A modified cluster sampling method for estimating the bacterial density in Xanthan gum

Suppose particles are randomly distributed in a certain medium, powder or liquid, which is conceptually divided into N cells. Let p_i denote the probability that a particle falls in the ith cell and Y_i denote the number of particles in the ith cell. Assume that the joint probability function of the Y_i follows a multinomial distribution with cell probabilities p_i respectively. Take n (≤N) cells at random without replacement and put each of the cells separately through a mixing mechanism of dilution and swirl. These n cells constitute the first stage samples and the number of particles in these cells are not observable. Now conceptually divide each of n cells into M subcells of equal size and let X_ij denote the number of particles in the jth subcell of the ith cell selected in the first stage; i=1,2,…,N and j=1,2,…,M. Consequently assume that the conditional joint probability function of the X_ij given Y_i=y_i follows a multinomial distribution with equal cell probabilities. Now take m (≤M) subcells at random from each of the cells selected in the first stage sample. Assume that the numbers of particles in M×N subcells are observable. The properties of the estimator of the particle density per sample unit are investigated under the modified two-stage cluster sampling method. A laboratory experiment for Xanthan Gum Products is analyzed in order to examine the appropriateness of the model assumed in this paper.     

Number of observations near the maximum in an F α-scheme

Let K _n (a) be the number of observations in the interval (M _n ,?a, M _n ), where M _n is the maximum value in a sequence of size n. We study the asymptotic properties of K _n (a) under the F ^α-scheme and discuss the influence of the associated sequence α _n on the limit behaviour of this random variable.     

Second order asymptotic relations of M-estimators and R-estimators in two-sample location model

Let T_M be an M-estimator (maximum likelihood type estimator) and T_R be an R-estimator (Hodges-Lehmann's estimator) of the shift parameter Δ in the two-sample location model. The asymptotic representation of √N(T_M-T_R) up to a term of the order O_p(N^{-$\text{1}\text{4}$}) is derived which is valid if the functions Ψ and ? generating T_M and T_R, respectively, decompose into an absolutely continuous and a step-function components; the order O_p(N^{-$\text{1}\text{4}$}) cannot be improved unless the discontinuous components vanish. As a consequence, the conditions under which √N(T_M-T_R)=O_p(N^{-$\text{1}\text{4}$}) are obtained. The main tool for obtaining the results is the second order asymptotic linearity of the pertaining linear rank statistics which is proved here under the assumption that the score-generating function ? has some jump-discontinuities.     

Number of occurrences in two-state Markov chains,with an application in linguistics

Let N_n be the number of occurrences in n trials of an event governed by a two-state Markov chain (of first order or second order). We obtain the distribution of N_n and apply it to a problem involving literary text.     

Multivariate signed rank statistics for shift alternatives

Let X ₁,X ₂,…,X _N be successive independent random P-vectors drawn from some continuous diagonally symmetric distribution. The problem of detecting a shift in level of the sequence at an unknown time point M, ≦M ≦ N-1, is studied. Test statistics based on multivariate analogues of the rank statistics derived by BHATTACHARYYA and JOHNSON (1888) are proposed, and their asymptotic properties are investigated.     

Two remarks on order statistics

Let X₍₁₎,…,X_(n) be the order statistics of n iid distributed random variables. We prove that (X_(i)) have a certain Markov property for general distributions and secondly that the order statistics have monotone conditional regression dependence. Both properties are well known in the case of continuous distributions.     

Simultaneous confidence bands for expectile functions

Expectile regression, as a general M smoother, is used to capture the tail behaviour of a distribution. Let (X ₁,Y ₁),…,(X _n ,Y _n ) be i.i.d. rvs. Denote by v(x) the unknown τ-expectile regression curve of Y conditional on X, and by v _n (x) its kernel smoothing estimator. In this paper, we prove the strong uniform consistency rate of v _n (x) under general conditions. Moreover, using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup_0≤x≤1|v _n (x)?v(x)|. According to the asymptotic theory, we construct simultaneous confidence bands around the estimated expectile function. Furthermore, we apply this confidence band to temperature analysis. Taking Berlin and Taipei as an example, we investigate the temperature risk drivers to these two cities.     

On the number of near-records after the maximum observation in a continuous sample

{X_n, n≥1} are independent and identically distributed random variables with continuous distribution function F(x). For j=1,…,n, X_j is called a near-record up to time n if X_j ∈ (M_na, M_n], where M_n = max_1≤j≤n {X_j} and a is a positive constant. Let Z_n(a) denote the number of near-records after, and including the maximum observation of the sequence. In this paper, the distributional results of Z_n(a) are considered and its asymptotic behaviours are studied.     

Density estimation for samples satisfying a certain absolute regularity condition

Let {ξ_i} be an absolutely regular sequence of identically distributed random variables having common density function f(x). Let H_k(x,y) (k=1, 2,…) be a sequence of Borel-measurable functions and f_n(x)=n^?1(H_n(x,ξ₁)+…+H_n(x,ξ_n)) the empirical density function. In this paper, the asymptotic property of the probability P(sup_x|f_n(x)?f(x)|>ε) (n→∞) is studied.     

DISTRIBUTIONS IN R WITH ROTATIONAL SYMMETRIES1

Let X ∈ R be a random vector with a distribution which is invariant under rotations within the subspaces V_j (dim V_j. = q_j) whose direct sum is R. The large sample distributions of the eigenvalues and vectors of M_n= n^-1Σⁿ_l x_ix_i are studied. In particular it is shown that several eigenvalue results of Anderson & Stephens (1972) for uniformly distributed unit vectors hold more generally.     

