Null distribution of the sum of squared z-transforms in testing complete independence

Brien et al. (1984, Biometrika, 71, 545–554; 1988, Biometrika, 75, 469–476) have proposed, illustrated and discussed advantages of using Fisher's z-transforms for analyzing correlation structures of multinormal data. Chen and Mudholkar (1988, Austral. J. Statist., 31, 105–110) have studied the sum of squared z-transforms of sample correlations as a test statistic for complete independence. In this paper Brown's (1987, Ann. Probab., 15, 416–422) graph-theoretic characterization of the dependence structure of sample correlations is used to evaluate moments of the test statistic. These moments are then used to approximate its null distribution accurately over a broad range of parameters, including the case where the population dimension exceeds the sample size.     

On the approximation of continuous time threshold ARMA processes

Threshold autoregressive (AR) and autoregressive moving average (ARMA) processes with continuous time parameter have been discussed in several recent papers by Brockwellet al. (1991,Statist. Sinica,1, 401–410), Tong and Yeung (1991,Statist. Sinica,1, 411–430), Brockwell and Hyndman (1992,International Journal Forecasting,8, 157–173) and Brockwell (1994,J. Statist. Plann. Inference,39, 291–304). A threshold ARMA process with boundary width 2>0 is easy to define in terms of the unique strong solution of a stochastic differential equation whose coefficients are piecewise linear and Lipschitz. The positive boundary-width is a convenient mathematical device to smooth out the coefficient changes at the boundary and hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equation from which the process is derived. In this paper we give a direct definition of a threshold ARMA processes with =0 in the important case when only the autoregressive coefficients change with the level of the process. (This of course includes all threshold AR processes with constant scale parameter.) The idea is to express the distributions of the process in terms of the weak solution of a certain stochastic differential equation. It is shown that the joint distributions of this solution with =0 are the weak limits as 0 of the distributions of the solution with >0. The sense in which the approximating sequence of processes used by Brockwell and Hyndman (1992,International Journal Forecasting,8, 157–173) converges to this weak solution is also investigated. Some numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the process obtained from the Cameron-Martin-Girsanov formula. It is used in particular to fit continuous-time threshold models to the sunspot and Canadian lynx series.Research partially supported by National Science Foundation Research Grants DMS 9105745 and 9243648.     

A Notion of α-Monotonicity with Generalized Multiplications

The multiplications of van Harn et al. (1982, Z. Wahrsch. Verw. Gebiete, 61, 97–118) are used to generalize the definition of -monotonicity of Olshen and Savage (1970, J. Appl. Probab., 7, 21–34) and Steutel (1988, Statist. Neerlandica, 42, 137–140) for distributions with support in Z ₊ and R ₊. Several characterizations are offered and a convolution property is established. Some relevant stability equations are solved and a relationship with the important concept of self-decomposability is noted. Poisson mixtures are used to deduce results for the R ₊-case from those for the Z ₊-case.     

Semi-empirical likelihood ratio confidence intervals for the difference of two sample means

We all know that we can use the likelihood ratio statistic to test hypotheses and construct confidence intervals in full parametric models. Recently, Owen (1988,Biometrika,75, 237–249; 1990,Ann. Statist.,18, 90–120) has introduced the empirical likelihood method in nonparametric models. In this paper, we combine these two likelihoods together and use the likelihood ratio to construct confidence intervals in a semiparametric problem, in which one model is parametric, and the other is nonparametric. A version of Wilks's theorem is developed.     

On a relationship between Uspensky's theorem and poisson approximations

In this paper we show that Uspensky's expansion theorem for the Poisson approximation of the distribution of sums of independent Bernoulli random variables can be rewritten in terms of the Poisson convolution semigroup. This gives rise to exact evaluations and simple remainder term estimations for the deviations of the distributions in study with respect to various probability metrics, generalizing results of Shorgin (1977, Theory Probab. Appl., 22, 846–850). Finally, we compare the sharpness of Poisson versus normal approximations.     

Fractional Poisson process

A fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov–Feller equation. We have found the probability of n arrivals by time t for fractional stream of events. The fractional Poisson process captures long-memory effect which results in non-exponential waiting time distribution empirically observed in complex systems. In comparison with the standard Poisson process the developed model includes additional parameter μ. At μ=1 the fractional Poisson becomes the standard Poisson and we reproduce the well known results related to the standard Poisson process.As an application of developed fractional stochastic model we have introduced and elaborated fractional compound Poisson process.     

