Orthogonal polynomials generated by random vectors

Every random q-vector with finite moments generates a set of orthonormal polynomials. These are generated from the basis functions xn = xn11…xnqq using Gram–Schmidt orthogonalization. One can cycle through these basis functions using any number of ways. Here, we give results using minimum cycling. The polynomials look simpler when centered about the mean of X, and still simpler form when X is symmetric about zero. This leads to an extension of the multivariate Hermite polynomial for a general random vector symmetric about zero. As an example, the results are applied to the multivariate normal distribution.     

Estimators of shift based on statistics of the Kolmogorov-Smirnov type

This paper is concerned with the estimation of a shift parameter δ_o, based on some nonnegative functional Hg₁ of the pair (D^δ_N(x), f?^δ_N(x)), where D^δ_N(x) = K_N/b {F_2,n(x)—F_1,m (x + δ)}, +^δ_N(x) = {mF_1,m (x + δ) + nF_2,n(x)}/N, where F_1,m and F_2,n are the empirical distribution functions of two independent random samples (N = m + n), and where K²_N = mn/N. First an estimator δ_N, is defined as a value of δ minimizing a functional H of the type of H₁. A second estimator δ¹_N is also defined which is a linearized version of the first. Finite and asymptotic properties of these estimators are considered. It is also shown that most well-known test statistics of the Kolmogorov-Smirnov type are particular cases of such functionals H₁. The asymptotic distribution and the asymptotic efficiency of some estimators are given.     

CRITICAL VALUES FOR INFERENCE ABOUT NORMAL DISPERSION1

Let F_N(.) be the density function of X²_N. Values of C₁/N, i= 1, 2, satisfying the twin conditions Pr (C₁≤X²_N≤C₂)=1-α and the conditional expectation of X²_N given C₁≤X²_N≤C₂ is N are tabulated for α=.2, .1, .05, .01, .005, .001, N=1(1)20(2)50(5)150(10)350. The second condition may be replaced by the condition f_N+2(C₁)=f_N+2V(C₂). The author has with him a bigger table giving C₁ and C₂ for α=.2, .1, .05, .01, .005, .001, N=1(1)350 to three decimals (to three significant digits, if some decimals are not significant). Several applications are mentioned. A practical application that is perhaps not obvious is to test whether two or more counts are distributed as independent Poisson variables. The new simple formulae used in the construction of the table are given and should prove useful in obtaining accurate values for omitted entries and in increasing the accuracy of entries.     

Uniform asymptotic linearity in a regression parameter of a process based on a rank statistic

For X₁, …, X_N a random sample from a distribution F, let the process S^δ_N(t) be defined as where K²_N = σ^N_i=1(c_i ? c?)² and R x_i, + Δd, is the rank of X_i + Δd_i, among X₁ + Δd₁, …, X_N + Δd_N. The purpose of this note is to prove that, under certain regularity conditions on F and on the constants c_i and d_i, S^Δ_N (t) is asymptotically approximately a linear function of Δ, uniformly in t and in Δ, |Δ| ≤ C. The special case of two samples is considered.     

On a problem of counting by weighing

Suppose it is desired to obtain a large number N_s of items for which individual counting is impractical, but one can demand a batch to weigh at least w units so that the number of items N in the batch may be close to the desired number N_s. If the items have mean weight ωTH, it is reasonable to have w equal to ωTHN_s when ωTH is known. When ωTH is unknown, one can take a sample of size n, not bigger than N_s, estimate ωTH by a good estimator ω_n, and set w equal to ω_nN_s. Let R_n = Kp²N²_s/n + K_sn be a measure of loss, where K_e and K_s are the coefficients representing the cost of the error in estimation and the cost of the sampling respectively, and p is the coefficient of variation for the weight of the items. If one determines the sample size to be the integer closest to pCN_s when p is known, where C is (K_e/K_s)^1/2, then R_n will be minimized. If p is unknown, a simple sequential procedure is proposed for which the average sample number is shown to be asymptotically equal to the optimal fixed sample size. When the weights are assumed to have a gamma distribution given ω and ω has a prior inverted gamma distribution, the optimal sample size can be found to be the nonnegative integer closest to pCN_s + p²A(pC – 1), where A is a known constant given in the prior distribution.     

