Longitudinal data analysis of mean passage time among malnutrition states: an application of Markov chains

Clinical prognosis of patients can be best described from a longitudinal study and a Markov regression model is an appropriate way of analyzing the prognosis of disease when the outcomes are serially dependent. Mean first passage time (MFPT) is a method to estimate the average number of transitions between the states of a Markov chain. The present study used the secondary data from a longitudinal study which was done during 1982–1986. This study was to illustrate the MFPT among the states of malnutrition, which were classified as Normal, Mild/Moderate and Severe among children aged 5–7 years, in South India. The 95% confidence interval (CI) for the MFPT was calculated using Monte Carlo simulation. Markov regression models were used to test for the association of state transitions across the risk factors. The average time taken for an underweight child to transit from Severe state of malnutrition to become Normal was nearly 2.73 (95% CI 2.60–2.86) years and 3.41 (95% CI 3.25–3.58) years in Rural area and 2.31(95% CI 2.20–2.42) in Urban area. The significant difference between the MFPT for some risk factors are useful to plan interventions. It will especially be useful to find the impact of duration among school-going children on their cognitive disorders.     

Locking the Wiener Process to its Level-Crossing Time

We consider the specific transformation of a Wiener process {X(t), t ≥ 0} in the presence of an absorbing barrier a that results when this process is “time-locked” with respect to its first passage time T _a through a criterion level a, and the evolution of X(t) is considered backwards (retrospectively) from T _a . Formally, we study the random variables defined by Y(t) ≡ X(T _a  ? t) and derive explicit results for their density and mean, and also for their asymptotic forms. We discuss how our results can aid interpretations of time series “response-locked” to their times of crossing a criterion level.     

On the connection between orthant probabilities and the first passage time problem

This article describes a new Monte Carlo method for the evaluation of the orthant probabilities by sampling first passage times of a non-singular Gaussian discrete time-series across an absorbing boundary. This procedure makes use of a simulation of several time-series sample paths, aiming to record their first crossing instants. Thus, the computation of the orthant probabilities is traced back to the accurate simulation of a non-singular Gaussian discrete-time series. Moreover, if the simulation is also efficient, this method is shown to be speedier than the others proposed in the literature. As example, we make use of the Davies–Harte algorithm in the evaluation of the orthant probabilities associated to the ARFIMA(0, d, 0) model. Test results are presented that compare this method with currently available software.     

A note on the admissibility of hammersley's estimator of an integer mean

Let X₁, X₂, …, X_n be identically, independently distributed N(i,1) random variables, where i = 0, ±1, ±2, … Hammersley (1950) showed that d = [X?_n], the nearest integer to the sample mean, is the maximum likelihood estimator of i. Khan (1973) showed that d is minimax and admissible with respect to zero-one loss. This note now proves a conjecture of Stein to the effect that in the class of integer-valued estimators d is minimax and admissible under squared-error loss.     

Variance of the asymmetric n-player gambler’s ruin problem with ties allowed

In this article, the variance of the duration of play in the asymmetric n-player gambler’s ruin problem is considered, when the players use equal initial fortunes of d dollars, 1 ? d ? n + 1, and ties allowed in each round. Some special games are simulated and the simulation results verify the validity of the proposed formulas. It is shown that when we do not have the possibility of tie in the game, the increase in the number of players will change the ruin time from a random variable to a degenerate random variable. Finally, the three-tower problem with one of its different definitions are introduced and their expected times as well as their variances of the duration are considered.     

