use the variogram rather than the autocovariance function in solving this problem.     

A NEW TEST FOR EQUALITY OF VARIANCES FOR k NORMAL POPULATIONS

ABSTRACT

A simple test based on Gini's mean difference is proposed to test the hypothesis of equality of population variances. Using 2000 replicated samples and empirical distributions, we show that the test compares favourably with Bartlett's and Levene's test for the normal population. Also, it is more powerful than Bartlett's and Levene's tests for some alternative hypotheses for some non-normal distributions and more robust than the other two tests for large sample sizes under some alternative hypotheses. We also give an approximate distribution to the test statistic to enable one to calculate the nominal levels and P-values.     

A REPARAMETERISATION METHOD FOR EMBEDDED MODELS

ABSTRACT

A reparameterisation procedure is investigated for embedded model problems. The procedure is given by solving differential equations determined by indeterminate forms of limit. Some properties are provided for the existence of an embedded model. Note that an embedded model may include another embedded model. We introduce the concept of embedded model of kth generation and discuss the use of one-by-one elimination procedure to construct graphs of embedded models. As examples, we derive embedded models for some distributions, to which existing method cannot be applied. Our method includes the method given by Cheng et al. [1] Cheng, R.C.H., Evans, B.E. and Iles, T.C. 1992. Embedded Models in Non-Linear Regression. J. R. Statist. Soc. B, 54: 877–888.  [Google Scholar] as a special case.