Change-Point Detection With Non-Parametric Regression

We consider the regression model y_i = ?(x_i ) + ε in which the function ? or its pth derivative ?^(p) may have a discontinuity at some unknown point τ. By fitting local polynomials from the left and right, we test the null that ?^(p) is continuous against the alternative that ?^(p)(τ?) ≠ ?^(p)(τ+). We obtain Darling-Erdös type limit theorems for the test statistics under the null hypothesis of no change, as well as their limits in probability under the alternative. Consistency of the related change-point estimators is also established.     

ON TRANSITION PROBABILITIES OF SKIP-FREE MARKOV CHAINS

Consider an ergodic Markov chain X(t) in continuous time with an infinitesimal matrix Q = (q_ij) defined on a finite state space {0, 1,…, N}. In this note, we prove that if X(t) is skip-free positive (negative, respectively), i.e., q_ij, = 0 for j > i+ 1 (i > j+ 1), then the transition probability p_ij(t) = Pr[X(t)=j | X(0) =i] can be represented as a linear combination of p_0N(t) (p^(m)_(N0)(t)), 0 ≤ m ≤N, where f^(m)(t) denotes the mth derivative of a function f(t) with f⁽⁰⁾(t) =f(t). If X(t) is a birth-death process, then p_ij(t) is represented as a linear combination of p_0N^(m)(t), 0 ≤m≤N - |i-j|.     

A uniform saddlepoint expansion for the null‐distribution of the wilcoxon‐mann‐whitney statistic

The authors give the exact coefficient of 1/N in a saddlepoint approximation to the Wilcoxon‐Mann‐Whitney null‐distribution. This saddlepoint approximation is obtained from an Edgeworth approximation to the exponentially tilted distribution. Moreover, the rate of convergence of the relative error is uniformly of order O (1/N) in a large deviation interval as defined in Feller (1971). The proposed method for computing the coefficient of 1/N can be used to obtain the exact coefficients of 1/Nⁱ, for any i. The exact formulas for the cumulant generating function and the cumulants, needed for these results, are those of van Dantzig (1947‐1950).     

On limiting distribution laws of statistics analogous to pearson's chi-square

Two test-statistics analogous to Pearson's chi-square test function - given in (1.6) and (1.7) - are investigated. These statistics utilize, apart from the number of sample elements lying in the respective intervals of the partition, their positions within the intervals too. It is shown that the test-statistics are asymptotically distributed - as the sample size N tends to infinity - according to the x ²distribution with parameter r, i.e. the number of intervals chosen. The limiting distribution of the test statistics under the null-hypothesis when N tends to the infinity and r =O(N ^α) (0<α<1), further the consistency of the tests based on these statistics is considered. Some remarks are made concerning the efficiency of the corresponding goodness of fit tests also; the authors intend to return to a more detailed treatment of the efficiency later.     

Explicit formulae and convergence rate for the system M α /G/1/N as N→∞

The paper deals with the system M ^α /G/1/N with a finite number of waiting places in which arrivals can occur in a group. The number of customers in the line and the virtual waiting time are studied both in the transient and in the stationary regime. Special attention is paid to the stationary distributions of these functionals as N→∞. The number of customers lost during a busy period is considered as well.     

A Two Sample Test for Mean Vectors with Unequal Covariance Matrices

In this paper, we consider testing the equality of two mean vectors with unequal covariance matrices. In the case of equal covariance matrices, we can use Hotelling’s T² statistic, which follows the F distribution under the null hypothesis. Meanwhile, in the case of unequal covariance matrices, the T² type test statistic does not follow the F distribution, and it is also difficult to derive the exact distribution. In this paper, we propose an approximate solution to the problem by adjusting the degrees of freedom of the F distribution. Asymptotic expansions up to the term of order N^{? 2} for the first and second moments of the U statistic are given, where N is the total sample size minus two. A new approximate degrees of freedom and its bias correction are obtained. Finally, numerical comparison is presented by a Monte Carlo simulation.     

