THE OPTIMAL TEST FOR SELECTING THE GREATER OF TWO BINOMIAL PROBABILITIES

The efficiency of a sequential test is related to the “importance” of the trials within the test. This relationship is used to find the optimal test for selecting the greater of two binomial probabilities, p_α and p_b, namely, the stopping rule is “gambler's ruin” and the optimal discipline when p_α+p_b≤ 1 (≥ 1) is play-the-winner (loser), i.e. an α-trial which results in a success is followed by an α-trial (b-trial) whereas an α-trial which results in a failure is followed by α b-trid (α-trial) and correspondingly for b-trials.     

ON CONSTRUCTION OF GENERALIZED GROUP DIVISIBLE DESIGNS WITH TWO GROUPS

ABSTRACT

Many simple and general methods of construction, using α-resolvable balanced incomplete block designs, for generalized group divisible designs with two groups (GGDD(2)) are given. The E-optimality of all constructed designs is also discussed.     

TESTING HYPOTHESES ABOUT THE COMMON MEAN OF TWO NORMAL POPULATIONS

Three tests are considered concerning the common mean of two normal populations: (1) an F test based on a sample from one population, (2) a test based on the addition of the F statistics from independent samples from two popultions (proposed), and (3) a test based on the maximum of the F statistics from two independent samples from two populations. A condition under which test (2) is locally more powerful than test (1) is given. As the test statistic in test (2) does not follow a standard distribution, a formula for approximating the observed significance level is provided. A simulation study is used to compare the power of these tests.     

PERMUTATION TEST FOR EQUALITY OF CORRELATION COEFFICIENTS IN TWO POPULATIONS

ABSTRACT

The purpose of this paper is to investigate the permutation tests for equality of correlation coefficients among two independent populations. We discuss how to apply permutation test to this problem and its asymptotic suitability. We also show some simulation studies and an example of the Iris data.     

A GALTON-WATSON BRANCHING PROCESS IN VARYING ENVIRONMENTS WITH ESSENTIALLY CONSTANT OFFSPRING MEANS AND TWO RATES OF GROWTH1

A Galton-Watson process in varying environments (Z_n), with essentially constant offspring means, i.e. E(Z_n)/mⁿ→α∈(0, ∞), and exactly two rates of growth is constructed. The underlying sample space Ω can be decomposed into parts A and B such that (Z_n)_n grows like 2ⁿon A and like mⁿon B (m > 4).     

THE OPTIMAL CLASSIFICATION RULE FOR EXPONENTIAL POPULATIONS

Let π₁ and π₂ be two exponential populations with location parameters α and β respectively. It is shown that the probability of misclassification for the optimal rule, R_E, depends on α and β only through their ratio α/β=r, and that the smaller the ratio r (i.e. the smaller α is compared to β), the greater the superiority of the optimal rule R_E over the commonly used rule R_N. When α and β are unknown, the sample-based version R_E(s) of R_E exhibits the same pattern of superiority over the sample-based R_N(s) of R_N.     

Nonlinear filters based on taylor series expansions ∗

The nonlinear filters based on Taylor series approximation are broadly used for computational simplicity, even though their filtering estimates are clearly biased. In this paper, first, we analyze what is approximated when we apply the expanded nonlinear functions to the standard linear recursive Kalman filter algorithm. Next, since the state variable α_t and α_t-t are approximated as a conditional normal distribution given information up to time t - 1 (i.e., I_t-1) in approximation of the Taylor series expansion, it might be appropriate to evaluate each expectation by generating normal random numbers of α_t and α_t-1 given I_t-1 and those of the error terms θ and η_t. Thus, we propose the Monte-Carlo simulation filter using normal random draws. Finally we perform two Monte-Carlo experiments, where we obtain the result that the Monte-Carlo simulation filter has a superior performance over the nonlinear filters such as the extended Kalman filter and the second-order nonlinear filter.     

