Limit Laws for Functions of Fringe trees for Binary Search Trees and Recursive Trees

We prove limit theorems for sums of functions of subtrees of binary search trees and random recursive trees. In particular, we give simple new proofs of the fact that the number of fringe trees of size $k=k_n$ in the binary search tree and the random recursive tree (of total size $n$) asymptotically has a Poisson distribution if $k\rightarrow\infty$, and that the distribution is asymptotically normal for $k=o(\sqrt{n})$. Furthermore, we prove similar results for the number of subtrees of size $k$ with some required property $P$, for example the number of copies of a certain fixed subtree $T$. Using the Cram\'er-Wold device, we show also that these random numbers for different fixed subtrees converge jointly to a multivariate normal distribution. As an application of the general results, we obtain a normal limit law for the number of $\ell$-protected nodes in a binary search tree or random recursive tree. The proofs use a new version of a representation by Devroye, and Stein's method (for both normal and Poisson approximation) together with certain couplings

Real Zeros and Partitions without singleton blocks

We prove that the generating polynomials of partitions of an $n$-element set into non-singleton blocks, counted by the number of blocks, have real roots only and we study the asymptotic behavior of the leftmost roots. We apply this information to find the most likely number of blocks.Comment: 16 page

Modelling International Tourist Arrivals and Volatility: An Application to Taiwan

International tourism is a major source of export receipts for many countries worldwide. Although it is not yet one of the most important industries in Taiwan (or the Republic of China), an island in East Asia off the coast of mainland China (or the People’s Republic of China), the leading tourism source countries for Taiwan are Japan, followed by USA, Republic of Korea, Malaysia, Singapore, UK, Germany and Australia. These countries reflect short, medium and long haul tourist destinations. Although the People’s Republic of China and Hong Kong are large sources of tourism to Taiwan, the political situation is such that tourists from these two sources to Taiwan are reported as domestic tourists. Daily data from 1 January 1990 to 30 June 2007 are obtained from the National Immigration Agency of Taiwan. The Heterogeneous Autoregressive (HAR) model is used to capture long memory properties in the data. In comparison with the HAR(1) model, the estimated asymmetry coefficients for GJR(1,1) are not statistically significant for the HAR(1,7) and HAR(1,7,28) models, so that their respective GARCH(1,1) counterparts are to be preferred. These empirical results show that the conditional volatility estimates are sensitive to the long memory nature of the conditional mean specifications. Although asymmetry is observed for the HAR(1) model, there is no evidence of leverage. The QMLE for the GARCH(1,1), GJR(1,1) and EGARCH(1,1) models for international tourist arrivals to Taiwan are statistically adequate and have sensible interpretations. However, asymmetry (though not leverage) was found only for the HAR(1)model, and not for the HAR(1,7) and HAR(1,7,28) models.

Daily Tourist Arrivals, Exchange Rates and Volatility for Korea and Taiwan

Both domestic and international tourism are a major source of service export receipts for many countries worldwide, and is also increasingly important in Taiwan. One of the three leading tourism source countries for Taiwan is the Republic of Korea, which is a source of short haul tourism. Daily data from 1 January 1990 to 31 December 2008 are used to model the Korean Won / New Taiwan $exchange rate and tourist arrivals from Korea to Taiwan, as well as their associated volatility. The sample period includes the Asian economic and financial crises in 1997, and a significant part of the global financial crisis of 2008-09. Inclusion of the exchange rate allows approximate daily price effects on Korean tourism arrivals to Taiwan to be captured. The Heterogeneous Autoregressive (HAR) model is used to capture long memory properties in exchange rates and Korean tourist arrivals, to test whether alternative estimates of conditional volatility are sensitive to the long memory in the conditional mean, and to examine asymmetry and leverage in volatility. The empirical results show that the conditional volatility estimates are not sensitive to the long memory nature of the conditional mean specifications. The QMLE for the GARCH(1,1), GJR(1,1) and EGARCH(1,1) models for Korean tourist arrivals to Taiwan and the Korean Won / New Taiwan$ exchange rate are statistically adequate and have sensible interpretations. Asymmetry (though not leverage) is found for several alternative HAR models.exchange rates;GARCH;leverage;asymmetry;long memory;EGARCH;HAR;Korean tourist arrivals;GJR;global financial crisis;approximate price effect