Optimal scheduling replacement policies for a system with multiple random works

From the economical viewpoint in reliability theory, this paper addresses a scheduling replacement problem for a single operating system which works at random times for multiple jobs. The system is subject to stochastic failure which results the imperfect maintenance activity based on some random failure mechanism: minimal repair due to type-I (repairable) failure, or corrective replacement due to type-II (non-repairable) failure. Three scheduling models for the system with multiple jobs are considered: a single work, N tandem works, and N parallel works. To control the deterioration process, the preventive replacement is planned to undergo at a scheduling time T or the job's completion time of for each model. The objective is to determine the optimal scheduling parameters (T^* or N^*) that minimizes the mean cost rate function in a finite time horizon for each model. A numerical example is provided to illustrate the proposed analytical model. Because the framework and analysis are general, the proposed models extend several existing results.     

Construction of minimum aberration blocked two-level regular factorial designs

ABSTRACT

In this article, we consider the construction of minimum aberration 2^{n ? k}: 2^p designs with respect to some existing combined wordlength patterns, where a 2^{n ? k}: 2^p design is a blocked two-level design with n treatment factors, 2^p blocks, and N = 2^q runs with q = n ? k. Two methods are proposed for two situations: n ? 2^{q ? p} ? 1 and n > N/2. These methods enable us to obtain some new minimum aberration 2^{n ? k}: 2^p designs from existing minimum aberration unblocked and blocked designs. Examples are included to illustrate the theory.     

Asymptotic normal tests for integration in panels with cross-dependent units

The asymptotically normal, regression-based LM integration test is adapted for panels with correlated units. The N different units may be integrated of different (fractional) orders under the null hypothesis. The paper first reviews conditions under which the test statistic is asymptotically (as T→∞) normal in a single unit. Then we adopt the framework of seemingly unrelated regression [SUR] for cross-correlated panels, and discuss a panel test statistic based on the feasible generalized least squares [GLS] estimator, which follows a χ ²(N) distribution. Third, a more powerful statistic is obtained by working under the assumption of equal deviations from the respective null in all units. Fourth, feasible GLS requires inversion of sample covariance matrices typically imposing T>N; in addition we discuss alternative covariance matrix estimators for T<N. The usefulness of our results is assessed in Monte Carlo experimentation.     

Journal Self-citation Analysis of Some Chinese Sci-tech Periodicals

This study investigates self-citation rates of 222 Chinese journals within seven groups including 76 journals of agronomy (34.2 percent), 57 of biology (25.7 percent), 28 of environmental science and technology (12.6 percent), 15 of forestry (6.8 percent), 24 of academic journals of agricultural university (10.8 percent), 9 of aquatic sciences (4.1 percent), and 13 of animal husbandry and veterinary medicine (5.9 percent). The average self-citation rates range from 2 percent to 67 percent in 2006, 1 percent to 68 percent in 2007 and 0 percent to 67 percent in 2008. There is a significant difference in self-citation rate between most groups of journals. The self-citation rate is positively and significantly correlated with the self-citation rate in 2006 for all 222 journals (N = 222, R² = 0.194, P = 0.004) (P < 0.05). However, the self-citation rate is not significantly correlated with the journal's impact factor in 2007 (N = 222, R² = 0.114, P = 0.091) and 2008 (N = 222, R² = 0.112, P = 0.096) (P < 0.05) for the 222 journals. The relationship between self-citation rate and journal impact factor is discussed.     

