Yi-Ching Yao

Academia Sinica · Institute of Statistical Science

In the classical gambler’s ruin problem, the gambler plays an adversary with initial capitals z and $a-z$ , respectively, where $a>0$ and $0< z < a$ are integers. At each round, the gambler wins or loses a dollar with probabilities p and $1-p$ . The game continues until one of the two players is ruined. For even a and $0 , the family of distributio...

We consider a birth–death process with killing where transitions from state i may go to either state $i-1$ or state $i+1$ or an absorbing state (killing). Stochastic ordering results on the killing time are derived. In particular, if the killing rate in state i is monotone in i , then the distribution of the killing time with initial state i is sto...

von Neumann [(1951). Various techniques used in connection with random digits. National Bureau of Standards Applied Math Series 12: 36–38] introduced a simple algorithm for generating independent unbiased random bits by tossing a (possibly) biased coin with unknown bias. While his algorithm fails to attain the entropy bound, Peres [(1992). Iteratin...

A man has a house with n doors. Initially he places k pairs of walking shoes at each door. For each walk, he chooses one door at random, and puts on a pair of shoes, returns after the walk to a randomly chosen door and takes off the shoes at the door. Let Tn be the first time a door is chosen to walk out but with no shoes available. We show that as...

In the literature on scan statistics, the distributions of continuous scan statistics for one-dimensional Poisson processes have been extensively studied, most of which deal with single window scan statistics under homogeneous Poisson processes. In this paper, we consider discrete approximations for the distributions of multiple window scan statist...

In the literature on optimal stopping, the problem of maximizing the expected discounted reward over all stopping times has been explicitly solved for some special reward functions (including $(x^+)^{\nu}$, $(e^x-K)^+$, $(K-e^{-x})^+$, $x\in\mathbb{R}$, $\nu\in(0,\infty)$ and $K>0$) under general random walks in discrete time and L\'evy processes i...

In the literature on optimal stopping, the problem of maximizing the expected discounted reward over all stopping times has been explicitly solved for some special reward functions (including $(x^+)^{\nu}$, $(e^x-K)^+$, $(K-e^{-x})^+$, $x\in\mathbb{R}$, $\nu\in(0,\infty)$ and $K>0$) under general random walks in discrete time and L\'evy processes i...

While attempting to better understand the 3-dimensional structure of the mammalian nucleus as well as a rigid-body kinematics application, the authors encountered a naturally arising generalized version of the Wahba (1965) problem concerned with bringing multiple sets of labeled points into close coincidence after making appropriate rotations of th...

In the subject of optimal stopping, the classical secretary problem is concerned with optimally selecting the best of $n$ candidates when their relative ranks are observed sequentially. This problem has been extended to optimally selecting the $k$-th best candidate for $k\ge 2$. While the optimal stopping rule for $k=1,2$ (and all $n\ge 2$) is know...

In the subject of optimal stopping, the classical secretary problem is concerned with optimally selecting the best of $n$ candidates when their relative ranks are observed sequentially. This problem has been extended to optimally selecting the $k$-th best candidate for $k\ge 2$. While the optimal stopping rule for $k=1,2$ (and all $n\ge 2$) is know...

The Auto-PARM (Automatic Piecewise AutoRegressive Modeling) procedure, developed by Davis, Lee, and Rodriguez-Yam (2006), uses the minimum description length (MDL) principle to estimate the number and locations of structural breaks in a non-stationary time series. Consistency of this model selection procedure has been established when using conditi...

We consider a game with K ≥ 2 players, each having an integer-valued fortune. On each round, a pair ( i , j ) among the players with nonzero fortunes is chosen to play and the winner is decided by flipping a fair coin (independently of the process up to that time). The winner then receives a unit from the loser. All other players' fortunes remain t...

The (conditional or unconditional) distribution of the continuous scan statistic in a one-dimensional Poisson process may be approximated by that of a discrete analogue via time discretization (to be referred to as the discrete approximation). With the help of a change-of-measure argument, we derive the first-order term of the discrete approximatio...

