Random Observation Times Research Articles

The last-success stopping problem with random observation times

AbstractSuppose N independent Bernoulli trials with success probabilities $$p_1, p_2,\ldots $$ p 1 , p 2 , … are observed sequentially at times of a mixed binomial process. The task is to maximise, by using a nonanticipating stopping strategy, the probability of stopping at the last success. The case $$p_k=1/k$$ p k = 1 / k has been studied by many authors as a version of the familiar best choice problem, where both N and the observation times are random. We consider a more general profile $$p_k=\theta /(\theta +k-1)$$ p k = θ / ( θ + k - 1 ) and assume that the prior distribution of N is negative binomial with shape parameter $$\nu $$ ν , so the arrivals occur at times of a mixed Poisson process. The setting with two parameters offers a high flexibility in understanding the nature of the optimal strategy, which we show is intrinsically related to monotonicity properties of the Gaussian hypergeometric function. Using this connection, we find that the myopic stopping strategy is optimal if and only if $$\nu \ge \theta $$ ν ≥ θ . Furthermore, we derive formulas to assess the winning probability and discuss limit forms of the problem for large N.

Determination of correlations in multivariate count data with informative observation times.

We consider there are various types of recurrent events and the total number of occurrences are collected at the random observation times. It has concerned that the observation process may not be independent to the multivariate event processes, hence the total counts and observation times may be correlated and the dependence may exist among different types of the event processes as well. Many methods have developed nonparametric models to accommodate such unknown structures; however, it is difficult to assess and directly quantify their correlation relationships. A multivariate frailty model is proposed to this study, in which the event and observation processes are linked by frailty variables whose joint distribution can be implicitly specified through the multivariate normal distribution with some unknown covariance matrix. The Bayesian inference method is conducted to obtain the estimates of the regression coefficients and correlation parameters. We use a form of trigonometric functions to represent the covariance matrix, so that it meets the positive-definiteness condition efficiently during the estimation schemes. The simulation studies demonstrate the utility of the proposed models. We apply the model to a skin cancer prevention study, and aim to determine the covariate and association effects. We found treatment is significant in determining the duration of examination times; prior-counts, age and gender are significant variables on the occurrence rates of tumor counts. Using the covariance matrix to access the underlying dependent structure, the mutual correlations among them are all positive, and the basal cell counts are more related to the examination times.

Robust estimation of integrated and spot volatility

We introduce a new method to estimate the integrated volatility (IV) and the spot volatility (SV) based on noisy high-frequency data. Our method employs the ReMeDI approach introduced by Li and Linton (2022) to estimate the moments of microstructure noise and thereby eliminate their influence, and the pre-averaging method to target the volatility parameter. The method is robust: it can be applied when the efficient price exhibits stochastic volatility and jumps, the observation times are random, and the noise process is nonstationary, autocorrelated, asymptotically vanishing and dependent on the efficient price. We derive the limit distributions for the proposed estimators under the infill asymptotics in a general setting. Our extensive simulation studies demonstrate the robustness, accuracy and computational efficiency of our estimators compared to several alternative estimators recently proposed in the literature. Empirically, we show that neglecting the complexities of noise and the random observation times generates substantial biases in volatility estimation and may yield a different intraday volatility pattern.

On the estimation of the jump activity index in the case of random observation times

We propose a nonparametric estimator of the jump activity index beta of a pure-jump semimartingale X driven by a beta -stable process when the underlying observations are coming from a high-frequency setting at irregular times. The proposed estimator is based on an empirical characteristic function using rescaled increments of X, with a limit that depends in a complicated way on beta and the distribution of the sampling scheme. Utilising an asymptotic expansion we derive a consistent estimator for beta and prove an associated central limit theorem.

Randomized observation periods for compound Poisson risk model with capital injection and barrier dividend

In this paper, we model the insurance company’s surplus by a compound Poisson risk model, where the surplus process can only be observed at random observation times. It is assumed that the insurer observes its surplus level periodically to decide on dividend payments and capital injection at the interobservation time having an operatorname{Erlang}(n) distribution. If the observed surplus level is greater than zero but less than injection line b_{1} > 0, the shareholders should immediately inject a certain amount of capital to bring the surplus level back to the injection line b_{1}. If the observed surplus level is larger than dividend line b_{2} (b_{2} > b_{1}), any excess of the surplus over b_{2} is immediately paid out as dividends to the shareholders of the company. Ruin is declared when the observed surplus level is negative. We derive the explicit expressions of the Gerber–Shiu function, the expected discounted capital injection, and the expected discounted dividend payments. Numerical illustrations are also given to analyze the effect of random observation times on actuarial quantities.

Periodic dividends and capital injections for a spectrally negative Lévy risk process under absolute ruin

The spectrally negative Levy risk model with random observation times is considered in this paper, in which both dividends and capital injections are made at some independent Poisson observation times. Under the absolute ruin, the expected discounted dividends and the expected discounted capital injections are discussed. We also study the joint Laplace transforms including the absolute ruin time and the total dividends or the total capital injections. All the results are expressed in scale functions.