Improved estimators for the selected location parameters

_i , i = 1, 2, ..., k be k independent exponential populations with different unknown location parameters θ _i , i = 1, 2, ..., k and common known scale parameter σ. Let Y _i denote the smallest observation based on a random sample of size n from the i-th population. Suppose a subset of the given k population is selected using the subset selection procedure according to which the population π _i is selected iff Y _i ≥Y ₍₁₎−d, where Y ₍₁₎ is the largest of the Y _i 's and d is some suitable constant. The estimation of the location parameters associated with the selected populations is considered for the squared error loss. It is observed that the natural estimator dominates the unbiased estimator. It is also shown that the natural estimator itself is inadmissible and a class of improved estimators that dominate the natural estimator is obtained. The improved estimators are consistent and their risks are shown to be O(kn ⁻²). As a special case, we obtain the coresponding results for the estimation of θ₍₁₎, the parameter associated with Y ₍₁₎. Received: January 6, 1998; revised version: July 11, 2000     

Maximizing survival time in a random walk on an interval

A gambler buys N tokens that enable him to play N rounds of the following game. A symmetric random walk on a discrete interval { ? r, …, r} starts from the point 0. The gambler knows only the number of steps made so far, but is unaware of the current position of the walk. Once the walk hits one of the barriers ? r or r for the first time in the current round, the round ends with no payoff. The gambler can start a new round by inserting a new token, if there are any tokens left. The gambler can end the game at any time getting the payoff equal to the number of steps made in the current round. We find the optimal stopping strategy for this game and calculate the expected payoff once the optimal strategy is applied.     

Uniform asymptotic linearity in a regression parameter of a process based on a rank statistic

For X₁, …, X_N a random sample from a distribution F, let the process S^δ_N(t) be defined as where K²_N = σ^N_i=1(c_i ? c?)² and R x_i, + Δd, is the rank of X_i + Δd_i, among X₁ + Δd₁, …, X_N + Δd_N. The purpose of this note is to prove that, under certain regularity conditions on F and on the constants c_i and d_i, S^Δ_N (t) is asymptotically approximately a linear function of Δ, uniformly in t and in Δ, |Δ| ≤ C. The special case of two samples is considered.     

Infant mortality rates: time trends and fractional integration

This paper examines the existence of time trends in the infant mortality rates in a number of countries in the twentieth century. We test for the presence of deterministic trends by adopting a linear model for the log-transformed data. Instead of assuming that the error term is a stationary I(0), or alternatively, a non-stationary I(1) process, we allow for the possibility of fractional integration and hence for a much greater degree of flexibility in the dynamic specification of the series. Indeed, once the linear trend is removed, all series appear to be I(d) with 0<d<1, implying long-range dependence. As expected, the time trend coefficients are significantly negative, although of a different magnitude from those obtained assuming integer orders of differentiation.     

A Subset Selection Procedure for Selecting the Exponential Population Having the Longest Mean Lifetime When the Guarantee Times are the Same

ABSTRACT

Consider k(≥ 2) independent exponential populations Π₁, Π₂, …, Π _k , having the common unknown location parameter μ ∈ (?∞, ∞) (also called the guarantee time) and unknown scale parameters σ₁, σ₂, …σ _k , respectively (also called the remaining mean lifetimes after the completion of guarantee times), σ _i  > 0, i = 1, 2, …, k. Assume that the correct ordering between σ₁, σ₂, …, σ _k is not known apriori and let σ_[i], i = 1, 2, …, k, denote the ith smallest of σ _j s, so that σ_[1] ≤ σ_[2] ··· ≤ σ_[k]. Then Θ _i  = μ + σ _i is the mean lifetime of Π _i , i = 1, 2, …, k. Let Θ_[1] ≤ Θ_[2] ··· ≤ Θ_[k] denote the ranked values of the Θ _j s, so that Θ_[i] = μ + σ_[i], i = 1, 2, …, k, and let Π_(i) denote the unknown population associated with the ith smallest mean lifetime Θ_[i] = μ + σ_[i], i = 1, 2, …, k. Based on independent random samples from the k populations, we propose a selection procedure for the goal of selecting the population having the longest mean lifetime Θ_[k] (called the “best” population), under the subset selection formulation. Tables for the implementation of the proposed selection procedure are provided. It is established that the proposed subset selection procedure is monotone for a general k (≥ 2). For k = 2, we consider the loss measured by the size of the selected subset and establish that the proposed subset selection procedure is minimax among selection procedures that satisfy a certain probability requirement (called the P*-condition) for the inclusion of the best population in the selected subset.     