The (conditional or unconditional) distribution of the continuous scan statistic in a one-dimensional Poisson process may be approximated by that of a discrete analogue via time discretization (to be referred to as the discrete approximation). With the help of a change-of-measure argument, we derive the first-order term of the discrete approximatio...

In the literature, the problem of maximizing the expected discounted reward over all stopping rules has been explicitly solved for a number of reward functions (including (max {x, 0} g)(nu), nu > 0, in particular) when the underlying process is either a random walk in discrete time or a Levy process in continuous time. All of such reward functions...

Consider two independent homogeneous Poisson point processes ΠΠ of intensity λλ and Π′Π′ of intensity λ′λ′ in dd-dimensional Euclidean space. Let qk,dqk,d, k=0,1,…k=0,1,…, be the fraction of ΠΠ-points which are the nearest ΠΠ-neighbor of precisely kk Π′Π′-points. It is known that as d→∞d→∞, the qk,dqk,d converge to the Poisson probabilities e−λ′/λ(...

We consider a discrete-time queueing system where the arrival process is general and each arriving customer brings in a constant amount of work which is processed at a deterministic rate. We carry out a sample-path analysis to derive an exact relation between the set of system size values and the set of waiting time values over a busy period of a g...

One version of Little’s law, written as $L = \lambda w$ , is a relation between averages along a sample path. There are two others in a stochastic setting; they readily extend to the case where the average waiting time $w$ is infinite. We investigate conditions for the sample-path version of this case to hold. Published proofs assume (our) Eq....

To deal with the compatibility issue of full conditional distributions of a (discrete) random vector, a graphical representation is introduced where a vertex corresponds to a configuration of the random vector and an edge connects two vertices if and only if the ratio of the probabilities of the two corresponding configurations is specified through...

Sengupta (1989) showed that, for the first-come–first-served (FCFS) G/G/1 queue, the workload and attained waiting time of a customer in service have the same stationary distribution. Sakasegawa and Wolff (1990) derived a sample path version of this result, showing that the empirical distribution of the workload values over a busy period of a given...

For a typical cell of a homogeneous Poisson-Voronoi tessellation in Rd, it is shown that the variance of the volume of the intersection of the typical cell with any measurable subset of Rd is bounded by the variance of the volume of the typical cell. It is also shown that the variance of the volume of the intersection of the typical cell with a tra...

For a typical cell of a homogeneous Poisson-Voronoi tessellation in ℝ d , it is shown that the variance of the volume of the intersection of the typical cell with any measurable subset of ℝ d is bounded by the variance of the volume of the typical cell. It is also shown that the variance of the volume of the intersection of the typical cell with a...

In the subfair red-and-black gambling problem, a gambler can stake any amount in his possession, winning an amount equal to the stake with probability w and losing the stake with probability 1 - w, where 0 < w < ½. The gambler seeks to maximize the probability of reaching a fixed fortune (to be normalized to unity) by gambling repeatedly with suita...

In the subfair red-and-black gambling problem, a gambler can stake any amount in his possession, winning an amount equal to the stake with probability w and losing the stake with probability 1 − w , where 0 < w < ½. The gambler seeks to maximize the probability of reaching a fixed fortune (to be normalized to unity) by gambling repeatedly with suit...

In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 - r)/r times the stake with probability w and losing the stake with probability 1 - w, 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune (the goal) by gambling repeatedly with...

In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 − r )/ r times the stake with probability w and losing the stake with probability 1 − w , 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune (the goal) by gambling repeatedly wi...

The Vardi casino with parameter 0 < c < 1 consists of infinitely many tables indexed by their odds, each of which returns the same (negative) expected winnings - c per dollar. A gambler seeks to maximize the probability of reaching a fixed fortune by gambling repeatedly with suitably chosen stakes and tables (odds). The optimal strategy is derived...

Corrected random walk approximations to continuous-time optimal stopping boundaries for Brownian motion, first introduced by Chernoff and Petkau, have provided powerful computational tools in option pricing and sequential analysis. This paper develops the theory of these second-order approximations and describes some new applications.

