Distribution of record statistics in a geometrically increasing population

The probability function and binomial moments of the number _Nn$N_n$ of (upper) records up to time (index) n in a geometrically increasing population are obtained in terms of the signless q-Stirling numbers of the first kind, with q   being the inverse of the proportion λ$\lambda$ of the geometric progression. Further, a strong law of large numbers and a central limit theorem for the sequence of random variables _Nn$N_n$, n=1,2,…,$n=1,2,\dots ,$ are deduced. As a corollary the probability function of the time _Tk$T_k$ of the kth record is also expressed in terms of the signless q  -Stirling numbers of the first kind. The mean of _Tk$T_k$ is obtained as a q  -series with terms of alternating sign. Finally, the probability function of the inter-record time _Wk=_Tk-_Tk-1$W_k=T_k-T_{k-1}$ is obtained as a sum of a finite number of terms of q  -numbers. The mean of _Wk$W_k$ is expressed by a q-series. As k   increases to infinity the distribution of _Wk$W_k$ converges to a geometric distribution with failure probability q. Additional properties of the q-Stirling numbers of the first kind, which facilitate the present study, are derived.     

Distribution of order statistics from selected subsets of concomitants

For a random sample of size n from an absolutely continuous bivariate population (X, Y), let X_i:n be the i th X-order statistic and Y_[i:n] be its concomitant. We study the joint distribution of (V_s:m, W_t:n−m), where V_s:m is the s th order statistic of the upper subset {Y_[i:n], i=n−m+1,…,n}, and W_t:n−m is the t th order statistic of the lower subset {Y_[j:n], j=1,…,n−m  } of concomitants. When m=⌈_np0⌉$m=\lceil {\mathit{np}}_0\rceil$, s=⌈_mp1⌉$s=\lceil {\mathit{mp}}_1\rceil$, and t=⌈(n−m)_p2⌉$t=\lceil (n-m)p_2\rceil$, 0<_pi<1,i=0,1,2$0<p_i<1,i=0,1,2$, and n→∞$n\to \infty$, we show that the joint distribution is asymptotically bivariate normal and establish the rate of convergence. We propose second order approximations to the joint and marginal distributions with significantly better performance for the bivariate normal and Farlie–Gumbel bivariate exponential parents, even for moderate sample sizes. We discuss implications of our findings to data-snooping and selection problems.     

Distribution of concomitants of order statistics and their order statistics

For a random sample of size n$n$ from an absolutely continuous random vector (X,Y)$(X,Y)$, let _Yi:n$Y_{i:n}$ be i$i$th Y$Y$-order statistic and _Y[j:n]$Y_{[j:n]}$ be the Y$Y$-concomitant of _Xj:n$X_{j:n}$. We determine the joint pdf of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$ for all i,j=1$i,j=1$ to n$n$, and establish some symmetry properties of the joint distribution for symmetric populations. We discuss the uses of the joint distribution in the computation of moments and probabilities of various ranks for _Y[j:n]$Y_{[j:n]}$. We also show how our results can be used to determine the expected cost of mismatch in broken bivariate samples and approximate the first two moments of the ratios of linear functions of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$. For the bivariate normal case, we compute the expectations of the product of _Yi:n$Y_{i:n}$ and _Y[i:n]$Y_{[i:n]}$ for n=2$n=2$ to 8 for selected values of the correlation coefficient and illustrate their uses.     

Testing spatial randomness based on empirical distribution function: A study on lattice data

In this article, we extended the empirical distribution function based test statistic _Ik$I_k$ of Skaug and Tjostheim [1993. Nonparametric test of serial independence based on the empirical distribution function. Biometrika 80, 591–602] in the time series setting to _Dn$D_n$ for spatial lattice data and derived the asymptotic distribution of the proposed test statistic _Dn$D_n$ under the null hypothesis of spatial independence. The size and power of the proposed test statistic under conditional autoregressive model (CAR) were simulated. We applied _Dn$D_n$, Moran's I and Geary's c   to the transformed and well-studied sudden infant death syndrome data from North Carolina and found that _Dn$D_n$ produced a much smaller p$p$-value in testing spatial independence.     

Asymptotic results in canonical discriminant analysis when the dimension is large compared to the sample size

This paper deals with the distributions of test statistics for the number of useful discriminant functions and the characteristic roots in canonical discriminant analysis. These asymptotic distributions have been extensively studied when the number p   of variables is fixed, the number q+1$q+1$ of groups is fixed, and the sample size N tends to infinity. However, these approximations become increasingly inaccurate as the value of p increases for a fixed value of N. On the other hand, we encounter to analyze high-dimensional data such that p is large compared to n. The purpose of the present paper is to derive asymptotic distributions of these statistics in a high-dimensional framework such that q   is fixed, p→∞$p\to \infty$, m=n-p+q→∞$m=n-p+q\to \infty$, and p/n→c∈(0,1)$p/n\to c\in (0,1)$, where n=N-q-1$n=N-q-1$. Numerical simulation revealed that our new asymptotic approximations are more accurate than the classical asymptotic approximations in a considerably wide range of (n,p,q)$(n,p,q)$.     