Markov regime switching in mean and in fractional integration parameter

We propose a specific general Markov-regime switching estimation both in the long memory parameter d and the mean of a time series. We employ Viterbi algorithm that combines the Viterbi procedures in two state Markov-switching parameter estimation. It is well-known that existence of mean break and long memory in time series can be easily confused with each other in most cases. Thus, we aim at observing the deviation and interaction of mean and d estimates for different cases. A Monte Carlo experiment reveals that the finite sample performance of the proposed algorithm for a simple mixture model of Markov-switching mean and d changes with respect to the fractional integrating parameters and the mean values for the two regimes.     

The Two-Dimensional Asymmetric Gambler's Ruin Problem with n Players and Simulations

In this article, the asymmetric n-player gambler's ruin problem is considered, when the players use equal initial fortunes of d dollars and d euros, 1 ≤ d ≤ n + 1. In each round an unfair coin is tossed to decide the currency. The expected ruin time and the individual ruin probabilities are computed. It is proved that the ruin time and which player is ruined are independent events. Finally, some special games are simulated. The simulation results verify the validity of the proposed formulas. As an innovation, the present study makes a combination of the n-player and multi dimensional games which can be viewed as a starting point for future studies.     

Why is one choice different?

Let X_i be nonnegative independent random variables with finite expectations and . The value is what can be obtained by a “prophet”. A “mortal” on the other hand, may use k1 stopping rules t₁,…,t_k yielding a return E[max_i=1,…,kX_ti]. For nk the optimal return is where the supremum is over all stopping rules which stop by time n. The well known “prophet inequality” states that for all such X_i's and one choice and the constant “2” cannot be improved on for any n2. In contrast we show that for k=2 the best constant d satisfying for all such X_i's depends on n. On the way we obtain constants c_k such that .     

Maximum likelihood estimation of stationary multivariate ARFIMA processes

This article considers the maximum likelihood estimation (MLE) of a class of stationary and invertible vector autoregressive fractionally integrated moving-average (VARFIMA) processes considered in Equation (26) of Luceño [A fast likelihood approximation for vector general linear processes with long series: Application to fractional differencing, Biometrika 83 (1996), pp. 603–614] or Model A of Lobato [Consistency of the averaged cross-periodogram in long memory series, J. Time Ser. Anal. 18 (1997), pp. 137–155] where each component y _{i, t} is a fractionally integrated process of order d _i , i=1, …, r. Under the conditions outlined in Assumption 1 of this article, the conditional likelihood function of this class of VARFIMA models can be efficiently and exactly calculated with a conditional likelihood Durbin–Levinson (CLDL) algorithm proposed herein. This CLDL algorithm is based on the multivariate Durbin–Levinson algorithm of Whittle [On the fitting of multivariate autoregressions and the approximate canonical factorization of a spectral density matrix, Biometrika 50 (1963), pp. 129–134] and the conditional likelihood principle of Box and Jenkins [Time Series Analysis, Forecasting, and Control, 2nd ed., Holden-Day, San Francisco, CA]. Furthermore, the conditions in the aforementioned Assumption 1 are general enough to include the model considered in Andersen et al. [Modeling and forecasting realized volatility, Econometrica 71 (2003), 579–625] for describing the behaviour of realized volatility and the model studied in Haslett and Raftery [Space–time modelling with long-memory dependence: Assessing Ireland's wind power resource, Appl. Statist. 38 (1989), pp. 1–50] for spatial data as its special cases. As the computational cost of implementing the CLDL algorithm is much lower than that of using the algorithms proposed in Sowell [Maximum likelihood estimation of fractionally integrated time series models, Working paper, Carnegie-Mellon University], we are thus able to conduct a Monte Carlo experiment to investigate the finite sample performance of the CLDL algorithm for the 3-dimensional VARFIMA processes with the sample size of 400. The simulation results are very satisfactory and reveal the great potentials of using the CLDL method for empirical applications.     

