Relative Potency Estimation in Parallel-Line Assays – Method Comparison and Some Extensions

Relative potency estimations in both multiple parallel-line and slope-ratio assays involve construction of simultaneous confidence intervals for ratios of linear combinations of general linear model parameters. The key problem here is that of determining multiplicity adjusted percentage points of a multivariate t-distribution, the correlation matrix R of which depends on the unknown relative potency parameters. Several methods have been proposed in the literature on how to deal with R . In this article, we introduce a method based on an estimate of R (also called the plug-in approach) and compare it with various methods including conservative procedures based on probability inequalities. Attention is restricted to parallel-line assays though the theory is applicable for any ratios of coefficients in the general linear model. Extension of the plug-in method to linear mixed effect models is also discussed. The methods will be compared with respect to their simultaneous coverage probabilities via Monte Carlo simulations. We also evaluate the methods in terms of confidence interval width through application to data on multiple parallel-line assay.     

Methods for calculating stationary distribution in linear models of time series

The article deals with methods for computing the stationary marginal distribution in linear models of time series. Two approaches are described. First, an algorithm based on approximation of solution of the corresponding integral equation is briefly reviewed. Then, we study the limit behaviour of the partial sums c ₁ η₁+c ₂ η₂+···+c _n η _n where η _i are i.i.d. random variables and c _i real constants. We generalize procedure of Haiman (1998) [Haiman, G., 1998, Upper and lower bounds for the tail of the invariant distribution of some AR(1) processes. Asymptotic Methods in Probability and Statistics, 45, 723–730.] to an arbitrary causal linear process and relax the assumptions of his result significantly. This is achieved by investigating the properties of convolution of densities.     

Upper bounds for ruin probabilities under model uncertainty

Abstract

In this paper, we investigate some ruin problems for risk models that contain uncertainties on both claim frequency and claim size distribution. The problems naturally lead to the evaluation of ruin probabilities under the so-called G-expectation framework. We assume that the risk process is described as a class of G-compound Poisson process, a special case of the G-Lévy process. By using the exponential martingale approach, we obtain the upper bounds for the two-sided ruin probability as well as the ruin probability involving investment. Furthermore, we derive the optimal investment strategy under the criterion of minimizing this upper bound. Finally, we conclude that the upper bound in the case with investment is less than or equal to the case without investment.     

Asymptotic distribution of data-driven smoothers in density and regression estimation under dependence

We consider automatic data-driven density, regression and autoregression estimates, based on any random bandwidth selector h/_T. We show that in a first-order asymptotic approximation they behave as well as the related estimates obtained with the “optimal” bandwidth h_T as long as h_T/h_T → 1 in probability. The results are obtained for dependent observations; some of them are also new for independent observations.     

A Note on Dirichlet One-Armed Bandits

One of the two independent stochastic processes (or ‘arms’) is selected and observed sequentially at each of n(≤ ∝) stages. Arm 1 yields observations identically distributed with unknown probability measure P with a Dirichlet process prior whereas observations from arm 2 have known probability measure Q. Future observations are discounted and at stage m, the payoff is a _m(≥0) times the observation Z _m at that stage. The objective is to maximize the total expected payoff. Clayton and Berry (1985) consider this problem when a _m equals 1 for m ≤ n and 0 for m > n(< ∝) In this paper, the Clayton and Berry (1985) results are extended to the case of regular discount sequences of horizon n, which may also be infinite. The results are illustrated with numerical examples. In case of geometric discounting, the results apply to a bandit with many independent unknown Dirichlet arms.     

Bonferroni-type Plug-in Procedure Controlling Generalized Familywise Error Rate

Consider the multiple hypotheses testing problem controlling the generalized familywise error rate k-FWER, the probability of at least k false rejections. We propose a plug-in procedure based on the estimation of the number of true null hypotheses. Under the independence assumption of the p-values corresponding to the true null hypotheses, we first introduce the least favorable configuration (LFC) of k-FWER for Bonferroni-type plug-in procedure, then we construct a plug-in k-FWER-controlled procedure based on LFC. For dependent p-values, we establish the asymptotic k-FWER control under some mild conditions. Simulation studies suggest great improvement over generalized Bonferroni test and generalized Holm test.     