Corrected random walk approximations to continuous-time optimal stopping boundaries for Brownian motion, first introduced by Chernoff and Petkau, have provided powerful computational tools in option pricing and sequential analysis. This paper develops the theory of these second-order approximations and describes some new applications.

The Vardi casino with parameter 0 < c < 1 consists of infinitely many tables indexed by their odds, each of which returns the same (negative) expected winnings -c per dollar. A gambler seeks to maximize the probability of reaching a fixed fortune by gambling repeatedly with suitably chosen stakes and tables (odds). The optimal strategy is derived e...

We consider the Gittins index for a normal distribution with unknown mean $\theta$ and known variance where $\theta$ has a normal prior. In addition to presenting some monotonicity properties of the Gittins index, we derive an approximation to the Gittins index by embedding the (discrete-time) normal setting into the continuous-time Wiener process...

In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 - r )/ r times the stake with probability w and losing the stake with probability 1 - w , 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune by gambling repeatedly with suitably...

In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 - r)/r times the stake with probability w and losing the stake with probability 1 - w, 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune by gambling repeatedly with suitably ch...

Optimal stopping has been one of Y. S. Chow’s major research areas in probability. This paper reviews the seminal work of Chow and Robbins on the optimal stopping problem for S n /n and subsequent developments and intercrosses with other areas of probability theory. It also uses certain ideas and techniques from these developments to devise relativ...

We discuss the quite remarkable global Markovian structure of the nucleotides in eukaryotic DNA strands with special emphasis on (i) the similarity property for intra- species chromosomes and (ii) the reversibility property for the two (complementary) strands of a chromosome.

Mr. G owes $100 000 to a loan shark, and will be killed at dawn if the loan is not repaid in full. Mr. G has $20 000, but partial payments are not accepted, and he has no other source of income or credit. The loan shark owns a primitive casino where one can stake any amount in one's possession, gaining r times the stake with probability w and losin...

Mr. G owes $100000 to a loan shark, and will be killed at dawn if the loan is not repaid in full. Mr. G has $20000, but partial payments are not accepted, and he has no other source of income or credit. The loan shark owns a primitive casino where one can stake any amount in one's possession, gaining r times the stake with probability w and losing...

A general set of distribution-free conditions is described under which an i.i.d. sequence of random variables is preserved under optional skipping. This work is motivated by theorems of J. L. Doob and Z. Ignatov, unifying and extending aspects of both.

Oblivious permutation routing in binary d-cubes has been well studied in the literature. In a permutation routing, each node initially contains a packet with a destination such that all the 2d destinations are distinct. Kaklamanis et al. (Math. Syst. Theory 24 (1991) 223–232) used the decomposability of hypercubes into Hamiltonian circuits to give...

Segregation ratio estimation has long been important in human genetics. A simple truncated binomial model is considered that assumes complete ascertainment and a deterministic genotype-phenotype relationship. A simple but intuitively appealing estimator of the segregation ratio, previously proposed, is shown to have a negative bias. It is also show...

We are concerned here with establishing the consistency and asymptotic normality for the maximum likelihood estimator of a “merit vector” $(u_0,\dots,u_t)$, representing the merits of $t +1$ teams (players, treatments, objects), under the Bradley–Terry model, as $t \to \infty$. This situation contrasts with the well-known Neyman–Scott problem under...

For (μ,σ2) ≠ (0,1), and 0 < z < ∞, we prove that where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depends only on μ and σ. In other words, for any given (μ,σ) ≠ (0,1), the infimum, over 0 < z...

For (μ,σ²) ≠ (0,1), and 0 < z < ∞, we prove that ( Φ((z-μ)/σ) - Φ((-z-μ)/σ) ) / ( Φ(z) - Φ(-z) ) > min {1, ((√(2π))/σ)φ(μ/σ) }, where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depen...

We extend the study on partition properties from the set partition to the graph partition, especially for the class of connected block graphs which includes trees. We introduce seventeen partition properties and determine their inter-relations. The notions of k-consistency and k-sortability were studied in the set partition to localize the properti...