A comparison of estimation methods in non-stationary ARFIMA processes

This paper reports an extensive Monte Carlo simulation study based on six estimators for the long memory fractional parameter when the time series is non-stationary, i.e., ARFIMA(p, d, q) process for d?>?0.5. Parametric and semiparametric methods are compared. In addition, the effect of the parameter estimation is investigated for small and large sample sizes and non-Gaussian error innovations. The methodology is applied to a well known data set, the so-called UK short interest rates.     

ON TRANSITION PROBABILITIES OF SKIP-FREE MARKOV CHAINS

Consider an ergodic Markov chain X(t) in continuous time with an infinitesimal matrix Q = (q_ij) defined on a finite state space {0, 1,…, N}. In this note, we prove that if X(t) is skip-free positive (negative, respectively), i.e., q_ij, = 0 for j > i+ 1 (i > j+ 1), then the transition probability p_ij(t) = Pr[X(t)=j | X(0) =i] can be represented as a linear combination of p_0N(t) (p^(m)_(N0)(t)), 0 ≤ m ≤N, where f^(m)(t) denotes the mth derivative of a function f(t) with f⁽⁰⁾(t) =f(t). If X(t) is a birth-death process, then p_ij(t) is represented as a linear combination of p_0N^(m)(t), 0 ≤m≤N - |i-j|.     

Markov-modulated infinite-server queues driven by a common background process

This paper studies a system with multiple infinite-server queues that are modulated by a common background process. If this background process, being modeled as a finite-state continuous-time Markov chain, is in state j, then the arrival rate into the i-th queue is λ_{i, j}, whereas the service times of customers present in this queue are exponentially distributed with mean μ^{? 1}_{i, j}; at each of the individual queues all customers present are served in parallel (thus reflecting their infinite-server nature).

Three types of results are presented: in the first place (i) we derive differential equations for the probability-generating functions corresponding to the distributions of the transient and stationary numbers of customers (jointly in all queues), then (ii) we set up recursions for the (joint) moments, and finally (iii) we establish a central limit theorem in the asymptotic regime in which the arrival rates as well as the transition rates of the background process are simultaneously growing large.     

On grouping sets of exponential populations

Suppose we have k random samples each of size n from a two parameter exponential distribution with location parameters μ _i i=1,…,k, and where each item has the same, unknown scale parameter. A multistage procedure is developed to determine t≦k groups such that in any one group the distributions have μ_i's that are not appreciably different. The method yields a unique grouping and extends the approach of the Kumar and Pate1 test.The emphasis is on the development of a procedure based on the null sampling distribution of the maximum gap of the ordered first order statistics from exponential distributions. The procedure is based on complete ordered samples or censored (of any or of all) samples.     

On multivariate variable-kernel density estimates for time series

Let X₁ be a strictly stationary multiple time series with values in R^d and with a common density f. Let X₁,.,.,X_n, be n consecutive observations of X₁. Let k = k_n, be a sequence of positive integers, and let H_ni be the distance from X_i to its kth nearest neighbour among X_j, j i. The multivariate variable-kernel estimate f_n, of f is defined by where K is a given density. The complete convergence of f_n, to f on compact sets is established for time series satisfying a dependence condition (referred to as the strong mixing condition in the locally transitive sense) weaker than the strong mixing condition. Appropriate choices of k are explicitly given. The results apply to autoregressive processes and bilinear time-series models.     

Two-queue polling systems with switching policy based on the queue that is not being served

We study a system of two non-identical and separate M/M/1/? queues with capacities (buffers) C₁ < ∞ and C₂ = ∞, respectively, served by a single server that alternates between the queues. The server’s switching policy is threshold-based, and, in contrast to other threshold models, is determined by the state of the queue that is not being served. That is, when neither queue is empty while the server attends Q_i (i = 1, 2), the server switches to the other queue as soon as the latter reaches its threshold. When a served queue becomes empty we consider two switching scenarios: (i) Work-Conserving, and (ii) Non-Work-Conserving. We analyze the two scenarios using Matrix Geometric methods and obtain explicitly the rate matrix R, where its entries are given in terms of the roots of the determinants of two underlying matrices. Numerical examples are presented and extreme cases are investigated.