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It is shown that in detecting sequentially a deterministic signal $ psi(t) $ in white noise $ eta(t) $ a similar identity (iii) in theorem 2.1, to the Wald's holds concerning a stopping time $ \tau $ determined by making use of a likelihood ratio. It is also shown that $ \tau $ has finite moments of any order under quite weak conditions over the signal. The exact A. S. N. $ E{\tau} $ in a constant signal case has been obtained and given by (2, 8). It is also considered a detection problem of a constant signal $ psi(t) equiv alpha $ in a coloured noise based on a sub-optimal statistic which become optimal when the noise were white. Similar properties of a stopping time $ \tau $ to those in the white noise case have been obtained in theorem 3.1.

For a wide class of stationary random sequences possessing a spectral density function, the variance of the best linear unbiased estimator for the mean is seen to depend asymptotically only on the behavior of the spectral density near the origin. Asymptotically efficient estimators based only on this behavior may be chosen. For spectral densities behaving like $\lambda^\nu$ at the origin, $\nu > -1$ a constant, the minimum variance decreases like $n^{-\nu-1}$, where $n$ is the sample size. Asymptotically efficient estimators depending on $\nu$ are given. Finally, the consequences of over- or under-estimating the value of $\nu$ in choosing an estimator are considered.

An asymptotic expression is given for the probability integral of range for samples from a symmetrical unimodal population. Its accuracy is investigated for the case of a normal parent population and for sample sizes from 20 to 100. Over this range errors are small, and by using a correction based on values given below the probability integral can be found with a maximum error of 0.0001. Percentage points of range in the normal case are tabled for $n =$ 20, 40, 60, 80 and 100.

A new concept of unbiasedness (density unbiasedness) for point estimates is introduced and the "best" density unbiased point estimate for the mean of any normal distribution is proved to be the ordinary sample mean $\bar{x} = \sum^n_{i = 1} x_i/n$. Under certain conditions on the form of the characteristic function of a family of probability density functions involving an unknown location parameter, $\bar{x}$ is shown to be a density unbiased point estimate of the location parameter.

Suppose $X$ is a chance variable taking values in $k$-dimensional Euclidean space. That is, $X = (Y_1, \cdots, Y_k),$ where $Y_i$ is a univariate chance variable. The joint distribution of $(Y_1, \cdots, Y_k)$ has density $f(y_1, \cdots, y_k),$ say. We shall call a function $h(y_1, \cdots, y_k)$ "piecewise continuous" if it is everywhere bounded, and $k$-dimensional Euclidean space can be broken into a finite number of Borel-measurable subregions, such that in the interior of each subregion $h(y_1, \cdots, y_k)$ is continuous, and the set of all boundary points of all subregions has Lebesgue measure zero. We assume that $f(y_1, \cdots, y_k)$ is piecewise continuous. Let $h(y_1, \cdots, y_k)$ be some given nonnegative piecewise continuous function, and let $X_1, \cdots, X_n$ be independent chance variables, each with the density $f(y_1, \cdots, y_k).$ Choose a nonnegative number $t,$ and for each $i$, construct a $k$-dimensional sphere with center at $X_i = (Y_{i1}, \cdots, Y_{ik})$ and of $k$-dimensional volume $$\frac{th(Y_{i1}, \cdots, Y_{ik})}{n}.$$ Such a sphere will be called "of type $s$" if it contains exactly $s$ of the $(n - 1)$ points $X_1, \cdots, X_{i-1}, X_{i+1}, \cdots, X_n.$ Let $R_n(t; s)$ denote the proportion of the $n$ spheres which are of type $s$. For typographical simplicity, we denote the vector $(y_1, \cdots, y_k)$ by $y.$ Let $S(t; s)$ denote the multiple integral $$(t^s/s!) \int^\infty_{-\infty} \cdots \int^\infty_{-\infty} h^s(y)f^{s+1}(y) \exp \{-th(y)f(y)\} dy_1 \cdots dy_k.$$ It is shown that $R_n(t; s)$ converges stochastically to $S(t; s)$ as $n$ increases. This result is then used to construct a test of the hypothesis that the unknown density $f(y)$ is equal to a given density $g(y).$

When proving results on the asymptotic behavior of estimates of the spectrum of a stationary time-series, it is invariably assumed that as the sample size $T$ tends to infinity, so does the truncation point $M_T$, but at a slower rate, so that $M_TT^{-1}$ tends to zero. This is a convenient assumption mathematically in that, in particular, it ensures consistency of the estimates, but it is unrealistic when such results are used as approximations to the finite case where the value of $M_TT^{-1}$ cannot be zero. We derive a formula for the asymptotic variance on the assumption that $M_TT^{-1}$ tends to a constant $\gamma$; a more accurate approximation to the variance in the finite case is then obtained by using this formula with $\gamma$ equal to the actual value of $M_TT^{-1}$. Numerical comparisons are made in the white noise case.