It is shown for an n × n symmetric positive definite matrix T = (ti, j with negative off-diagonal elements, positive row sums and satisfying certain bounding conditions that its inverse is well approximated, uniformly to order l/n2, by a matrix S = (si, j), where si,j = δi,j/ti,j + 1/tδi,j being the Kronecker delta function, and t.. being the sum o...

Kaklamanis, Krizanc, and Tsantilas (1991) gave an asymptotically optimal oblivious algorithm for many—one routing in hypercubes. It is shown that their argument needs to be modified in order for the algorithm to attain the asymptotic lower bound. They also applied the algorithm to permutation routing via the many—one and one—many routing phases. Th...

For an infinite sequence of independent and identically distributed (i.i.d.) random variables, the k-record process consists of those terms that are the kth largest at their appearance. Ignatov's theorem states that the k-record processes, $k = 1, 2, \dots ,$ are i.i.d. A new proof is given which is based on a "continualization" argument. An advant...

Optimal packet routing algorithms for all binary d-cubes of dimension d ⩽ 7 are presented. The algorithms given are synchronous, offer distributed control, and assume d-port, multiaccepting communication. While the previous best known packet routing algorithm [3]on the 7-cube takes 11 time-units, our algorithm has reduced the worst-case time comple...

The purpose of this paper is to develop a framework for the analysis of combinatorial properties of partitions. Our focus is on the relation between global properties of partitions and their localization to subpartitions. First, we study properties that are characterized by their local behavior. Second, we determine sufficient conditions for classe...

In his recent monograph on Poisson processes, Kingman generalized Rényi's characterization of Poisson processes and also suggested a characteristic functional approach. A direct proof is given along this line.

For an arbitrary point of a homogeneous Poisson point process in a d-dimensional Euclidean space, consider the number of Poisson points that have that given point as their rth nearest neighbor $(r = 1, 2, \dots)$. It is shown that as d tends to infinity, these nearest neighbor counts $(r = 1, 2, \dots)$ are iid asymptotically Poisson with mean 1. T...

In the last century, Désiré André obtained many remarkable properties of the numbers of alternating permutations, linking them to trigonometric functions among other things. By considering the probability that a random permutation is alternating and that a random sequence (from a uniform distribution) is alternating, and by conditioning on the firs...

The so-called finite fuel follower problem, solved by V. Benĕs etai., is to minimize the (discounted) integral of the square of X(t), i.e. , where is Brownian motion starting at x, and the non-anticipating control u must satisfy the total fuel constraint, pointwise: Benĕ etal. indicated that the so-called “principle of smooth fit” was really needed...

The problem of testing whether or not a change has occurred in the parameter values and order of an autoregressive model is considered. It is shown that if the white noise in the AR model is weakly stationary with finite fourth moments, then under the null hypothesis of no changepoint, the normalized Gaussian likelihood ratio test statistic converg...

Knockout tournaments are often used in sports (or experiments where preferences are registered by comparisons instead of measurements) to determine the champion of an event. A knockout tournament plan (KTP) for n players is a rooted binary tree with n leaves to be labeled by the n players. Each subtree of two leaves represents a match between the t...

Various Markov chains on hypercubes ℒare considered and their spectral representations are presentend in terms of Kronecker products. Special attention is given to random walks on the graphs (l = 1,n − 2), where the vertex set is ℒ and two vertices are connected if and only if their Hamming distance is at most l. It is shown that λ(1)>λ(1)>λ(n−1)>λ...

Suppose you have u units of ammunition and want to destroy as many as possible of a sequence of attacking enemy aircraft. If you fire v = v(u), 0 , units of your ammunition at the first enemy, it survives with probability qv , where 0 < q < 1 is given, and then kills you. With the complementary probability, 1 – qv , you destroy the aircraft and you...

Kaplansky gave a formula which determines the number of ways of selecting j objects, no two consecutive, from n objects arrayed in a cycle. Possibly due to a lack of intuitive and direct argument for this result, generalization to ‘no k consecutive’ has not been obtained as it has for the line case. This paper gives such an argument and consequentl...

