Sung Nok Chiu

The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion

Abstract The paper studies the joint distribution of the time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process that is perturbed by diffusion. We prove that the expected discounted penalty satisfies an integro-differential equation of renewal type, the solution of which can be expressed as a convolution formula. The asymptotic behaviour of the expected discounted penalty as the initial capital tends to infinity is discussed.

Tone Perception Ability of Cantonese-Speaking Children

Studies have shown that while children acquire all Cantonese tones by age two, they are not able to label them reliably until approximately age 10. One possible explanation for the large age discrepancy may be the different methodologies used. This study aimed to (1) investigate a new research design for the collection of reliable tone perception data from young children; (2) compare lexical and nonlexical items for testing tone perception ability; and (3) identify the relative ease of perceiving the three basic tone contrasts in Cantonese, that is, high level/high rising (T1/T2), high level/low falling (T1/T4), and, high rising/low falling tones (T2/T4). The three tone pairs were presented to 31 children in the form of word and nonword stimuli. It was found that the research design could be used to assess the tone perception knowledge of children as young as 2;09. Significant differences were found between word and nonword stimuli and also in the identification of the T2/T4 contrast in comparison with the other two pairs. Childrens overall tone perception abilities are discussed in detail with reference to the role of lexical knowledge and the potential for tone perception confusions arising from differences in fundamental frequencies for tone onset and offset.Studies have shown that while children acquire all Cantonese tones by age two, they are not able to label them reliably until approximately age 10. One possible explanation for the large age discrepancy may be the different methodologies used. This study aimed to (1) investigatea new research design for the collection of reliable tone perception data from young children; (2) compare lexical and nonlexical items for testing tone perception ability; and (3) identify the relative ease of perceiving the three basic tone contrasts in Cantonese, that is, high level/high rising (T1 /T2), high level/low falling (T1/T4), and, high rising/low falling tones (T2/T4). The three tone pairs were presented to 31 children in the form of word and nonword stimuli. It was found that the research design could be used to assess the tone perception knowledge of children as young as 2;09. Significant differences were found between word and nonword stimuli and also in the identification of the T2/T4 contrast in comparison with the other two pairs. Childrens overall tone perception abilities are discussed in detail with reference to the role of lexical knowledge and the potential for tone perception confusions arising from differences in fundamental frequencies for tone onset and offset.

A limit theorem for the replication time of a DNA molecule

The DNA of higher animals replicates by an interesting mechanism. Enzymes recognise specific sites randomly scattered on the molecule and establish a bidirectional process of unwinding and replication from these sites. We investigate the limiting distribution of the completion time for this process by considering related coverage problems investigated by Janson (1983) and Hall (1988).

Estimators of distance distributions for spatial patterns

Baddeley and Gill (1994, 1996) have introduced an edge‐corrected Kaplan–Meier type estimator of the empty space function, which is very important in point process statistics. The present paper suggests a further estimator of this function, which is based on a method used by Hanisch (1984) for unbiased edge‐corrected estimation of the nearest neighbour distance distribution function. Moreover, it turns out that the Kaplan–Meier and the new estimator are closely related, since their densities are border method or minus‐sampling type estimators.

Generalized Cramér-von Mises goodness-of-fit tests for multivariate distributions

A class of statistics for testing the goodness-of-fit for any multivariate continuous distribution is proposed. These statistics consider not only the goodness-of-fit of the joint distribution but also the goodness-of-fit of all marginal distributions, and can be regarded as generalizations of the multivariate Cramer-von Mises statistic. Simulation shows that these generalizations, using the Monte Carlo test procedure to approximate their finite-sample p-values, are more powerful than the multivariate Kolmogorov-Smirnov statistic.

Testing the complete spatial randomness by Diggle's test without an arbitrary upper limit

Diggles test for complete spatial randomness of a given point pattern uses the discrepancy between the estimated and the theoretical form of a summary function as the test statistic. One commonly used discrepancy measure is the supremum of the pointwise differences over a suitably chosen range; the upper bound of the range is an arbitrary but sometimes crucial parameter. This paper shows that when we use Ripleys K-function as the summary function, it is possible to avoid using an arbitrary upper bound by using adapted distance dependent intensity estimators.

On occupation times for a risk process with reserve-dependent premium

Consider a risk reserve process under which the reserve can generate interest. For constants a and b such that a<b, we study the occupation time T a,b (t), which is the total length of the time intervals up to time t during which the reserve is between a and b. We first present a general formula for piecewise deterministic Markov processes, which will be used for the computation of the Laplace transform of T a,b (t). Explicit results are then given for the special case that claim sizes are exponentially distributed. The classical model is discussed in detail. *Research supported by RGC of Hong Kong SAR (Grant No. HKBU/2075/98P). **Research supported by NSF of China(Grant No. 19801020).

Homogeneity tests for several Poisson populations

In this paper we compare the size distortions and powers for Pearsons @g^2-statistic, likelihood ratio statistic LR, score statistic SC and two statistics, which we call UT and VT here, proposed by [Potthoff, R.F., Whittinghill, M., 1966. Testing for homogeneity: II. The Poisson distribution. Biometrika 53, 183-190] for testing the equality of the rates of K Poisson processes. Asymptotic tests and parametric bootstrap tests are considered. It is found that the asymptotic UT test is too conservative to be recommended, while the other four asymptotic tests perform similarly and their powers are close to those of their parametric bootstrap counterparts when the observed counts are large enough. When the observed counts are not large, Monte Carlo simulation suggested that the asymptotic test using SC, LR and UT statistics are discouraged; none of the parametric bootstrap tests with the five statistics considered here is uniformly best or worst, and the asymptotic tests using Pearsons @g^2 and VT always have similar powers to their bootstrap counterparts. Thus, the asymptotic Pearsons @g^2 and VT tests have an advantage over all five parametric bootstrap tests in terms of their computational simplicity and convenience, and over the other four asymptotic tests in terms of their powers and size distortions.

A NEW GRAPH RELATED TO THE DIRECTIONS OF NEAREST NEIGHBOURS IN A POINT PROCESS

This paper introduces a new graph constructed from a point process. The idea is to connect a point with its nearest neighbour, then to the second nearest and continue this process until the point belongs to the interior of the convex hull of these nearest neighbours. The number of such neighbours is called the degree of a point. We derive the distribution of the degree of the typical point in a Poisson process, prove a central limit theorem for the sum of degrees, and propose an edge-corrected estimator of the distribution of the degree that is unbiased for a stationary Poisson process. Simulation studies show that this degree is a useful concept that allows the separation of clustering and repulsive behaviour of point processes.

A regularity condition and strong limit theorems for linear birth–growth processes

A linear birth–growth process is generated by an inhomogeneous Poisson process on ℝ × [0, ∞). Seeds are born randomly according to the Poisson process. Once a seed is born, it commences immediately to grow bidirectionally with a constant speed. The positions occupied by growing intervals are regarded as covered. New seeds continue to form on the uncovered part of ℝ. This paper shows that the total number of seeds born on a very long interval satisfies the strong invariance principle and some other strong limit theorems. Also, a gap (an unproved regularity condition) in the proof of the central limit theory in [5] is filled in.