In [4], Doob shows that $F^\ast(x) = \mu^{-1} \int^x_0 \lbrack 1 - F(u)\rbrack du$ is a stationary probability measure for a renewal process when the common distribution function $F$ has a finite mean $\mu$. In [2], Derman shows that an irreducible, null recurrent Markov chain (MC) has a unique positive stationary measure. In this paper, similar results are obtained for a class of irreducible recurrent Markov renewal processes (MRP). Since MRP's are generalizations of MC's and renewal processes these results generalize those mentioned above. Stationary measures are also derived for a class of MRP's with auxiliary paths.

Let $H$ be a set and $\{T_n, n = 1, 2, \cdots\}$ a sequence of transformations of $H$ into itself. Let $X_1$ and $\{U_n\}$ be random elements in $H$ and generate the sequence $\{X_n\}$ by $X_{n + 1} = T_n(X_n) + U_n.$ Theorems giving conditions under which $\{X_n\}$ is "stochastically attracted" towards a given subset of $H$ and will eventually be within or arbitrarily close to this set in an appropriate sense, are called Dvoretzky stochastic approximation theorems. The main results of this paper (Theorems 1, 2 and 3) are of this type. They generalize the work of Dvoretzky [6] and Wolfowitz [12] for the case $H$ equal to the real line, of Derman and Sacks [4] for the case $H$ a finite dimensional Euclidian space and Schmetterer [11] for the case $H$ a Hilbert space.

Let $\{T_i: i = 1, 2, \cdots\}$ be a stochastic process of Poisson type, with $\lambda(t)$, the rate of occurrence of the events, depending on time. We may interpret $T_i$ as the time of occurrence of the $i$th event. In Section 2, starting with the joint density function of $T_1, \cdots, T_n$, the maximum likelihood estimate of $\lambda(t)$ subject to $0 \leqq \lambda (t) \leqq M$ being a non-decreasing function of time ($M$ some positive number) is found. In Section 3, starting with the conditional joint density of $T_1, \cdots, T_n$ given there are $n$ events in $(0, t^\ast\rbrack$, the conditional maximum likelihood estimate of $\lambda(t)$ subject to $0 \leqq \lambda(t)$ being a non-decreasing function of time is found. In Section 4, the conditional likelihood ratio test of the hypothesis that $\lambda(t)$ is constant against the alternate hypothesis that $\lambda(t)$ is not constant but is nondecreasing is found, and a limiting distribution is found which may be used to approximate the probability of a type I error for large sample size. Theorem 2.1 (Brunk-van Eeden), I believe is important in its own right. It is contained in the works of Brunk and van Eeden, although it is not explicitly stated. This theorem can be used as a basis for tests of hypotheses for constant parameters against increasing parameters or for increasing parameters against all other alternatives.

The property ($W$) of the class of decision functions, which corresponds to the concept of weak compactness in the intrinsic sense in [6], is discussed and several sufficient conditions for it are given in this article. Some examples concerning the non-sequential case are discussed.

For the general statistical decision problem, Wald suggested the minimax solution as one possible choice of an optimum solution. Sufficient conditions for the existence of minimax solutions are well known. However, minimax solutions have been calculated only for isolated problems. A drawback of the methods that have been used to obtain minimax solutions is that there is no guarantee that they will yield a minimax solution when they are applied to a specific new problem. An exception to this is M. N. Ghosh's method [2] of approximating minimax estimators for a certain class of problems. Since his method differs from the method presented here, no further mention of his work will be made. Presented in Section 3 is an iterative method of calculating minimax solutions that is applicable to a class of problems. The method is based on the result that, under certain conditions, if, for a sequence of a priori distributions on the parameter space, the corresponding sequence of Bayes risks converges to the supremum of all Bayes risks, then the corresponding sequence of Bayes decision functions converges to the minimax decision function and the corresponding sequence of risk functions converges uniformly to the minimax risk function. A method for iteratively constructing such a sequence of a priori distributions is given. The derivation in Section 3 of the iterative method of calculating minimax solutions depends on some well-known results of Wald [5]. These results are stated in Section 2 for convenience of reference and are proved for the sake of completeness, although many of the proofs are available in [5] or elsewhere. Also, previously no exposition of the results of Section 2 was available in the literature with the mathematical presentation used here. The assumptions used here to obtain Wald's results are weaker than those used in [5] and are similar to Wald's Assumptions 5.1 through 5.6 in [6]. However, in [6] Wald did not give all of the results of Section 2. Also, the results hold for a larger class of problems than Wald indicated. Wald's restriction to fixed sample size problems is unnecessary; the results apply to sequential problems with prescribed sequential experimentation rule of the type considered by Wald in [6]. In [6], Wald treats discrete and continuous random variables separately. Here the results are given in a generality that includes discrete and continuous random variables as special cases.