Characterizations of geometric distribution and discrete IFR (DFR) distributions using order statistics

Let X be a discrete random variable the set of possible values (finite or infinite) of which can be arranged as an increasing sequence of real numbers a₁<a₂<a₃<…. In particular, a_i could be equal to i for all i. Let X_1n≦X_2n≦?≦X_nn denote the order statistics in a random sample of size n drawn from the distribution of X, where n is a fixed integer ≧2. Then, we show that for some arbitrary fixed k(2≦k≦n), independence of the event {X_kn=X_1n} and X_1n is equivalent to X being either degenerate or geometric. We also show that the montonicity in i of P{X_kn = X_1n | X_1n = a_i} is equivalent to X having the IFR (DFR) property. Let a_i = i and $\text{G(i) = P(X≧i), i = 1, 2, …}$. We prove that the independence of {X_2n ? X_1n ∈B} and X_1n for all i is equivalent to X being geometric, where B = {m} (B = {m,m+1,…}), provided G(i) = q^i?1, 1≦i≦m+2 (1≦i≦m+1), where 0<q<1.     

The number of solutions to the alternate matrix equation over a finite field and a q-identity

Let F_q be a finite field with q elements, where q is a power of a prime. In this paper, we first correct a counting error for the formula N(K_2ν,0^(m)) occurring in Carlitz (1954. Arch. Math. V, 19–31). Next, using the geometry of symplectic group over F_q, we have given the numbers of solutions X of rank k and solutions X to equation XAX′=B over F_q, where A and B are alternate matrices of order n, rank 2ν and order m, rank 2s, respectively. Finally, an elementary q-identity is obtained from N(K_2ν,0⁽⁰⁾), and the explicit results for N(K_n,2ν,K_m,2s) is represented by terminating q-hypergeometric series.     

Poisson approximations in selected metrics by coupling and semigroup methods with applications

Let S_n = X₁ + … + X_n, where X₁,…, X_n are independent Bernoulli random variables. In this paper, we evaluate probability metrics of the Wasserstein type between the distribution of S_n and a Poisson distribution. Our results show that, if E(S_n) = O(1) and if the individual probabilities of success of the X_i's tend uniformly to zero, then the general rate of convergence of the above mentioned metrics to zero is O(∑ⁿ_i = ₁P²_i). We also show that this rate is sharp and discuss applications of these results.     

Orthogonal [k − 1, k + 1]-factorizations in graphs

Let G be a graph. Let F={F₁,F₂,...F_d} be a factorization of G and H be a subgraph of G. If H has exactly one edge in common with F_i for all i = 1,2,…,d, then we say that F is orthogonal to H. In this paper it is proved that for any d-matching M of a [kd − 1, kd + 1]-graphG, there is a [k − 1, k + 1]-factorization of G orthogonal to M where k ⩾ 2 is an integer.     

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.     

On tail parameter estimation in certain point process models

Let μ(ds, dx) denote Poisson random measure with intensity dsG(dx) on (0, ∞) × (0, ∞), for a measure G(dx) with tails varying regularly at ∞. We deal with estimation of index of regular variation α and weight parameter ξ if the point process is observed in certain windows K_n = [0, T_n] × [Y_n, ∞), where Y_n → ∞ as n → ∞. In particular, we look at asymptotic behaviour of the Hill estimator for α. In certain submodels, better estimators are available; they converge at higher speed and have a strong optimality property. This is deduced from the parametric case G(dx) = ξαx^−α−1 dx via a neighbourhood argument in terms of Hellinger distances.     

A RENEWAL THEOREM IN MULTIDIMENSIONAL TIME

Let Y_l, Y₂,… be i.i.d., positive, integer-valued random variables with means, μ. Let the sequences {Y_ij, j= 1,2,…}, i= 1,…, r be independent copies of {Y₁, Y₂,…}. For n={n₁,…, n_r.}, n₁≥1, let S_n=S?ⁿ¹_k1=1= 1 …S?^nr_kr=1 Y_ik1… Y_rkr. We show that S?^N_k=1S?^∞_k1=1…S?^∞_nr=1 P[[S_n= k] ? [μ^-r N log^r-1 (N)/(r-1)!] as N →∞.     

Estimation of a standard measuring error by repeated measurements and sortings

A sorting-and-measuring machine (SMM) measures and sorts (classifies) on-line produced items into several groups according to their size. The measuring devices of the SMM perceive the actual item size with a random error ε and classify the item as being smaller than b iff z+ε<b. Here ε is a normal zero-mean r.v. with unknown standard deviation σ which is the main parameter characterizing the precision and technical condition of an SMM. The paper gives the following method of estimating σ. N₀ items are measured and N₁ of them are recognized by the SMM as belonging to the group a<z≤b. These N₁ items are sorted again and N₂ of them return to this group, these are sorted again, and so on. The estimation of σ is based on the statistics N_m/N_n. Moments of the ratio statistics N_m/N_n and their distributional properties are investigated. It turns out that the expected value of N_m/N_n depends almost linearly on σ which allows us to construct ‘almost’ unbiased estimators of type $\text{σ}\text{?}{}_{\mathrm{mn}}\text{=A}\textcolor{red}{}\text{N}{}_{\mathrm{m}}\text{/N}{}_{\mathrm{n}}\text{+B}$ with good propert including robustness with respect to the distribution of item size. Convex combinations of $\text{σ}\text{?}{}_{\mathrm{mn}}$ statistics are considered to obtain an estimator with minimal variance.     

A note on L1 consistent estimation

Let (??, ??) be a space with a σ-field, M = {P_s; s o} a family of probability measures on A, Θ arbitrary, X₁,…,X_n independently and identically distributed P random variables. Metrize Θ with the L₁ distance between measures, and assume identifiability. Minimum-distance estimators are constructed that relate rates of convergence with Vapnik-Cervonenkis exponents when M is “regular”. An alternative construction of estimates is offered via Kolmogorov's chain argument.