Israel has shown the surprising and counter-intuitive result that there exist knockout tournament plans (rooted binary trees with as many terminal nodes as players) for which, under a random assignment of players to terminal nodes, a weaker player has a greater likelihood of winning the tournament than a stronger player. Chung and Hwang have shown,...

Consider a multiaccess channel shared by an infinite set of users, each of which, independently, has a message to transmit with probability p . Pairwise enabling is the scheduling algorithm which enables the users to transmit by pairs, and, if a collision occurs, it lets the two users transmit separately in the next two time slots. M.L. Molle (see...

It is shown that Bern's probabilistic asymptotic results on rectilinear Steiner trees remain valid for the model that there are exactlyN nodes uniformly distributed in a square of side √N.

The problem of optimally allocating partially effective, defensive weapons against randomly arriving enemy aircraft so that a bomber maximizes its probability of reaching its designated target is considered in the usual continuous-time context, and in a discrete-time context. The problem becomes that of determining the optimal number of missiles K(...

The problem of optimally allocating partially effective, defensive weapons against randomly arriving enemy aircraft so that a bomber maximizes its probability of reaching its designated target is considered in the usual continuous-time context, and in a discrete-time context. The problem becomes that of determining the optimal number of missiles K...

A class of linear rank statistics is considered for testing a sequence of independent random variables with common distribution against alternatives involving a change in distribution at an unknown time point. It is shown that, under the null hypothesis and suitably normalized, this class of statistics converges in distribution to the double expone...

Group testing was first proposed for blood testing although it has many industrial applications as well. Most of the group testing literature has studied a naturally defined class of algorithms called nested algorithms. Optimal nested algorithms are usually defined by recursive equations which do not seem to have general closed-form solutions, and...

It is known that the lifetime of a k−out−of−n system has increasing failure rate (IFR) if all of its components have independent and identically distributed IFR lifetimes. Derman, Lieberman, and Ross raised the same question for consecutive−k−out−of−n systems. But the scarcity of results gives no clue as to whether most of such systems have IFR's....

The classical binomial group testing problem studies plans to sort defectives from good items using a minimum number of group tests where a group test is a test applicable to any subset of items with a yes and no answer to the question of whether the subset contains any defectives. In a multinomial group testing problem, each item can be in one of...

G. Shanthikumar (ibid., vol.R-36, p.546-550, Dec. 1987) has proposed a system called the consecutively connected system, which is a generalization of the consecutive- k -out-of- n :F system. He gave an O( n ²) algorithm for computing the reliability of the consecutively connected system. In the present work, the authors further generaliz...

It is well known that optimally stopping the sample mean of a standard Wiener process is associated with a square root boundary. It is shown that when W(t) is replaced by X(t) = W(t) + [theta]t with [theta] normally distributed N([mu], [sigma]2) and independently of the Wiener process, the optimal stopping problem is equivalent to the time-truncate...

The binomial group-testing problem consists of finding by group tests all defectives in a given set of items, each of which independently has probability p of being defective. The conjecture that the expected number of tests under an optimal testing algorithm is nondecreasing in p has recently been proved by transplanting the probability structure...

Methods are described which permit one to work with continuous-time optimal stopping problems, using the heat equation, even when the prior placed on the drift parameter of a Wiener process is not normal. The details of the method are worked out for Chernoff's problem of testing the sign of the drift parameter when the prior is "smooth."

A consecutive−k−out−of−n:F system is an n−vertex graph where the system fails if and only if some k consecutive vertices all fail. Assuming that the n vertices have, independently, the respective failure probabilities q1,…, qn, the problem is to find, subject to πqi = Q (a constant), a set of q1,…, qn so as to maximize the system reliability. In th...

Consider the problem of estimating step functions in the presence of additive measurement noise. In the case that the number of jumps is known, the least-squares estimators for the locations of the jumps and the levels of the step function are studied and their limiting distributions are derived. When the number of jumps is unknown, an estimator is...