It is well-known that there is a close connection between linear functionals on an appropriate Banach space and unbiased estimators. In Section 2 we prove some results concerning unbiased estimation of location and scale parameters. As application of these results we consider the case of Cauchy density with unknown location [scale] but known scale [location] parameter. We show that there exists no unbiased estimator for the location parameter, and none with finite variance for the scale parameters. If the Cauchy density involves both location and scale parameters, then it is shown that neither of these parameters has an unbiased estimator. Some information about other parametric functions is also given. The present results for the location parameter case were obtained previously by H. Pollard; we are grateful to Professor Kiefer for informing us of Pollard's work.

The problem of estimating a linear transformation between two finite dimensional vector spaces is considered, when the observed vectors in both spaces are subject to error, and there is an indefinitely increasing number of replications of a fixed number of treatments. A general class of ordinary estimators is defined, and it is shown that, in the case of homogeneity of variances, the simple least squares estimators are asymptotically efficient within the class of ordinary estimators, in the sense that they minimize, within that class, the asymptotic mean square error of prediction. When the error variances are unequal, however, the asymptotically efficient estimators are weighted least squares estimators, whose weights are based on preliminary estimators of the linear transformation and the error variances.

It is a common practice, in engineering and applied sciences, to ask for an estimator of a parameter of a statistical distribution which, with high probability, does not deviate from the value of the parameter by more than a certain percentage of its absolute value. In other words, if $\theta$ is a parameter under consideration, and $\hat{\theta}$ is an estimator of $\theta$ it is required that, for given $0 < \delta < 1$, and $0 < \gamma < 1$, \begin{equation*}\tag{0.1}P_\theta\lbrack|\hat{\theta} - \theta| < \delta|\theta|\rbrack \geqq \gamma,\quad\text{for all} \theta.\end{equation*} This probability is called the proportional closeness of $\hat\theta$ (see Ehrenfeld and Littauer [5] p. 339). In the present paper we study the problem of estimating the mean of a log-normal distribution, by a procedure which guarantees a prescribed proportional closeness. When the variance, $\sigma^2$, of the corresponding normal distribution is known, there is an efficient fixed sample estimation procedure having the required closeness property. The sample size required in this case is, \begin{equation*}\tag{0.2}n_0 = \text{smallest integer} \geqq \chi_\gamma^2\lbrack 1\rbrack\sigma^2 \log^{-2} (1 + \delta),\end{equation*} where $\chi_\gamma^2\lbrack 1\rbrack$ denotes the $\gamma$th fractile of the $\chi^2$-distribution, with 1 degree of freedom. As indicated in the sequel, there is no such fixed sample procedure, when $\sigma^2$ is unknown. The prescribed closeness property can be, however, guaranteed if the estimation is based on at least two stages of sampling. The properties of two sequential estimation procedures, which asymptotically (as $\delta \rightarrow 0$) guarantee the prescribed proportional closeness, are presented in the present paper. One procedure is based on the maximum likelihood estimator of the mean, and is called the sequential M.L. procedure. The other procedure is based on the sample mean, and is called the sequential S.M. procedure. Let $v_n$ denote the maximum likelihood estimator of $\sigma^2$. The stopping rule for the sequential M.L. procedure defines the sample size, $K$, to be the first integer $k \geqq 2$, for which the following inequality is satisfied: \begin{equation*}\tag{0.3}v_k \leqq (1 + (2/c_k)k \log^2 (1 + \delta))^{\frac{1}{2}} - 1,\quad 0 < \delta < 1\end{equation*} where $\{c_k\}$ is a sequence of bounded, positive constants, approaching $\chi_\gamma^2\lbrack 1\rbrack$ as $k \rightarrow \infty$. The sample size, $N$, in the sequential S.M. procedure, is the first integer $n \geqq 2$, such, \begin{equation*}\tag{0.4}v_n \leqq \log (1 + (\delta/c_n)n),\quad 0 < \delta < 1\end{equation*} It is proven that both stopping rules, (0.3) and (0.4), yield well-defined stopping variables, which are decreasing functions of $\delta$, and have finite expectations for every $0 < \delta < 1$. The asymptotic orders of magnitude (a.s.) of $K$ and of $N$ are given, as well as the asymptotic order of magnitudes of their expectations (as $\delta \rightarrow 0$). It is shown that the efficiency of the sequential S.M. procedure, relative to that of the sequential M.L. procedure, decreases to zero as $\sigma^2 \rightarrow \infty$. That is, \begin{equation*}\tag{0.5}\lim_{\delta \rightarrow 0, \sigma^2 \rightarrow \infty} E\{K\}/E\{N\} = 0.