"Cruising is Risky Business"

As the fastest growing sector within the international tourism industry, having grown at roughly double the rate of international tourism as a whole, the cruise liner business has shown impressive growth in the North American and European markets. For port management purposes, as well as for transport policy, it is essential to be able to forecast accurately cruise passenger arrivals and their variability. In the presence of time-varying variances (or volatility), it is crucial to model such volatility in order to provide sensible forecast intervals in addition to the forecast themselves. Time-varying volatility in port management is important because governments and businesses need to be aware of the uncertainty associated with the number of cruise passenger arrivals and their associated growth. In calculating income elasticities, port taxes and tourist taxes, it is essential to obtain accurate estimates of cruise passenger arrivals and their volatility. Moreover, in an international context in which natural disasters, terrorism, crime and ethnic conflicts, among others, have significant impacts on tourism, it is crucial to assess the persistence of shocks on cruise passenger arrivals for effective crisis management plans, including different forms of co-operation among ports facing similar shocks. Appropriate models are required to enable optimal private and public decision making in designing ports for cruise ships. Daily cruise passenger arrivals data for the three major ports in the Balearic Islands, Spain, namely Palma, Ibiza and Mahon, for the period 1997-2006, as well as for the high cruise season for each island,are analyzed using alternative conditional mean and conditional volatility models in order to provide empirical support for purposes of optimal decision making. Four different types of asymmetries are analyzed according to the positive and negative shocks to daily cruise passenger arrivals, as well as from distinctions between the high and low cruise seasons. The estimates of cruise passenger arrivals and their volatility are generally found to be sensible and to have valid statistical properties. Likelihood ratio tests of the constancy of coefficients in the high and low cruise seasons indicate that the weekly delayed response of cruise passenger arrivals differ significantly spatially across islands and temporally across seasons.

Model Assessment Tools for a Model False World

A standard goal of model evaluation and selection is to find a model that approximates the truth well while at the same time is as parsimonious as possible. In this paper we emphasize the point of view that the models under consideration are almost always false, if viewed realistically, and so we should analyze model adequacy from that point of view. We investigate this issue in large samples by looking at a model credibility index, which is designed to serve as a one-number summary measure of model adequacy. We define the index to be the maximum sample size at which samples from the model and those from the true data generating mechanism are nearly indistinguishable. We use standard notions from hypothesis testing to make this definition precise. We use data subsampling to estimate the index. We show that the definition leads us to some new ways of viewing models as flawed but useful. The concept is an extension of the work of Davies [Statist. Neerlandica 49 (1995) 185--245].Comment: Published in at http://dx.doi.org/10.1214/09-STS302 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Daily Tourist Arrivals, Exchange Rates and Volatility for Korea and Taiwan

Both domestic and international tourism are a major source of service export receipts for many countries worldwide, and is also increasingly important in Taiwan. One of the three leading tourism source countries for Taiwan is the Republic of Korea, which is a source of short haul tourism. Daily data from 1 January 1990 to 31 December 2008 are used to model the Korean Won / New Taiwan $exchange rate and tourist arrivals from Korea to Taiwan, as well as their associated volatility. The sample period includes the Asian economic and financial crises in 1997, and a significant part of the global financial crisis of 2008-09. Inclusion of the exchange rate allows approximate daily price effects on Korean tourism arrivals to Taiwan to be captured. The Heterogeneous Autoregressive (HAR) model is used to capture long memory properties in exchange rates and Korean tourist arrivals, to test whether alternative estimates of conditional volatility are sensitive to the long memory in the conditional mean, and to examine asymmetry and leverage in volatility. The empirical results show that the conditional volatility estimates are not sensitive to the long memory nature of the conditional mean specifications. The QMLE for the GARCH(1,1), GJR(1,1) and EGARCH(1,1) models for Korean tourist arrivals to Taiwan and the Korean Won / New Taiwan$ exchange rate are statistically adequate and have sensible interpretations. Asymmetry (though not leverage) is found for several alternative HAR models.