A note on the uniformly most powerful tests in the presence of nuisance parameters

When a testing problem has nuisance parameters, the uniformly most powerful (UMP) tests do not generally exist. Exceptional examples were given by Dubey (1962, Skand. Aktuarietidskr., 45, 25–38; 1963, Skand. Aktuarietidskr., 46, 1–24) and Takeuchi (1968, Ann. Math. Statist., 40, 1838–1839) for the exponential distributions. What is essential for proving the existence of UMP tests lies in a special relationship between null hypothesis and the alternative. Assuming a similar relationship between them, a similar kind of result can be shown under more general situation.     

Empirical Bayes detection of a change in distribution

The problem of detection of a change in distribution is considered. Shiryayev (1963, Theory Probab. Appl., 8, pp. 22–46, 247–264 and 402–413; 1978, Optimal Stopping Rules, Springer, New York) solved the problem in a Bayesian framework assuming that the prior on the change point is Geometric (p). Shiryayev showed that the Bayes solution prescribes stopping as soon as the posterior probability of the change having occurred exceeds a fixed level. In this paper, a myopic policy is studied. An empirical Bayes stopping time is investigated for detecting a change in distribution when the prior is not completely known.Research was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant GP 7987.     

Characterization based on conditional distributions

The field of application of a result given by Singh and Vasudeva (1984, J. Indian Statist. Assoc., 22, 93–96) which provides a way of characterizing the distribution of a random variable X, through conditional distributions of a second variable Z, given X, is extended.     

Asymptotic theorems for estimating the distribution function under random truncation

Representation theorem and local asymptotic minimax theorem are derived for nonparametric estimators of the distribution function on the basis of randomly truncated data. The convolution-type representation theorem asserts that the limiting process of any regular estimator of the distribution function is at least as dispersed as the limiting process of the product-limit estimator. The theorems are similar to those results for the complete data case due to Beran (1977, Ann. Statist., 5, 400–404) and for the censored data case due to Wellner (1982, Ann. Statist., 10, 595–602). Both likelihood and functional approaches are considered and the proofs rely on the method of Begun et al. (1983, Ann. Statist., 11, 432–452) with slight modifications.Division of Biostatistics, School of Public Health, Columbia Univ.     

First-Quantized Fermions in Compact Dimensions

We discuss a path integral representation for fermionic particles and strings in the spirit of V. Ya. Fainberg and the author (Nucl. Phys. B, 306, 659–676, (1998); Phys. Lett. B, 211, 81–85, (1988)). We concentrate on the problems arising when some target-space dimensions are compact. We consider the partition function for a fermionic particle at a finite temperature or in compact time in detail as an example. We demonstrate that a self-consistent definition of the path integral generally requires introducing nonvanishing background Wilson loops and that modulo some common problems for real fermions in the Grassmannian formulation, these loops can be interpreted as condensates of world-line fermions. Properties of the corresponding string-theory path integrals are also discussed.     

On the joint distribution of studentized order statistics

In this paper, the joint distribution of some special linear combinations of the (internally) studentized order statistics are derived for both normal and exponential populations; the exact relationship between their pdf's is also obtained. The exact sampling distributions of studentized extreme deviation statistic, which has been proposed by Pearson and Chandra Sekar (1936,Biometrika,28, 308–320), are derived for these two populations. An application to the most powerful location and scale invariant test is discussed briefly.     

Recurrence relations for single and product moments of progressive Type-II right censored order statistics from exponential and truncated exponential distributions

In this paper, we establish several recurrence relations satisfied by the single and product moments of progressive Type-II right censored order statistics from an exponential distribution. These relations may then be used, for example, to compute all the means, variances and covariances of exponential progressive Type-II right censored order statistics for all sample sizes n and all censoring schemes (R ₁, R ₂, ..., R _m ), mn. The results presented in the paper generalize the results given by Joshi (1978, Sankhy Ser. B, 39, 362–371; 1982, J. Statist. Plann. Inference, 6, 13–16) for the single moments and product moments of order statistics from the exponential distribution.To further generalize these results, we consider also the right truncated exponential distribution. Recurrence relations for the single and product moments are established for progressive Type-II right censored order statistics from the right truncated exponential distribution.     