When a curve is fitted to a set of data by moments, the usual procedure used in testing the hypothesis that the population is of the given form with the parameters as computed from the moments is to compare the higher moments with their expected values as determined by the hypothesis. Generally speaking, moments about the mean are computed although the reason for this is not clear. To shed some light on this question, the sample given in the introduction is fitted to two curves. Moments about various points are compared with their expected values and the discrepancy in standard units examined. This discrepancy is found to vary widely and to have a maximum. The notion of equivalent moment tests is introduced, and on this basis the most efficient moment test is defined in such a way that of all equivalent moment tests, this one is most likely to reject a false hypothesis. For any moment it is shown that there is a point about which its variance is a minimum. The conditions are found which determine the position of this point for second and third moments. It is proved that for symmetrical populations the variance is minimal when the moments are computed about the mean of the population. If the population is an asymmetrical Pearson frequency function, it is proved that the point about which the third moment variance is minimal differs more from the mean that does the corresponding point for second moments. The condition is pointed out for which this is true in the general case. The third and fourth standard semi-invariants of second moments of minimal variance are computed and compared to those of the second moment about the mean. The ratios of these are displayed for some populations to illustrate how this may be used to investigate when the approach to normality is more rapid in one case than in the other. Some examples are presented to contrast these and other tests.

The life distribution $H(t)$ of a device subject to shocks governed by a Poisson process is considered as a function of the probabilities $P_k$ of not surviving the first $k$ shocks. Various properties of the discrete failure distribution $P_k$ are shown to be reflected in corresponding properties of the continuous life distribution $H(t)$. As an example, if $P_k$ has discrete increasing hazard rate, then $H(t)$ has continuous increasing hazard rate. Properties of $P_k$ are obtained from various physically motivated models, including that in which damage resulting from shocks accumulates until exceedance of a threshold results in failure. We extend our results to continuous wear processes. Applications of interest in renewal theory are obtained. Total positivity theory is used in deriving many of the results.

The probability that a random process crosses an arbitrary level for the first time is expressed as a Gram—Charlier series, the leading term of which is the Poisson approximation. The coefficients of this series are related to the moments of the number of level crossings. The results are applicable to both stationary and non-stationary processes. Some numerical results are presented for the response process of a linear single-degree-of-freedom oscillator under Gaussian white noise excitation.

In this paper it is shown that in order for the trimmed mean to be asymptotically normal, it is necessary and sufficient that the sample be trimmed at sample percentiles such that the corresponding population percentiles are uniquely defined. (The sufficiency of this condition is well known.) In addition, the (non-normal) limiting distribution of the trimmed mean when this condition is not satisfied is derived, and it is shown that in some situations the use of the trimmed mean may lead to severely biased inferences. Some possible remedies are briefly discussed, including the use of "smoothly" trimmed means.

A table is given to simplify the estimation of the parameters of an incomplete Gamma or type-III distribution. A new procedure is suggested for estimating the parameters of a truncated gamma distribution. The method is considered applicable for doubly truncated gamma distributions, singly or doubly truncated normal distributions, and a beta distribution with known range, either truncated or not. The technique is also considered useful in estimating the parameters of the normal curve for the case of systematic gaps in the observations. The method does not appear to be satisfactory for distributions with finite but unknown ranges.

Let $u_n$ denote the sample correlation coefficient for $n$ observations from a bivariate, equal variance, complex Gaussian distribution. In this note we derive the exact distribution of $u_n$ by extending a method of Mehta and Gurland to the complex case. The asymptotic behavior of $E|u_n|^k$ as $n \rightarrow \infty$ is determined via the method of steepest descent. Applicability of the results to the analysis of certain estimators of spectral parameters of stationary time series is discussed.

It is well known that, if the fourth moment about the mean of a frequency distribution equals the square of the variance, then the frequencies are piled up at exactly two points, namely, the two points that are one standard deviation away from the mean. In this paper is developed a general inequality which describes the piling up of frequency around these two points for the case where the fourth moment exceeds the square of the variance. In a sense, it is shown how "U-shaped" a distribution must be according to its second and fourth moments.

The distribution of the latent vectors of a sample covariance matrix was found by T. W. Anderson [1] (1951) when the population covariance matrix is a scalar matrix, $\Sigma = \sigma^2I.$ The asymptotic distribution for arbitrary $\Sigma$, also, was obtained by T. W. Anderson [3] in 1963. The elements of each latent vector are the coefficients of a principal component (with sum of squares of coefficients being unity). The object of the paper is to obtain the exact distribution of the latent vectors when the observations are obtained from bivariate normal distribution.