Rita Aggarwala

PROGRESSIVE INTERVAL CENSORING: SOME MATHEMATICAL RESULTS WITH APPLICATIONS TO INFERENCE

In this paper, we will introduce a union of two methods of collecting Type-I censored data, namely interval censoring and progressive censoring. We will call the resulting sample a progressively Type-I interval censored sample.We will discuss likelihood point and interval estimation, and simulation of such a censored sample from a random sample of units put on test whose lifetime distribution is continuous. An illustrative example will also be presented. *Dr. Aggarwala has formerly published as R. A. Sandhu.

Some properties of progressive censored order statistics from arbitrary and uniform distributions with applications to inference and simulation

Abstract In this paper, we first establish three properties of progressive Type-II censored order statistics from arbitrary continuous distributions. These properties are then used to develop an algorithm to simulate general progressive Type-II censored order statistics from any continuous distribution, by generalizing the algorithm given recently by Balakrishnan and Sandhu (Sankhya Series B58 (1995), 1–9). We then establish an independence result for general progressive Type-II censored samples from the uniform (0,1) population, which generalizes a result given by Balakrishnan and Sandhu (1995) for progressive Type-II right censored samples. This result is used in order to obtain moments for general progressive Type-II censored order statistics from the uniform (0,1) distribution. This independence result also gives rise to a second algorithm for the generation of general progressive Type-II censored order statistics from any continuous distribution. Finally, best linear unbiased estimators (BLUEs) for the parameters of one- and two-parameter uniform distributions are derived, and the problem of maximum-likelihood estimation is discussed.

RECURRENCE RELATIONS FOR SINGLE AND PRODUCT MOMENTS OF PROGRESSIVE TYPE-II RIGHT CENSORED ORDER STATISTICS FROM EXPONENTIAL AND TRUNCATED EXPONENTIAL DISTRIBUTIONS

In this paper, we establish several recurrence relations satisfied by the single and product moments of progressive Type-II right censored order statistics from an exponential distribution. These relations may then be used, for example, to compute all the means, variances and covariances of exponential progressive Type-II right censored order statistics for all sample sizes n and all censoring schemes (R1, R2, ..., Rm), m≤n. The results presented in the paper generalize the results given by Joshi (1978, Sankhyā Ser. B, 39, 362–371; 1982, J. Statist. Plann. Inference, 6, 13–16) for the single moments and product moments of order statistics from the exponential distribution.To further generalize these results, we consider also the right truncated exponential distribution. Recurrence relations for the single and product moments are established for progressive Type-II right censored order statistics from the right truncated exponential distribution.

Inverting the Pascal Matrix Plus One

Immediately we notice this is basically the same matrix, except with every other subdiagonal multiplied by a factor of negative one. Obviously this result generalizes to any size Pascal matrix, and there is something so beautiful and natural about this result that it hardly needs a proof (see Call and Velleman [2] for a particularly elegant demonstration). In fact, it is interesting to note that there is no reason to stop at a finite matrix, for one can extend the Pascal matrix to an infinite lower triangular matrix and verify row-by-row that the inverse is given as

RELATIONSHIPS FOR MOMENTS OF ORDER STATISTICS FROM THE RIGHT-TRUNCATED GENERALIZED HALF LOGISTIC DISTRIBUTION

In this paper, we establish several recurrence relations satisfied by the single and the product moments for order statistics from the right-truncated generalized half logistic distribution. These relationships may be used in a simple recursive manner in order to compute the single and the product moments of all order statistics for all sample sizes and for any choice of the truncation parameter P. These generalize the corresponding results for the generalized half logistic distribution derived recently by Balakrishnan and Sandhu (1995, J. Statist. Comput. Simulation, 52, 385–398).

Conditional Inference for the Parameters of Pareto Distributions when Observed Samples are Progressively Censored

In this paper we develop procedures for obtaining confidence intervals for the location and scale parameters of a Pareto distribution as well as upper and lower γ probability tolerance intervals for a proportion β when the observed samples are progressively censored. The intervals are exact, and are obtained by conditioning on the observed values of the ancillary statistics. Since the procedures assume that the shape parameter ν is known, a sensitivity analysis is also carried out to see how the procedures are affected by changes in ν.

Folding Distributions for Order Statistics

Abstract Govindarajulus classic result in order statistics, which relates the moments of order statistics from folded distributions to those from symmetric distributions, is inverted to provide explicit formulae for the reverse relationship.

A Robust Algorithm For Constant-Q Wavelet Estimation Using Gabor Analysis

Seismic attenuation can be modeled macroscopically via an exponential amplitude decay in both time and frequency, at a rate determined by a single dimensionless quantity, Q. Current deconvolution methods, based on the convolutional model, attempt to estimate and remove the embedded causal wavelet. We propose a nonstationary seismic model, expressed in the time-frequency domain, in which (1) the embedded causal wavelet factors as the product of a stationary seismic signature and a nonstationary exponential decay; and (2) a nonstationary impulse response for the earth is tractable. Least squares fitting our model to the Gabor-transformed seismic trace yields a Q-value and an estimate of the source signature, hence an estimate of the nonstationary wavelet. These estimates lead to a smoothed version of the magnitude of the Gabor spectrum of the seismic trace, from which a least-squares nonstationary minimum-phase deconvolution filter is easily constructed. Our preliminary results on synthetic data are very promising.

Moments of Order Statistics—From Even to Odd Sample Sizes

Abstract A method is demonstrated to compute the complete set of first moments of order statistics for an arbitrary distribution, given only the first moments of the maximal order statistics either for all even sample sizes, or for all odd samples sizes.

Recursive Computation and Algorithms

In the next two chapters, considerable emphasis will be placed on obtaining moments of progressively Type-II censored order statistics. One reason for this will become clear in Chapters 6 and 10 when the focus is on developing efficient inferential procedures based on progressive Type-II censoring. We have seen in the last chapter that moments of progressively censored order statistics from some distributions can be obtained explicitly. However, very often, it is not possible to obtain explicit expressions for these moments. The idea of obtaining moments of usual order statistics in a recursive manner has been explored for a number of distributions; for example, see Balakrishnan, Malik and Ahmed (1988), and Balakrishnan and Sultan (1998). In this chapter, we establish several recurrence relations satisfied by the single and product moments of progressively Type-II right censored order statistics from the exponential, Pareto and power function distributions as well as their truncated forms. It is important when establishing such recurrence relations that the relations be complete,in the sense that they may be used in a simple recursive manner in order to compute all the single and product moments of all progressively T-II right censored order statistics from the distributions of interest for all sample sizes n and all censoring schemes (R1,R2, ⋯, R), m ≤ n. This, in fact, is the case for all the distributions mentioned above.