Yongcheng Qi

Limit distributions for products of sums

Let {X,Xn, n[greater-or-equal, slanted]1} be a sequence of independent and identically distributed positive random variables and set Sn=[summation operator]j=1n Xj for n[greater-or-equal, slanted]1. This paper proves that properly normalized products of the partial sums, ([product operator]j=1nSj/n![mu]n)[mu]/An, converges in distribution to some nondegenerate distribution when X is in the domain of attraction of a stable law with index [alpha][set membership, variant](1,2].

Confidence regions for high quantiles of a heavy tailed distribution

Estimating high quantiles plays an important role in the context of risk management. This involves extrapolation of an unknown distribution function. In this paper we propose three methods, namely, the normal approximation method, the likelihood ratio method and the data tilting method, to construct confidence regions for high quantiles of a heavy tailed distribution. A simulation study prefers the data tilting method.

A note on the number of maxima in a discrete sample

Consider {Xj, j [greater-or-equal, slanted] 1}, a sequence of i.i.d., positive, integer-valued random variables. Let Kn denote the number of the integer j [set membership, variant] {1,2,...,n} for which Xj = max1[less-than-or-equals, slant]m[less-than-or-equals, slant]n Xm. In this paper we prove that limn-->[infinity] EKn = 1 if and only if Kn converges in probability to one, if and only if limn-->[infinity] P(X1=n)/P(X1[greater-or-equal, slanted]n)=0 and prove that Kn converges almost surely to one, if and only if . Some of the results were shown by Baryshnikov et al. (1995) and Brands et al. (1994).

Strong convergence in nonparametric regression with truncated dependent data

In this paper we derive rates of uniform strong convergence for the kernel estimator of the regression function in a left-truncation model. It is assumed that the lifetime observations with multivariate covariates form a stationary @a-mixing sequence. The estimation of the covariates density is considered as well. Under the assumption that the lifetime observations are bounded, we show that, by an appropriate choice of the bandwidth, both estimators of the covariates density and regression function attain the optimal strong convergence rate known from independent complete samples.

The limit behavior of maxima modulo one and the number of maxima

Asymptotic uniformity of fractional parts of maxima, and the limit behavior of the number of maxima in a discrete sample have been recently studied in the literature. We consider the relationship between these two subjects. Furthermore, we study the convergence in distribution of fractional parts of maxima more generally.

Asymptotic independence of correlation coefficients with application to testing hypothesis of independence

Abstract: This paper first proves that the sample based Pearson’s productmoment correlation coefficient and the quotient correlation coefficient are asymptotically independent, which is a very important property as it shows that these two correlation coefficients measure completely different dependencies between two random variables, and they can be very useful if they are simultaneously applied to data analysis. Motivated from this fact, the paper introduces a new way of combining these two sample based correlation coefficients into maximal strength measures of variable association. Second, the paper introduces a new marginal distribution transformation method which is based on a rank-preserving scale regeneration procedure, and is distribution free. In testing hypothesis of independence between two continuous random variables, the limiting distributions of the combined measures are shown to follow a max-linear of two independent χ random variables. The new measures as test statistics are compared with several existing tests. Theoretical results and simulation examples show that the new tests are clearly superior. In real data analysis, the paper proposes to incorporate nonlinear data transformation into the rank-preserving scale regeneration procedure, and a conditional expectation test procedure whose test statistic is shown to have a non-standard limit distribution. Data analysis results suggest that this new testing procedure can detect inherent dependencies in the data and could lead to a more meaningful decision making.

On sequential estimation for branching processes with immigration

Abstract Consider a Galton–Watson process with immigration. The limiting distributions of the nonsequential estimators of the offspring mean have been proved to be drastically different for the critical case and subcritical and supercritical cases. A sequential estimator, proposed by Sriram et al. (Ann. Statist. 19 (1991) 2232), was shown to be asymptotically normal for both the subcritical and critical cases. Based on a certain stopping rule, we construct a class of two-stage estimators for the offspring mean. These estimators are shown to be asymptotically normal for all the three cases. This gives, without assuming any prior knowledge, a unified estimation and inference procedure for the offspring mean.

Some results on covariance of function of order statistics

Let X1:n [less-than-or-equals, slant] X2:n [less-than-or-equals, slant] ... [less-than-or-equals, slant] Xn:n denote order statistics of n i.i.d. samples. It is proven that g(Xi:n) and g(Xi + 1:n) are nonnegatively correlated for any function g with finite variance. Some examples are also constructed to show that Mas (1992b) conjecture, that g(Xi:n) and g(Xj;n) are nonnegatively correlated for any function g, is not true.

Empirical Distributions of Eigenvalues of Product Ensembles

Assume a finite set of complex random variables form a determinantal point process; we obtain a theorem on the limit of the empirical distribution of these random variables. The result is applied to two types of n-by-n random matrices as n goes to infinity. The first one is the product of m i.i.d. (complex) Ginibre ensembles, and the second one is the product of truncations of m independent Haar unitary matrices with sizes