Multinomial logistic regression algorithm

The lower bound principle (introduced in Böhning and Lindsay 1988, Ann. Inst. Statist. Math., 40, 641–663), Böhning (1989, Biometrika, 76, 375–383) consists of replacing the second derivative matrix by a global lower bound in the Loewner ordering. This bound is used in the Newton-Raphson iteration instead of the Hessian matrix leading to a monotonically converging sequence of iterates. Here, we apply this principle to the multinomial logistic regression model, where it becomes specifically attractive.Supplement to Monotonicity of quadratic-approximation algorithms by Böhning and Lindsay (1988). Ann. Inst. Statist. Math., 40, 641–663.This research was supported by the German Research Foundation.     

On a Multiparameter Version of Tukey's Linear Sensitivity Measure and its Properties

A multiparameter version of Tukey's (1965, Proc. Nat. Acad. Sci. U.S.A., 53, 127–134) linear sensitivity measure, as a measure of informativeness in the joint distribution of a given set of random variables, is proposed. The proposed sensitivity measure, under some conditions, is a matrix which is non-negative definite, weakly additive, monotone and convex. Its relation to Fisher information matrix and the best linear unbiased estimator (BLUE) are investigated. The results are applied to the location-scale model and it is observed that the dispersion matrix of the BLUE of the vector location-scale parameter is the inverse of the sensitivity measure. A similar property was established by Nagaraja (1994, Ann. Inst. Statist. Math., 46, 757–768) for the single parameter case when applied to the location and scale models. Two illustrative examples are included.     

Symbolic computing the exact distributions of L-statistics from a uniform distribution

The exact probability density function of linear combinations of k=k(n) order statistics selected from the whole order statistics (L-statistic) based on a random sample of size n from the uniform distribution on [0, 1] was derived by Matsunawa (1985, Ann. Inst. Statist. Math., 37, 1–16). As the main expression for the density function given by Matsunawa is not complete for the general situation, we first provide the corrections for this formula. Second, we propose a simple scheme involving symbolic computing for evaluating the corrected version of the density function. The cumulative distribution function and the r-th mean of his L-statistic are also derived.     

A bivariate exponential distribution arising in random geometry

Summary In this paper a new bivariate exponential distribution, arising naturally in the theory of Poisson line processes, is studied. The distribution has some interesting and useful properties which renders it suitable for use in statistical modelling work. It is presented in the spirit of adding to the repertoire of bivariate exponential forms. It joins other models, such as those of Downton (1970,J. R. Statist. Soc., B,32, 408–417), Marshall and Olkin (1967,J. Appl. Prob.,4, 291–302) and Nagao and Kadoya (1971,Bulletin of the Disaster Prevention Research Institute,20, 3, 183–215), which have their origins in the theory of stochastic processes.     

Asymptotic expansions for the distribution of quadratic forms in normal variables

Higher order asymptotic expansions for the distribution of quadratic forms in normal variables are obtained. The Cornish-Fisher inverse expansions for the percentiles of the distribution are also given. The resulting formula for a definite quadratic form guarantees accuracy almost up to fourth decimal place if the distribution is not very skew. The normalizing transformation investigated by Jensen and Solomon (1972, J. Amer. Statist. Assoc., 67, 898–902) is reconsidered based on the rate of convergence to the normal distribution.Faculty of Science, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812 Japan     

Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution

Chen and Bhattacharyya (1988,Comm. Statist. Theory Methods,17, 1857–1870) derived the exact distribution of the maximum likelihood estimator of the mean of an exponential distribution and an exact lower confidence bound for the mean based on a hybrid censored sample. In this paper, an alternative simple form for the distribution is obtained and is shown to be equivalent to that of Chen and Bhattacharyya (1988). Noting that this scheme, which would guarantee the experiment to terminate by a fixed timeT, may result in few failures, we propose a new hybrid censoring scheme which guarantees at least a fixed number of failures in a life testing experiment. The exact distribution of the MLE as well as an exact lower confidence bound for the mean is also obtained for this case. Finally, three examples are presented to illustrate all the results developed here.