An identity for the Fisher information and Mahalanobis distance

Consider a mixture problem consisting of k classes. Suppose we observe an s-dimensional random vector X   whose distribution is specified by the relations P(X∈A|Y=i)=_Pi(A)$P(X\in A|Y=i)=P_i(A)$, where Y   is an unobserved class identifier defined on {1,…,k}$\{1,\dots ,k\}$, having distribution P(Y=i)=_pi$P(Y=i)=p_i$. Assuming the distributions _Pi$P_i$ having a common covariance matrix, elegant identities are presented that connect the matrix of Fisher information in Y   on the parameters _p1,…,_pk$p_1,\dots ,p_k$, the matrix of linear information in X, and the Mahalanobis distances between the pairs of P  's. Since the parameters are not free, the information matrices are singular and the technique of generalized inverses is used. A matrix extension of the Mahalanobis distance and its invariant forms are introduced that are of interest in their own right. In terms of parameter estimation, the results provide an independent of the parameter upper bound for the loss of accuracy by esimating _p1,…,_pk$p_1,\dots ,p_k$ from a sample of X^′$X^{\prime }$s, as compared with the ideal estimator based on a random sample of Y^′$Y^{\prime }$s.     

Optimal mixed-level k-circulant supersaturated designs

Supersaturated designs (SSDs) offer a potentially useful way to investigate many factors with only few experiments in the preliminary stages of experimentation. This paper explores how to construct E(_fNOD)$E(f_{\mathrm{NOD}})$-optimal mixed-level SSDs using k-cyclic generators. The necessary and sufficient conditions for the existence of mixed-level k-circulant SSDs with the equal occurrence property are provided. Properties of the mixed-level k  -circulant SSDs are investigated, in particular, the sufficient condition under which the generator vector produces an E(_fNOD)$E(f_{\mathrm{NOD}})$-optimal SSD is obtained. Moreover, many new E(_fNOD)$E(f_{\mathrm{NOD}})$-optimal mixed-level SSDs are constructed and listed. The method here generalizes the one proposed by Liu and Dean [2004. k$k$-circulant supersaturated designs. Technometrics 46, 32–43] for two-level SSDs and the one due to Georgiou and Koukouvinos [2006. Multi-level k-circulant supersaturated designs. Metrika 64, 209–220] for the multi-level case.     

On the maximum of periodic integer-valued sequences with exponential type tails via max-semistable laws

In this work we study the limiting distribution of the maximum term of periodic integer-valued sequences with marginal distribution belonging to a particular class where the tail decays exponentially. This class does not belong to the domain of attraction of any max-stable distribution. Nevertheless, we prove that the limiting distribution is max-semistable when we consider the maximum of the first k_n observations, for a suitable sequence {_kn}$\{k_n\}$ increasing to infinity. We obtain an expression for calculating the extremal index of sequences satisfying certain local conditions similar to conditions D^(m)(_un)$D^{(m)}(u_n)$, m∈N$m\in \mathbb{N}$, defined by Chernick et al. (1991). We apply the results to a class of max-autoregressive sequences and a class of moving average models. The results generalize the ones obtained for the stationary case.     

Improving density estimators of discretely observed processes by interpolation

We consider density estimation for a smooth stationary process _Xt$X_t$, t∈R$t\in \mathbf{R}$, based on a discrete sample _Yi=_XΔi$Y_i=X_{\mathrm{\Delta }i}$, i=0,…,n=T/Δ$i=0,\dots ,n=T/\Delta$. By a suitable interpolation scheme of order p  , we augment data to form an approximation _Xp,t$X_{p,t}$, t∈[0,T]$t\in [0,T]$, of the continuous-time process and base our density estimate on the augmented sample path. Our results show that this can improve the rate of convergence (measured in terms of n) of the density estimate. Among other things, this implies that recording n   observations using a small Δ$\Delta$ can be more efficient than recording n independent observations.     

Comparing several treatments with a control

Consider the partially balanced one-way layout for comparing k   treatments _μi,${\mu }_i,$1?i?k,$1?i?k,$ with a control _μ0${\mu }_0$. We propose a new test which is similar to the test statistics of Marcus [1976. The powers of some tests of the equality of normal means against an ordered alternative. Biometrika 63, 177–183]. By simulation we find that the proposed test has a good power performance when compared with other tests. Moreover, it can produce confidence intervals for _μi-_μ0,1?i?k.${\mu }_i-{\mu }_0,1?i?k.$