Decision-Making for the Lifetime Performance Index

The purpose of this research is to develop a maximum likelihood estimator (MLE) for lifetime performance index CL for the parameter of mixture Rayleigh-Half Normal distribution (RHN) under progressively type-II right-censored samples under the constraint of knowing the lower specification limit (L). Additionally, we suggest an asymptotic normal distribution for the MLE for CL in order to construct a mechanism for evaluating products’ lifespan efficiency. We have specified all the steps to carry out the test. Additionally, not only does hypothesis testing successfully assess the lifetime performance of items, but it also functions as a supplier selection criterion for the consumer. Finally, we have added two real data examples as illustration examples. These two applications are provided to demonstrate how the results can be applied.

Asymptotic Normality of Sequential Stopping Times with Applications: Confidence Intervals for an Exponential Mean

In sequential methodologies, finally accrued data customarily look like [Formula: see text] where [Formula: see text] is the total number of observations collected through termination. Under mild regulatory conditions, a standardized version of [Formula: see text] follows an asymptotic normal distribution (Ghosh–Mukhopadhyay theorem) which we highlight with a number of illustrations from the recent literature for completeness. Then, we emphasize the role of such asymptotic normality results along with second-order approximations for stopping times in the construction of sequential fixed-width confidence intervals for the mean in an exponential distribution. Two kinds of confidence intervals are developed: (a) one centred at the randomly stopped sample mean [Formula: see text] and (b) the two other centred at appropriate constructs using the stopping variable [Formula: see text] alone. Ample comparisons among all three proposed methodologies are summarized via simulations. We emphasize our finding that the two fixed-width confidence intervals centred at appropriate constructs using the stopping variable [Formula: see text] alone perform as well or better than the customary one centred at the randomly stopped sample mean.

DRAWING MULTISETS OF BALLS FROM TENABLE BALANCED LINEAR URNS

We investigate the evolution of an urn of balls of two colors, where one chooses a pair of balls and observes rules of ball addition according to the outcome. A nonsquare ball addition matrix of the form $\left( \matrix{a & b \cr c & d \cr e & f}\right)$ corresponds to such a scheme, in contrast to pólya urn models that possess a square ball addition matrix. We look into the case of constant row sum (the so-called balanced urns) and identify a linear case therein. Two cases arise in linear urns: the nondegenerate and the degenerate. Via martingales, in the nondegenerate case one gets an asymptotic normal distribution for the number of balls of any color. In the degenerate case, a simpler probability structure underlies the process. We mention in passing a heuristic for the average-case analysis for the general case of constant row sum.

A CONSISTENT NONPARAMETRIC TEST FOR CAUSALITY IN QUANTILE

This paper proposes a nonparametric test of Granger causality in quantile. Zheng (1998, Econometric Theory 14, 123–138) studied the idea to reduce the problem of testing a quantile restriction to a problem of testing a particular type of mean restriction in independent data. We extend Zheng’s approach to the case of dependent data, particularly to the test of Granger causality in quantile. Combining the results of Zheng (1998) and Fan and Li (1999, Journal of Nonparametric Statistics 10, 245–271), we establish the asymptotic normal distribution of the test statistic under a β-mixing process. The test is consistent against all fixed alternatives and detects local alternatives approaching the null at proper rates. Simulations are carried out to illustrate the behavior of the test under the null and also the power of the test under plausible alternatives. An economic application considers the causal relations between the crude oil price, the USD/GBP exchange rate, and the gold price in the gold market.

On the Variety of Shapes on the Fringe of a Random Recursive Tree

We consider a variety of subtrees of various shapes lying on the fringe of a recursive tree. We prove that (under suitable normalization) the number of isomorphic images of a given fixed tree shape on the fringe of the recursive tree is asymptotically Gaussian. The parameters of the asymptotic normal distribution involve the shape functional of the given tree. The proof uses the contraction method.

On the Variety of Shapes on the Fringe of a Random Recursive Tree

We consider a variety of subtrees of various shapes lying on the fringe of a recursive tree. We prove that (under suitable normalization) the number of isomorphic images of a given fixed tree shape on the fringe of the recursive tree is asymptotically Gaussian. The parameters of the asymptotic normal distribution involve the shape functional of the given tree. The proof uses the contraction method.

Derangement Polynomials and Excedances of Type $B$

Based on the notion of excedances of type $B$ introduced by Brenti, we give a type $B$ analogue of the derangement polynomials. The connection between the derangement polynomials and Eulerian polynomials naturally extends to the type $B$ case. Using this relation, we derive some basic properties of the derangement polynomials of type $B$, including the generating function formula, the Sturm sequence property, and the asymptotic normal distribution. We also show that the derangement polynomials are almost symmetric in the sense that the coefficients possess the spiral property.

The Precision of Gain Scores Under an Item Response Theory Perspective: A Comparison of Asymptotic and Exact Conditional Inference About Change

The precision of simple difference or “gain” scores is described in terms of their confidence intervals on the latent trait scale and of significance probabilities under the H₀ of no change. For this, two approaches are compared: one employs the asymptotic normal distribution of the maximum likelihood estimator of the person parameter, the other is based on the exact conditional distribution of the gain score, given the total number-correct score over the two time points. In either case, a detailed assessment of the precision of change measurements results. For illustration, results are presented of three test scales. The present methods yield more relevant and much more detailed psychometric information than the traditional estimation of reliability as a sole indicator of measurement precision. Other areas of application, namely, the comparison of the abilities of two examinees or the aggregation of individual signi.cance levels within groups of examinees, are also mentioned.

Central limit theorem for germination-growth models in ℝd with non-Poisson locations

Seeds are randomly scattered in ℝ d according to an m-dependent point process. Each seed has its own potential germination time. From each seed that succeeds in germinating, a spherical inhibited region grows to prohibit germination of any seed with later potential germination time. We show that under certain conditions on the distribution of the potential germination time, the number of germinated seeds in a large region has an asymptotic normal distribution.

Central limit theorem for germination-growth models in ℝd with non-Poisson locations

Seeds are randomly scattered in ℝd according to an m-dependent point process. Each seed has its own potential germination time. From each seed that succeeds in germinating, a spherical inhibited region grows to prohibit germination of any seed with later potential germination time. We show that under certain conditions on the distribution of the potential germination time, the number of germinated seeds in a large region has an asymptotic normal distribution.