The group-testing problem for a binomial set of items is considered. It is desired to classify all items as good or defective with a minimum expected number of group tests. An improvement over the information lower bound, via a weak concavity property, is made for the minimum expected number of group tests.

The group testing problem is to find by group tests all defectives in a given set of items each of which independently has probability p of being defective. The group testing problem originated with the famous Dorfman’s blood testing problem and has since found many industrial applications including a recent one on multiple-access communication. Un...

An estimator of the number of change-points in an independent normal sequence is proposed via Schwarz' criterion. Weak consistency of this estimator is established.

A class of multiple sample tests based on empirical coverages is proposed which is a generalization of Greenwood's and Sherman's one-sample goodness-of-fit test statistics. The asymptotic normality of the tests is established by embedding the empirical coverages into a stationary process satisfying the strong mixing condition.

We consider the group testing problem for a set of independent items I = [I1,… In] where Ii, has probability pi, of being defective and probability qi = 1 – pi of being good. The problem is to classify all items as good or defective with a minimum expected number of group tests where a group test is a test on a subset S of I with two possible outco...

Suppose that Z1 …Z are independent and normally distributed with common mean u and variance σ.When σ = λμwith given λ> 0, it is well known that the minimal sufficient statistic is not complete for μ if n ≥ 2. The question of completeness seems to be unresolved when n = 1. In this note, we consider this case and, by using the heat equation, establis...

This chapter reviews the Markov decision approach to nuclear materials safeguard. One basic problem in safeguarding special nuclear materials is that of the early detection of diversion. In the case of nuclear safeguards, the use of the difference of successive inventories in estimating the materials balance introduces dependence among successive e...

The problem of estimating the change-point in a sequence of independent random variables is considered. As the sample sizes before and after the change-point tend to infinity, Hinkley (1970) showed that the maximum likelihood estimate of the change-point converges in distribution to that of the change-point based on an infinite sample. Letting the...

The problem of testing for constant hazard against a change-point alternative is considered. It is shown that this problem is related to another one in quality control. Based on this relationship, a test is proposed. The main advantages of this test are its computational simplicity and the ready availability of small and large sample distribution t...

The problem of estimation of parameters in hazard rate models with a change-point is considered. An interesting feature of this problem is that the likelihood function is unbounded. A maximum likelihood estimator of the change-point subject to a natural constraint is proposed, which is shown to be consistent.The limiting distributions are also deri...

Let X 1 ,···,X n be an independent sequence of random variables such that X 1 ,···,X r ∼i·i·d. N(μ,σ 2 ) and X r+1 ,···,X n ∼i·i·d. N(μ+θ,σ 2 ) where μ,θ≠0, σ 2 and r are unknown parameters. The asymptotic properties of the likelihood ratio in testing H 0 : r=n (no change point) vs. H 1 : r<n are derived. It is shown, using a result of D. A. Darlin...

In the estimation problem of a two-state stationary Markov process with Gaussian white noise added, the optimal smoother is a two-filter smoother. In a special case, the performance of the optimal nonlinear filter and smoother is evaluated analytically. Some asymptotic results are also derived.

Consider the problem of estimating, in a Bayesian framework and in the presence of additive Gaussian noise, a signal which is a step function. Best linear estimates and Bayes estimates are derived, evaluated and compared. A characterization of the Bayes estimates is presented. This characterization has an intuitive interpretation and also provides...

Let $x(t)$ be a Wiener process with drift $\mu$ and variance 1 per unit time. The following problem is treated; test $H:\mu \leq 0$ vs. $A:\mu > 0$ with the loss function $|\mu|$ if the wrong decision is made and 0 otherwise, and with $c =$ cost of observation per unit time, where $\mu$ has a prior distribution which is normal with mean 0 and varia...

The Nyquist frequency is half the sampling frequency when a continuous-time function is sampled at equally spaced time points. That is, the Nyquist frequency is pi/delta (in radians per unit time) where delta is the time interval between two successive sampled data. This report discusses the basic ideas of the Nyquist frequency and explains its rel...