\end{equation*} Moreover, $\lim_{\delta \rightarrow \infty} E\{K\}/E\{N\} < 1$ for all $0 < \sigma^2 < \infty$. This result establishes the uniform asymptotic superiority (with respect to all $0 < \sigma^2 < \infty$) of the sequential M.L. procedure over the sequential S.M. procedure. The sequential M.L. procedure studied in the present paper is not, however, asymptotically efficient in the Chow-Robbins sense. Chow and Robbins defined in [3] a sequential procedure to be asymptotically efficient, if \begin{equation*}\tag{0.6}\lim_{\delta \rightarrow 0}\frac{E\{\text{sample size in sequential procedure}\}}{\text{minimum sample size required for} \sigma^2 \text{known}} = 1.\end{equation*} It is proven that, for the sequential M.L. procedure \begin{equation*}\tag{0.7}\lim_{\delta \rightarrow 0} E\{K\}/n_0 = 1 + \frac{1}{2}\sigma^2,\quad 0 < \sigma^2 < \infty.\end{equation*} This limit is always greater than 1, and approaches infinity as $\sigma^2 \rightarrow \infty$. A sequential procedure for the log-normal case, which satisfies asymptotically the prescribed closeness condition, and is asymptotically efficient in the Chow-Robbins sense is still unavailable. The reason for this shortcoming is that we have actually to determine a fixed-width confidence interval for $(\mu + \frac{1}{2}\sigma^2)$, where $(\mu, \sigma^2)$ are the mean and variance of the normally distributed $Y = \log X$. Chow and Robbins [3], Gleser, Robbins and Starr [6], and Starr [7] show that a sequential estimation of the mean, based on the sample mean and the sample variance, provides an asymptotically efficient fixed width confidence procedure. This result is in contrast to the main result of the present paper, which shows that a prescribed proportional closeness sequential estimator of the meanof a log-normal distribution based on the sample mean is inefficient, and that there exists a more efficient sequential procedure, which is the one based on the M.L. estimator.

Bell and Doksum (1965) proposed a new class of nonparametric tests which have the advantage of possessing exact and well tabulated distributions under the null hypothesis. The basic departure from the usual nonparametric tests, suggested by the above authors is that of taking an additional sample from a known distribution such as normal, uniform, exponential, etc. and use those observations as the rank scores. Throughout this paper such procedures are called `Randomized Rank Score' (RRS) as opposed to the usual Rank Score (RS) procedures. Bell and Doksum (1965) have shown that the RRS tests have the same Pitman efficiency behaviour as the corresponding RS tests. One could interpret this result by saying that the effect of the superimposed noise of the additional sample dissipates near the null hypothesis as the sample size tends to infinity. However, as will be shown here, for the finite sample sizes the noise does create undesirable properties for the RRS tests. Firstly, in most of the familiar testing problems the power of the RRS test remains bounded away from unity as the parameter varies over the entire region of the alternative. Secondly, the confidence sets based on RRS procedure have the following peculiar property. With positive probability, the confidence set becomes the whole parameter space and the procedure completely disregards the observations of the experiment.

The point of this paper is that likelihood ratios, especially in conjunction with suitable symmetry, yield at least partial operating characteristic (OC) information for procedures other than the sequential probability ratio test (SPRT). Unpublished work of W. J. Hall ([12], [13]) apparently contains related observations to this effect. We find it useful to discuss OC computations in terms of mutually "conjugate" parametric values, which are considered at some length in Section 2; this term appears first in [17], and the concept also elsewhere ([11], [4], [6]), all in the context of the SPRT. In Section 3 the ideas are applied to the Wiener process, for the sorts of two-decision "wedge" procedures discussed in [15], [8], [1], and [12], and, in Section 4, to modifications of the sorts of three-decision procedures discussed in [2], [20], and [3]. Section 5 contains certain special absorption probabilities for Brownian motion; Section 6 is devoted to the binomial case.