A random weighting approach for posterior distributions

ABSTRACT

In Bayesian theory, calculating a posterior probability distribution is highly important but typically difficult. Therefore, some methods have been proposed to deal with such problem, among which, the most popular one is the asymptotic expansions of posterior distributions. In this paper, we propose an alternative approach, named a random weighting method, for scaled posterior distributions, and give an ideal convergence rate, o(n^{( ? 1/2)}), which serves as the theoretical guarantee for methods of numerical simulations.     

A modified bootstrap procedure for cluster sampling variance estimation of species richness

Variance estimators for probability sample-based predictions of species richness (S) are typically conditional on the sample (expected variance). In practical applications, sample sizes are typically small, and the variance of input parameters to a richness estimator should not be ignored. We propose a modified bootstrap variance estimator that attempts to capture the sampling variance by generating B replications of the richness prediction from stochastically resampled data of species incidence. The variance estimator is demonstrated for the observed richness (SO), five richness estimators, and with simulated cluster sampling (without replacement) in 11 finite populations of forest tree species. A key feature of the bootstrap procedure is a probabilistic augmentation of a species incidence matrix by the number of species expected to be ‘lost’ in a conventional bootstrap resampling scheme. In Monte-Carlo (MC) simulations, the modified bootstrap procedure performed well in terms of tracking the average MC estimates of richness and standard errors. Bootstrap-based estimates of standard errors were as a rule conservative. Extensions to other sampling designs, estimators of species richness and diversity, and estimates of change are possible.     

Sub-optimal investment for insurers

Abstract

We consider the investment problem for a non-life insurance company seeking to minimize the ruin probability. Its reserve is described by a perturbed risk process possibly correlated with the financial market. Assuming exponential claim size, the Hamilton-Jacobi-Bellman equation reduces to a first order nonlinear ordinary differential equation, which seems hard to solve explicitly. We study the qualitative behavior of its solution and determine the Cramér-Lundberg approximation. Moreover, our approach enables to find very naturally that the optimal investment strategy is not constant. Then, we analyze how much the company looses by adopting sub-optimal constant (amount) investment strategies.     

Asymptotic ruin probabilities of a two-dimensional renewal risk model with dependent inter-arrival times

Abstract

This article mainly considers the uniform asymptotics for the finite-time ruin probabilities of a two-dimensional renewal risk model with heavy-tailed claims. In this model, the two claim-number processes are arbitrarily dependent and each of them is generated by widely orthant dependent claim inter-arrival times. Two types of ruin are studied and for each type of ruin, an asymptotic formula for the finite-time ruin probability is established. These formulae possess a certain uniformity feature in the time horizon.     

The uniform local asymptotics for a Lévy process and its overshoot and undershoot

ABSTRACT

In this article, we obtain the uniform local asymptotics for a Lévy process with a heavy-tailed Lévy measure and for the overshoot and undershoot of the Lévy process. As applications, we get the uniform asymptotics of the finite-time ruin probability and the local ruin probability for the Lévy risk model with a heavy-tailed Lévy measure. By the above results, we find that in the compound Poisson model perturbed by a Brownian motion, the effect of the Brownian component on the asymptotics of the finite-time ruin probability and the local ruin probability washes out.     

Pseudo confidence regions for the solution of a multivariate linear functional relationship

The problem of finding confidence regions (CR) for a q-variate vector γ given as the solution of a linear functional relationship (LFR) Λγ = μ is investigated. Here an m-variate vector μ and an m × q matrix Λ = (Λ₁, Λ₂,…, Λ_q) are unknown population means of an m(q+1)-variate normal distribution $\text{N}{}_{\mathrm{m(q+1)}}\text{(ζΩ?Σ)}$, where ζ′ = (μ′, Λ₁′, Λ₂′,…, Λ_q′Σ is an unknown, symmetric and positive definite m × m matrix and Ω is a known, symmetric and positive definite (q+1) × (q+1) matrix and ? denotes the Kronecker product. This problem is a generalization of the univariate special case for the ratio of normal means.A CR for γ with level of confidence 1 ? α, is given by a quadratic inequality, which yields the so-called ‘pseudo’ confidence regions (PCR) valid conditionally in subsets of the parameter space. Our discussion is focused on the ‘bounded pseudo’ confidence region (BPCR) given by the interior of a hyperellipsoid. The two conditions necessary for a BPCR to exist are shown to be the consistency conditions concerning the multivariate LFR. The probability that these conditions hold approaches one under ‘reasonable circumstances’ in many practical situations. Hence, we may have a BPCR with confidence approximately 1 ? α. Some simulation results are presented.     

