Brad C. Johnson

Design based estimation for ranked set sampling in finite populations

In this paper, we consider design-based estimation using ranked set sampling (RSS) in finite populations. We first derive the first and second-order inclusion probabilities for an RSS design and present two Horvitz–Thompson type estimators using these inclusion probabilities. We also develop an alternate Hansen–Hurwitz type estimator and investigate its properties. In particular, we show that this alternate estimator always outperforms the usual Hansen–Hurwitz type estimator in the simple random sampling with replacement design with comparable sample size. We also develop formulae for ratio estimator for all three developed estimators. The theoretical results are augmented by numerical and simulation studies as well as a case study using a well known data set. These show that RSS design can yield a substantial improvement in efficiency over the usual simple random sampling design in finite populations.

Radix- b extensions to some common empirical tests for pseudorandom number generators

Empirical testing of computer generated pseudo-random sequences is widely practiced. Extensions to the <italic>coupon collectors</italic> and <italic>gap</italic> tests are presented that examine the distribution and independence of radix-<italic>b</italic> digit patterns in sequences with modulo of the form <italic>b<supscrpt>w</supscrpt></italic>. An algorithm is given and the test is applied to a number of popular generators. Theoretical expected values are derived for a number of defects that may be present in a pseudorandom sequence and additional empirical evidence is given to support these values. The test has a simple model and a known distribution function. It is easily and efficiently implemented and easily adaptable to testing only the bits of interest, griven a certain application.

APPROXIMATE PROBABILITIES FOR RUNS AND PATTERNS IN I.I.D. AND MARKOV-DEPENDENT MULTISTATE TRIALS

Let X n (Λ) be the number of nonoverlapping occurrences of a simple pattern Λ in a sequence of independent and identically distributed (i.i.d.) multistate trials. For fixed k, the exact tail probability P{Xn (∧) < k} is difficult to compute and tends to 0 exponentially as n → ∞. In this paper we use the finite Markov chain imbedding technique and standard matrix theory results to obtain an approximation for this tail probability. The result is extended to compound patterns, Markov-dependent multistate trials, and overlapping occurrences of Λ. Numerical comparisons with Poisson and normal approximations are provided. Results indicate that the proposed approximations perform very well and do significantly better than the Poisson and normal approximations in many cases.

Efficient generation of exponential and normal deviates

We present efficient procedures for generating random exponential and normal deviates based on the acceptance-complement method [Kronmal, R.A. and Peterson, A.V., 1981, Journal of the American Statistical Association, 76, 446–451.]. We provide comparisons with the corresponding Ziggurat procedures proposed by Marsaglia and Tsang [Marsaglia, G. and Tsang, W.W., 1984, SIAM Journal on Scientific and Statistical Computing, 5, 349–359; Marsaglia, G. and Tsang, W.W., 2000, Journal of Statistical Software, 5(8), 1–7.]. The proposed procedures maintain good precision over the entire support of the respective densities and are very easy to set up and implement. The proposed exponential procedure compares favourably with the Ziggurat procedure in terms of speed, running up to 24% faster on some platform/compiler/uniform generator combinations tested.

The distribution of increasing l-sequences in random permutations: a Markov chain approach

This paper examines the distribution of increasing l-sequences in a random permutation generated by the integers 1,...,n; an increasing l-sequence being a sequence of l consecutive integers. The exact distribution is obtained by a finite Markov chain embedding technique in combination with a conditioning argument. The result is easy to implement, intuitively simple and can be generalized.

Approximating the distributions of runs and patterns

The distribution theory of runs and patterns has been successfully used in a variety of applications including, for example, nonparametric hypothesis testing, reliability theory, quality control, DNA sequence analysis, general applied probability and computer science. The exact distributions of the number of runs and patterns are often very hard to obtain or computationally problematic, especially when the pattern is complex and n is very large. Normal, Poisson and compound Poisson approximations are frequently used to approximate these distributions. In this manuscript, we (i) study the asymptotic relative error of the normal, Poisson, compound Poisson and finite Markov chain imbedding and large deviation approximations; and (ii) provide some numerical studies to comparing these approximations with the exact probabilities for moderately sized n. Both theoretical and numerical results show that, in the relative sense, the finite Markov chain imbedding approximation performs the best in the left tail and the large deviation approximation performs best in the right tail.AMS Subject ClassificationPrimary 60E05; Secondary 60J10

The distribution of increasing 2-sequences in random permutations of arbitrary multi-sets

The distribution of increasing 2-sequences in random permutations of the first n integers is generalized to random permutations of arbitrary multi-sets using a finite Markov chain embedding technique. A numerical example is provided to aid in understanding and some applications are briefly discussed.

Distribution of Increasing ℓ-sequences in a Random Permutation

AbstractThis paper examines the distribution of the number, k, of increasing ℓ-sequences in a random permutation of

Confidence intervals for quantiles in finite populations with randomized nomination sampling

\left\{ {1,...,n} \right\}