In the theory of estimation it is well known that when all the observations in a sample are available, it is sometimes possible to find estimators that are the most efficient linear combinations of a given number of order statistics. In many practical situations, we encounter censored samples, that is, samples where values of some of the observations are not available, and it is desired to obtain linear estimators based on a few optimal order statistics from such a sample. The present study concerns the determination of the optimal set of order statistics for a given integer $k$ (where $k$ is much less than the number of observations in the censored sample), in estimating the parameters of the exponential distribution when the sample is censored. The study is based on the asymptotic theory and under Type II censoring scheme. The problem of estimation of the parameters of the exponential distribution in censored samples has been considered by Sarhan and Greenberg (1957). The choice of optimal set of order statistics for the scale parameter alone in a left censored sample has been studied numerically by Sarhan, Greenberg and Ogawa (1963). For the estimation of the parameters of exponential distribution based on optimal set of order statistics, we present in Section 2 the asymptotically best linear unbiased estimates (BLUE's) of the parameters based on $k$ sample quantiles of given orders when all the sample values are available and define the censored samples considered. In Sections 3-5, the detailed treatment for the determination of the $k$ optimum order statistics in singly and doubly censored sample is presented. In Section 6 some extremal properties of a related function are given. The results are always referred to in the text of Sections 2 to 5 to establish uniqueness of the optimal order statistics so determined. Further, for $k = 2(1)4$ and proportion of censoring the right $1 - \beta = .05(.05).40$, Table I has been prepared for the estimation of the scale parameter (assuming the location parameter known) furnishing the coefficients of the BLUE and the spacings corresponding to the optimum order statistics. For $k = 2(1)4$ and equal proportions of censoring on both sides from .05 to .25 at steps of .05, Table II has been prepared for the simultaneous estimation of the location and the scale parameters.

A theory of simultaneous confidence bounds on certain parametric functions, and their `partials,' associated with some problems in multivariate normal statistical analysis has been developed by S. N. Roy and his associates in a series of publications over the past decade (e.g., [6], [7], [8], also see the references in [8]). In this paper we have obtained simultaneous confidence bounds on the members of a class of parametric functions, together with their `partials,' associated with the two problems mentioned in the title. Our results, not only contain Roy's results as particular cases but, also throw a new light on the parametric functions used by Roy. Furthermore, confidence bounds, which can be considered as being associated with Hotelling's trace criterian for the MANOVA problem can be obtained as an example from our results.

This paper deals with some aspects of nonparametric confidence intervals for shift and scale parameters which may be obtained from a celebrated class of rank order tests for the location and scale problems. This also provides a distribution-free method of estimating asymptotic efficiency of a class of tests and estimates (point as well as intervals) that may be derived from the same class of rank order statistics. Further, the proposed method is also applicable for estimating certain functionals of the distribution function which may not be otherwise estimated in a simple manner.

A strictly distribution-free test has been proposed for testing that several independent pairs of random variables have locations as specified. Under the null hypothesis, the test statistic has asymptotically a chi-square distribution with degrees of freedom two. The test has been shown to be unbiased and consistent against reasonable classes of alternatives. Asymptotic power and efficiency have been examined.

To every matrix $A$ of real valued payoffs there corresponds a unique quantity $V(A)$, the value of a game with payoff matrix $A$, defined by $V(A) = \min_y \max_x x^t Ay = \max_x \min_y x^t Ay,$ where the maximum and minimum are taken over all $x$ and $y$ having nonnegative components summing to one. We wish to show that, for a relatively wide class of probability distributions on the class of all $m \times n$ game matrices, the probability $P(m, n)$ that the value of a random game is greater than zero is given by the cumulative binomial distribution $P(m, n) = (\frac{1}{2})^{m + n - 1} \sum^{m - 1}_{k = 0} \binom{m + n -1}{k}.$ The observation that, for any $\epsilon > 0$, $\lim_{m \rightarrow \infty} P(m, (1 + \epsilon)m) = 0, \\ P(m, m) = \frac{1}{2},\quad \text{for all m}, \\ \lim_{m \rightarrow \infty} P(m, (1 - \epsilon)m) = 1,$ suggests that large rectangular games tend to be strongly biased in favor of the player having the greater number of alternatives. Thus we characterize the extent to which the "nonsquareness" of a game is reflected in a bias for one of the players. Our method involves an application of a theorem in combinatorial geometry due to Schlafli [5]. A discussion of the consequences of Schlafli's theorem in geometrical probability is given in [2], [3], [9].

