Search

Let X(1)≤X(2)≤...≤X(2m+1) be an ordered sample of a random variable X, which has a median μ and a continuous probability distribution function. We consider the statistic Sm,r=(X(m+1)−μ)/(X(m+1+r)−X(m+1−r)), for some integer r, 1≤r≤m. This statistic is independent of the scale parameter and has other properties similar to Student’s t, and can be computed when the values of X(m+1), X(m+1+r), X(m+1−r) are known while some or all other sample values are not available (e.g. are censored out). This paper presents tables of exact critical values of Sm,r for small sample sizes, under the assumption that X has a normal probability distribution. Furthermore, a table is also given of asymptotic critical values for Sm,r, which can be used for large sample sizes even when X is not normally distributed. These tables may make possible the practical use of Sm,r in some situations in which Student’s t cannot be used.

Let X(1) ≤ X (2) ≤ … ≤ X (n) be an ordered sample of a random variable X which has continuous probability distribution function F (x), and let Fn (x) be the corresponding empirical distribution function. The following three statistics, introduced by A. Renyi, are considered: The paper presents table of exact probabilities for these statistics for finite sample sizes. The limiting distributions of these statistics for sample size n ∞ are discussed, and sample sizes are indicated for which these limiting distributions can be used instead of the exact distributions. Numerical examples for the use of the tables are presented, as well as applications to testing hypotheses on life distributions and to one-sided estimation of probability distribution functions from censored data.

SummaryA new two parameter family of life length distributions is presented which is derived from a model for fatigue. This derivation follows from considerations of renewal theory for the number of cycles needed to force a fatigue crack extension to exceed a critical value. Some closure properties of this family are given and some comparisons made with other families such as the lognormal which have been previously used in fatigue studies.

A statistical model for life-lengths of structures under dynamic loading is derived. The model makes it possible to express the probability distribution of life-lengths in terms of the load given as a function of time and of deterioration occurring in time independently of loading. The special case of a constant load (or of periodic loading with constant amplitude) leads to gamma distributions for life lengths.

A technique is presented for the treatment of errors introduced into sampling surveys due to the non-availability of respondents. The expected cost and variance of the sample survey are expressed as functions of sample size and of the number of call-backs made on the non-availables. A method is then presented which optimizes precision for a given cost by playing sampling error against the bias resulting from non-availables. * Research under the sponsorship of the Office of Naval Research. Presented to the Institute of Mathematical Statistics, November 27, 1948.

SummaryThe estimation problem is studied for a new two-parameter family of life length distributions which has been previously derived from a model of fatigue crack growth. Maximum likelihood estimates of both parameters are obtained and iterative computing procedures are given and examined. A simple estimate of the median life is exhibited, shown to be consistent and then compared, favorably, with the maximum likelihood estimate. More important, the asymptotic distribution of this estimate is shown to be within the same class of distributions as the observations themselves. This model, and these estimation procedures, are tried by fitting this distribution to several extensive sets of fatigue data and then some comparisons of practical significance are made.

A class of statistics is considered, which are based on order-statistics and have properties analogous to Student's $t$. They can be used to estimate a required quantile of a random variable or to test hypotheses about a quantile; they are simple to compute, and can be calculated when not all sample values are available, e.g. for censored samples. The limiting distributions of these statistics are derived and shown to be independent of the distribution of the underlying random variable. A numerical tabulation of the limiting distribution is included, for the special case when the quantile considered is the median.