On Erlang(2) Risk Process Perturbed by Diffusion

ABSTRACT

In this article, we consider an Erlang(2) risk process perturbed by diffusion. From the extreme value distribution of Brownian motion with drift and the renewal theory, we show that the survival probability satisfies an integral equation. We then give the bounds for the ultimate ruin probability and the ruin probability caused by claim. By introducing a random walk associated with the proposed risk process, we define an adjustment-coefficient. The relation between the adjustment-coefficient and the bound is given and the Lundberg-type inequality for the bound is obtained. Also, a formula of Pollaczek–Khinchin type for the bound is derived. Using these results, the bound can be calculated when claim sizes are exponentially distributed.     

One-Sample One-Sided Student T-Test Under Contaminants

A paradoxical behavior of the t-test under ε-contamination is presented. The paradox consists in that under a fixed distribution of contaminants an increasing of the probability of the appearance of a contaminant may decrease the violation of the size of the test! A simple explanation of the phenomenon is given. It is revealed which contaminants make the test conservative and which make it liberal: it appears that, in spite of the established opinion, conservatism or liberalism of the test depends not so much on the tails of the contaminating distribution as on where its support is located.     

Approximating M/G/1 Waiting Time Tail Probabilities

Abstract

We propose a new approximation formula for the waiting time tail probability of the M/G/1 queue with FIFO discipline and unlimited waiting space. The aim is to address the difficulty of obtaining good estimates when the tail probability has non-exponential asymptotics. We show that the waiting time tail probability can be expressed in terms of the waiting time tail probability of a notional M/G/1 queue with truncated service time distribution plus the tail probability of an extreme order statistic. The Cramér–Lundberg approximation is applied to approximate the tail probability of the notional queue. In essence, our technique extends the applicability of the Cramér–Lundberg approximation to cases where the standard Lundberg condition does not hold. We propose a simple moment-based technique for estimating the parameters of the approximation; numerical results demonstrate that our approximation can yield very good estimates over the whole range of the argument.     

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.     

Estimating the probability of winning (losing) in a gambler's ruin problem with applications

In a classical gambler's ruin problem, the distribution of the number of games lost till ruin is considered, which we call the lost game distribution (LGD). Some applications of LGD in the theory of queues, in the theory of epidemic and in certain clustering and branching models are mentioned. The maximum likelihood estimation of LGD in the framework of modified power series distribution (MPSD), introduced by the author (1974), is studied. The variance and bias of the MLE are given and the actual mean of the MLE is obtained by discussing the negative moments of the MPSD in general. The minimum variance unbiased estimator of θ^k (k≥1) is obtained employing the technique developed by the author (1977) for the class of MPSD.     

A note on asymptotic parametric prediction

This paper is devoted to asymptotic behaviour of plug-in statistical predictors obtained by replacing the unknown parameter in a conditional expectation by a suitable estimator. We derive the L²$L^2$-convergence rate and limit in distribution for the predictors. Applications to ARMA processes and diffusion processes are considered.     

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.     

Asymptotic power comparison of T2-type test and likelihood ratio test for a mean vector based on two-step monotone missing data

Abstract

In a 2-step monotone missing dataset drawn from a multivariate normal population, T²-type test statistic (similar to Hotelling’s T² test statistic) and likelihood ratio (LR) are often used for the test for a mean vector. In complete data, Hotelling’s T² test and LR test are equivalent, however T²-type test and LR test are not equivalent in the 2-step monotone missing dataset. Then we interest which statistic is reasonable with relation to power. In this paper, we derive asymptotic power function of both statistics under a local alternative and obtain an explicit form for difference in asymptotic power function. Furthermore, under several parameter settings, we compare LR and T²-type test numerically by using difference in empirical power and in asymptotic power function. Summarizing obtained results, we recommend applying LR test for testing a mean vector.