Estimation of the mean of the selected gamma population

Let л₁ and л₂ denote two independent gamma populations G(α₁, p) and G(α₂, p) respectively. Assume α(i=1,2)are unknown and the common shape parameter p is a known positive integer. Let Y_i denote the sample mean based on a random sample of size n from the i-th population. For selecting the population with the larger mean, we consider, the natural rule according to which the population corresponding to the larger Y_i is selected. We consider? in this paper, the estimation of M, the mean of the selected population. It is shown that the natural estimator is positively biased. We obtain the uniformly minimum variance unbiased estimator(UMVE) of M. We also consider certain subclasses of estikmators of the form c₁x₍₁₎ +c₁x₍₂₎ and derive admissible estimators in these classes. The minimazity of certain estimators of interest is investigated. Itis shown that p(p+1)^-1x₍₁₎ is minimax and dominates the UMVUE. Also UMVUE is not minimax.     

The multivariate generalized waring distribution

The robust procedure given by Balakrishnan, Tiku and Shaarawi (1985) for classifying an observation in one of the two univariate populations _πand _πare generalized to situations where the two populations differ not only in their means but their variances also.     

A distribution function estimator for the difference of order statistics from two independent samples

For two independent populations X and Y we develop the empirical distribution function estimator for the difference of order statistics of the form X _(i)−Y _(j). The key practical application for this estimator pertains to inference between quantiles from two independent populations.     

MODELING THE LIFETIME OF LONGITUDINAL ELEMENTS

In the study of the stochastic behaviour of the lifetime of an element as a function of its length, it is often observed that the failure time (or lifetime) decreases as the length increases. In probabilistic terms, such an idea can be expressed as follows. Let T be the lifetime of a specimen of length x, so the survival function, which denotes the probability that an element of length x survives till time t, will be given by S_T (t, x) = P(T > t/α(x), where α(x) is a monotonically decreasing function. In particular, it is often assumed that T has a Weibull distribution. In this paper, we propose a generalization of this Weibull model by assuming that the distribution of T is Generalized gamma (GG). Since the GG model contains the Weibull, Gamma and Lognormal models as special and limiting cases, a GG regression model is an appropriate tool for describing the size effect on the lifetime and for selecting among the embedded models. Maximum likelihood estimates are obtained for the GG regression model with α(x) = cx^b . As a special case this provide an alternative to the usual approach to estimation for the GG distribution which involves reparametrization. Related parametric inference issues are addressed and illustrated using two experimental data sets. Some discussion of censored data is also provided.     

A two stage procedure for selecting δ∗-optimal means in the normal model

Suppose π₁,…,π_k are k normal populations with π_i having unknown mean μ_i and unknown variance σ². The population π_i will be called δ^?-optimal (or good) if μ_i is within a specified amountδ? of the largest mean. A two stage procedure is proposed which selects a subset of the k populations and guarantees with probability at least P^? that the selected subset contains only δ?-optimal π_i ’s. In addition to screening out non-good populations the rule guarantees a high proportion of sufficiently good π_i’S will be selected.     

Testing stability in a spatial unilateral autoregressive model

ABSTRACT

Least squares estimator of the stability parameter ? ? |α| + |β| for a spatial unilateral autoregressive process X_{k, ?} = αX_{k ? 1, ?} + βX_{k, ? ? 1} + ?_{k, ?} is investigated and asymptotic normality with a scaling factor n^5/4 is shown in the unstable case ? = 1. The result is in contrast to the unit root case of the AR(p) model X_k = α₁X_{k ? 1} + ??? + α_pX_{k ? p} + ?_k, where the limiting distribution of the least squares estimator of the unit root parameter ? ? α₁ + ??? + α_p is not normal.     

ESTIMATING THE ANGLE BETWEEN THE MEAN DIRECTIONS OF TWO SPHERICAL DISTRIBUTIONS

This paper considers the problem of calculating a confidence interval for the angular difference between the mean directions of two spherical random variables with rotationally symmetric unimodal distributions. For large sample sizes, it is shown that the asymptotic distribution of 1 – cos α, where α is the sample angular difference, is approximately exponential if the true difference is zero, and approximately normal for a ‘large’ true difference; a scaled beta approximation is determined for the general case. For small sample sizes, a bootstrap approach is recommended. The results are applied to two sets of palaeomagnetic data.     