ON THE RESIDUAL AND PAST LIFETIMES OF COHERENT SYSTEMS UNDER RANDOM MONITORING

This paper discusses stochastic comparisons for the residual and past lifetimes of coherent systems with dependent and identically distributed (d.i.d.) components under random monitoring in terms of the hazard rate, the reversed hazard rate, and the likelihood ratio orders. Some stochastic comparisons results are also established on the residual lifetimes of coherent systems under random observation times when all of the components are alive at that time. Sufficient conditions are established in terms of the aging properties of the components and the distortion functions induced from the system structure and dependence among components lifetimes. Numerical examples are provided to illustrate the theoretical results as well.

Daily (In)Activities of Nursing Home Residents in Their Wards: An Observation Study

ObjectivesResearch shows that nursing home residents are largely inactive. This inactivity negatively influences physical fitness, and participation in daily activities is known to have a positive influence on physical function and quality of life. Existing research does not provide sufficient insight into the daily activities in which nursing home residents participate. This insight is needed to develop future interventions so as to encourage nursing home residents to participate in daily activities and, thereby, decrease inactivity. The purpose of this study was to obtain insight into daily (in)activities of psychogeriatric and somatic nursing home residents during the day and their body positions during these (in)activities. DesignCross-sectional observation study. SettingNursing homes in the Netherlands (19 psychogeriatric and 11 somatic wards). ParticipantsParticipants were 723 home residents in 7 nursing homes. MeasurementsObservations were conducted using a self-developed observation list. Residents were observed in their wards during 5 random observation times between 7:00 am and 11:00 pm, in which the daily activity and position of the resident during this activity were scored. Percentages of activities and positions were calculated for each observation time. ResultsIn total, 3282 observations (91% of the intended 3615 observations) were conducted. Nursing home residents of both psychogeriatric and somatic wards were mainly observed partaking in inactivities, such as sleeping, doing nothing, and watching TV (range: 45%–77% of the 5 observation times). Furthermore, residents were engaged in activities of daily living (ADLs) (range: 15%–38%) that mainly comprised activities related to mobility (range: 10%–19%) and eating and drinking (range: 2%–17%). Engagement of residents in instrumental ADLs (IADLs) was rarely observed (up to 3%). Residents were largely observed in a lying or sitting position (range: 89%–92%). ConclusionMost of the psychogeriatric and somatic nursing home residents spend their day inactive in a lying or sitting position in the ward. To encourage nursing home residents in daily activities in the wards, interventions are needed that (1) focus on increasing ADLs and IADLs, and (2) encourage standing and walking.

Efficient Estimation of Integrated Volatility and Related Processes

We derive nonparametric bounds for inference about functionals of high-frequency volatility, in particular, integrated power variance. In the absence of microstructure noise, we find that standard Realized Variance attains the nonparametric efficiency bound, also in case of unequally spaced random observation times. For higher powers, e.g., integrated quarticity, the block-based procedures of Mykland and Zhang (2009) can get arbitrarily close to the nonparametric bounds in case of equally spaced observations. The estimator in Jacod and Rosenbaum (2013) is efficient, also at non-constant volatility, still for equally spaced data. For unequally spaced data, we provide an estimator, similar to that of Kristensen (2010), that can get arbitrarily close to the nonparametric bound. Finally, contrary to public opinion, we demonstrate that parametric information about the functional form of volatility generally leads to a decreased lower bound, unless the volatility process is piecewise constant.

A joint model for multistate disease processes and random informative observation times, with applications to electronic medical records data.

Multistate models are used to characterize individuals' natural histories through diseases with discrete states. Observational data resources based on electronic medical records pose new opportunities for studying such diseases. However, these data consist of observations of the process at discrete sampling times, which may either be pre-scheduled and non-informative, or symptom-driven and informative about an individual's underlying disease status. We have developed a novel joint observation and disease transition model for this setting. The disease process is modeled according to a latent continuous-time Markov chain; and the observation process, according to a Markov-modulated Poisson process with observation rates that depend on the individual's underlying disease status. The disease process is observed at a combination of informative and non-informative sampling times, with possible misclassification error. We demonstrate that the model is computationally tractable and devise an expectation-maximization algorithm for parameter estimation. Using simulated data, we show how estimates from our joint observation and disease transition model lead to less biased and more precise estimates of the disease rate parameters. We apply the model to a study of secondary breast cancer events, utilizing mammography and biopsy records from a sample of women with a history of primary breast cancer.

On a risk model with randomized dividend-decision times

In this paper, we consider a perturbed compound Poisson risk modelwith a randomized dividend strategy. Assume that decisions on payingoff dividends are made at some random observation times. Whenever the observedvalue of the surplus process exceeds a given barrier $b$, the excessvalue will be paid off as dividends. We assume that the Laplacetransform of the individual claim size belongs to the rationalfamily. When the time intervals between successive decision timesfollow exponential distribution, we present explicit expressions forthe Gerber-Shiu function. We also extend the exponentialassumption to Erlang and discuss the solution procedure.