Commuting Matrices in the Queue Length and Sojourn Time Analysis of MAP/MAP/1 Queues

Queues with Markovian arrival and service processes, i.e., MAP/MAP/1 queues, have been useful in the analysis of computer and communication systems and different representations for their stationary sojourn time and queue length distribution have been derived. More specifically, the class of MAP/MAP/1 queues lies at the intersection of the class of QBD queues and the class of semi-Markovian queues. While QBD queues have a matrix exponential representation for their queue length and sojourn time distribution of order N and N², respectively, where N is the size of the background continuous time Markov chain, the reverse is true for a semi-Markovian queue. As the class of MAP/MAP/1 queues lies at the intersection, both the queue length and sojourn time distribution of a MAP/MAP/1 queue has an order N matrix exponential representation. The aim of this article is to understand why the order N² distributions of the sojourn time of a QBD queue and the queue length of a semi-Markovian queue can be reduced to an order N distribution in the specific case of a MAP/MAP/1 queue. We show that the key observation exists in establishing the commutativity of some fundamental matrices involved in the analysis of the MAP/MAP/1 queue.     

Strassen type limit points for moving averages of a wiener process

Let {W(s); 8 ≥ 0} be a standard Wiener process, and let β_N = (2a_N (log (N/a_N) + log log N)^-1/2, 0 < α_N ≤ N < ∞, where α_N↑ and α_N/N is a non-increasing function of N, and define γ_N(t) = β_N[W(n_N + ta_N) ? W(n_N)), 0 ≥ t ≥ 1, with n_N = N – a_N. Let K = {x ? C[0,1]: x is absolutely continuous, x(0) = 0 and }. We prove that, with probability one, the sequence of functions {γ_N(t), t ? [0,1]} is relatively compact in C[0,1] with respect to the sup norm ||·||, and its set of limit points is K. With a_N = N, our result reduces to Strassen's well-known theorem. Our method of proof follows Strassen's original, direct approach. The latter, however, contains an oversight which, in turn, renders his proof invalid. Strassen's theorem is true, of course, and his proof can also be rectified. We do this in a somewhat more general context than that of his original theorem. Applications to partial sums of independent identically distributed random variables are also considered.     

Bias of the lse estimator of the first order autoregressive model under tukey contamination

Results of an exhaustive study of the bias of the least square estimator (LSE) of an first order autoregression coefficient α in a contaminated Gaussian model are presented. The model describes the following situation. The process is defined as X_t = α X_t-1 + Y_t . Until a specified time T, Y_t are iid normal N(0, 1). At the moment T we start our observations and since then the distribution of Y_t, t≥T, is a Tukey mixture T(εσ) = (1 – ε)N(0,1) + εN(0, σ²). Bias of LSE as a function of α and ε, and σ² is considered. A rather unexpected fact is revealed: given α and ε, the bias does not change montonically with σ (“the magnitude of the contaminant”), and similarly, given α and σ, the bias is not growing with ε (“the amount of contaminants”).     

PIECEWISE POLYNOMIAL APPROXIMATIONS FOR HEAVY-TAILED DISTRIBUTIONS IN QUEUEING ANALYSIS

ABSTRACT

A basic difficulty in dealing with heavy-tailed distributions is that they may not have explicit Laplace transforms. This makes numerical methods that use the Laplace transform more challenging. This paper generalizes an existing method for approximating heavy-tailed distributions, for use in queueing analysis. The generalization involves fitting Chebyshev polynomials to a probability density function g(t) at specified points t ₁, t ₂, …, t _N . By choosing points t _i , which rapidly get far out in the tail, it is possible to capture the tail behavior with relatively few points, and to control the relative error in the approximation. We give numerical examples to evaluate the performance of the method in simple queueing problems.     

Estimators of location based on Kolmogorov-Smirnov-type statistics

Two classes of estimators of a location parameter ø₀ are proposed, based on a nonnegative functional H₁^* of the pair (D₁^ø_N, G^ø_N), where and where F_N is the sample distribution function. The estimators of the first class are defined as a value of ø minimizing H₁^*; the estimators of the second class are linearized versions of those of the first. The asymptotic distribution of the estimators is derived, and it is shown that the Kolmogorov-Smirnov statistic, the signed linear rank statistics, and the Cramérvon Mises statistics are special cases of such functionals H₁^*;. These estimators are closely related to the estimators of a shift in the two-sample case, proposed and studied by Boulanger in B2 (pp. 271–284).