An Estimation Problem with Poisson Processes

For n independent Poisson processes such that the i th process has intensity function lM_i(t) =δ_iρ(t; α) we consider estimation of p(t; α) =∫_oρ:(u; α) du. Two procedures are developed, one using exact arrival times, the other using categorical arrival times. Two instances where p(t; α) =p(α t) are investigated further. An example applying the methodology to the active life of a judicial opinion is described.     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

UNBIASED ESTIMATION OF PARAMETRIC FUNCTIONS IN SAMPLING FROM TWO ONE-TRUNCATION PARAMETER FAMILIES

We consider the problem of minimum variance unbiased estimation of a U-estimable function of two unknown truncation parameters based on independent random samples from two one-truncation parameter families. In particular, we obtain the UMVU estimator of the probability that Y > X.     

RATIO-TO-SIZE ESTIMATORS OF MEAN PER SUBUNIT IN TWO-STAGE SAMPLING OVER TWO OCCASIONS

We propose here some strategies for estimating the population mean per subunit at the current occasion and change in mean from one occasion to the next based on two-stage sampling on two successive occasions. Estimators of the mean in two-stage sampling over successive occasions have so far been based on the knowledge of the total number of subunits (elements), M⁰ in the population or on assumed equal sizes for the primary units. We have therefore given the ratio-to-size estimators for the population mean per subunit on the current occasion and the change in mean over the two occasions where M⁰ is not known. The results obtained also apply to the situation where M¹ is correlated with the variable of interest y.     

THE CONVERGENCE RATE OF SEQUENTIAL FIXED-WIDTH CONFIDENCE INTERVALS FOR REGULAR FUNCTIONALS

Let X₁X₂,.be i.i.d. random variables and let U_n= (n r)^-1S?^(n,r) h (X_i1,., X_ir,) be a U-statistic with EU_n= v, v unknown. Assume that g(X₁) =E[h(X₁,.,X_r) - v |X₁]has a strictly positive variance s?². Further, let a be such that φ(a) - φ(-a) =α for fixed α, 0 < α < 1, where φ is the standard normal d.f., and let S²_n be the Jackknife estimator of n Var U_n. Consider the stopping times N(d)= min {n: S²_n: + n^-1≤²a^-2},d > 0, and a confidence interval for v of length 2d,of the form I_n,d= [U_n,-d, Un + d]. We assume that Var U_n is unknown, and hence, no fixed sample size method is available for finding a confidence interval for v of prescribed width 2d and prescribed coverage probability α Turning to a sequential procedure, let I_N(d),d be a sequence of sequential confidence intervals for v. The asymptotic consistency of this procedure, i.e. lim_{d → 0}P(v ∈ I_N(d),d)=α follows from Sproule (1969). In this paper, the rate at which |P(v ∈ I_N(d),d) converges to α is investigated. We obtain that |P(v ∈ I_N(d),d) - α| = 0 (d^{1/2-(1+k)/2(1+m)}), d → 0, where K = max {0,4 - m}, under the condition that E|h(X₁, X_r)|^m < ∞m > 2. This improves and extends recent results of Ghosh & DasGupta (1980) and Mukhopadhyay (1981).     

BAYESIAN ANALYSIS FOR THE SUPERPOSITION OF TWO DEPENDENT NONHOMOGENEOUS POISSON PROCESSES

ABSTRACT

A Bayesian analysis for the superposition of two dependent nonhomogenous Poisson processes is studied by means of a bivariate Poisson distribution. This particular distribution presents a new likelihood function which takes into account the correlation between the two nonhomogenous Poisson processes. A numerical example using Markov Chain Monte Carlo method with data augmentation is considered.