Randomized observation periods for the compound Poisson risk model: the discounted penalty function

In the framework of collective risk theory, we consider a compound Poisson risk model for the surplus process where the process (and hence ruin) can only be observed at random observation times. For Erlang(n) distributed inter-observation times, explicit expressions for the discounted penalty function at ruin are derived. The resulting model contains both the usual continuous-time and the discrete-time risk model as limiting cases, and can be used as an effective approximation scheme for the latter. Numerical examples are given that illustrate the effect of random observation times on various ruin-related quantities.

Alternating branching processes

We introduce the idea of controlling branching processes by means of another branching process, using the fractional thinning operator of Steutel and van Harn. This idea is then adapted to the model of alternating branching, where two Markov branching processes act alternately at random observation and treatment times. We study the extinction probability and limit theorems for reproduction by n cycles, as n → ∞.

Alternating branching processes

We introduce the idea of controlling branching processes by means of another branching process, using the fractional thinning operator of Steutel and van Harn. This idea is then adapted to the model of alternating branching, where two Markov branching processes act alternately at random observation and treatment times. We study the extinction probability and limit theorems for reproduction by n cycles, as n → ∞.

Nonparametric Estimation under Censoring and Passive Registration

The classical random censorship model assumes that we follow an individual continuously up to the time of failure or censoring, so observing this time as well as the indicator of its type. Under passive registration we only get information on the state of the individual at random observation or registration times. In this paper we assume that these registration times are the times of events in an independent Poisson process, stopped at failure or censoring; the time of failure is also observed if not censored. This problem turns up in historical demography, where the survival time of interest is the life‐length, censoring is by emigration, and the observation times are times of births of children, and other life‐events. (Church registers contain dates of births, marriages, deaths, but not emigrations.) The model is shown to be related to the problem of estimating a density known to be monotone. This leads to an explicit description of the non‐parametric maximum likelihood estimator of the survival function (based on i.i.d. observations from this model) and to an analysis of its large sample properties.

Modelling of repairable systems with various degrees of repair

The availability of a stochastic repairable system depends on the failure behaviour and on repair strategies. In this paper, we deal with a general repair model for a system using auxiliary counting processes and corresponding intensities which include various degrees of repair (between minimal repair and perfect repair). For determining the model parameters we need estimators depending on failure times and repair times: maximum likelihood (ML) estimator and Bayes estimators are considered. Special results are obtained by the use of Weibull-type intensities and random observation times.

A monte carlo comparison of tests of parallelism of the ponses over time of two populations vith non-uniform observation times

Repeated measures data collected at random observation times are quite common in clinical studies and are often difficult to analyze. A Monte Carlo comparison of four analysis procedures with respect to significance level and power is presented. The basic procedures compared are successive difference analyses and three procedures using the data as summarized in the estimated quadratic polynomial regression coefficients for each subject. These three procedures are (1) Hotelling's T-square, (2) Multivariate Multisample Rank Sum Test (MMRST) and (3) Multivariate Multisample Median Test (MMMT). For the variety of dispersion structures, sample sizes and treatement groups simulated the MMRST and successive difference analysis were the most satisfactory.

A monte carlo study of successive difference analysis of growth curve data at random observation times

Simulations of the analysis of successive differences, that is rhe changes observed on a subject with random observation times suggest this procedure is a viable alternative tn other methods The successive differences approach focuses on the change that occur over time and allows the inclusion of subjects with as few as two observations. Furthermore, it measures the change over the actual observation period and does not attempt to extrapolate over a larger time span. The simulations indicate that with a proper degree of freedom adjustment and partitioning of the time span the successive difference has approximately the nominal significance level and seems to be reasonably powerful in detecting non—parallelism.

A Successive Differences Method for Growth Curves with Missing Data and Random Observation Times

Abstract Incomplete growth curve data can be analyzed by the successive differences (SD) method, which uses the difference in consecutive pairs of observations for all subjects having two or more repeated measures. We have generalized it to handle varying observation times as well by partitioning the time interval spanned by all repeated measures into subintervals. The model is developed, including test statistics for the hypotheses of parallelism and of no change in response level over time, assuming parallelism. This generalized SD method is then applied to repeated serum cholesterol measurements on a subsample of 1,072 men from a prospective cohort study of 8,006 Hawaiian Japanese men living on Oahu.

A Successive Differences Method for Growth Curves with Missing Data and Random Observation Times

Abstract Incomplete growth curve data can be analyzed by the successive differences (SD) method, which uses the difference in consecutive pairs of observations for all subjects having two or more repeated measures. We have generalized it to handle varying observation times as well by partitioning the time interval spanned by all repeated measures into subintervals. The model is developed, including test statistics for the hypotheses of parallelism and of no change in response level over time, assuming parallelism. This generalized SD method is then applied to repeated serum cholesterol measurements on a subsample of 1,072 men from a prospective cohort study of 8,006 Hawaiian Japanese men living on Oahu.