In the theory of sequential analysis developed by Wald [6] there appear sums of the form $X = \sum^N_1 x_i$ where both the $x_i$ and $N$ are random variables. In this note we shall consider conditions for the existence of $E(X^k)$ when the $x_i$ are independent random variables and the event $N \geqq i$ is independent of $x_i, x_{i + 1}, \cdots$. Let $|x_i| = y_i, Y = \sum^N_1 y_i$. We show that sufficient conditions for $E(Y^k) < \infty$ are that $E(y^k_i) \leqq \beta_k < \infty, E(N^k) < \infty, k = 1, 2, \cdots$ (proved for $k = 1$ in [7]), and that if we can find a constant $c < \infty$ such that $P(y_i \leqq c)E(y_i \mid y_i \leqq c) \geqq \alpha > 0$ for $i = 1, 2, \cdots$, a necessary condition for $E(Y^k) < \infty$ is $E(N^k) < \infty$. We also show that $E(x_i) = 0, E(x^{2k}_i) \leqq \beta_{2k} < \infty, E(N^k) < \infty$ imply that $E(X^{2k}) = \lim_{m \rightarrow \infty} E(X^{2k}_m) < \infty$ for $k = 1, 2, \cdots$ (proved for $k = 1, 2$ in [1], for $k \geqq 3$ proved independently in [5]).

Let the random variable $X$ be distributed according to the probability density $p(x,\omega) = \beta(\omega) \exp (\omega x)$ with respect to a $\sigma$-finite measure $\mu$ defined on the real line, and $\omega$, the unknown state of nature, belongs to the natural parameter space:$\Omega = \{\omega\mid\int^\infty_{-\infty}\exp (\omega x) d\mu(x) < \infty\}$ which is an interval of the real line. Let $\bar\omega$ and $\underline{\omega}$ be the upper and lower end points of $\Omega$, respectively. $\bar\omega$ and $\underset{\bar}{\omega}$ may or may not belong to $\Omega$, and in some cases $\bar\omega = |\infty$ or $\underline{\omega} = -\infty$. The problem under consideration is the estimation of the quantity $\theta(\omega) = E_\omega(x) = -\beta'(\omega)/\beta(\omega)$ from a single observation $x$ on $X$. The loss-function is the standard squared-error loss function. Karlin [1] considered the admissibility of the linear estimates of the form $a_\gamma(x) = \gamma x = x/(\lambda + 1)$ for $0 \leqq \gamma \leqq 1$ and the following theorem of his gives sufficient conditions for the admissibility of $a_\gamma(x)$. Theorem 1.1 (Karlin). Let $p(x,\omega) = \beta(\omega) \exp (\omega x)$ describe the density of the exponential family wrt a measure $\mu$. If \begin{equation*}\tag{1.2)}\int^{\bar\omega}_c \beta^{-\lambda} (\omega) d\omega = +\infty\end{equation*} and \begin{equation*}\tag{1.2}\int^c_\omega \beta^{-\lambda}(\omega) d\omega = +\infty,\end{equation*} where $c$ is an interior point of $\Omega = (\underset{\bar}{\omega}, \bar\omega)$, then $\gamma x = x/(\lambda + 1)$ is an admissible estimate of $\theta(\omega) = E_\omega(X)$. In the sequel we shall refer to the integrals in (1.1) and (1.2) as Karlin's integrals. Karlin [1] conjectured that conditions (1.1) and (1.2) are also necessary for the admissibility of $x/(\lambda + 1)$. We shall, in the following, refer to this as Karlin's conjecture. Let us denote by $I^2(\omega)$ the square of the coefficient of variation given by $\lbrack(\beta'(\omega))^2 - \beta(\omega)\beta''(\omega)\rbrack/(\beta'(\omega))^2$. Suppose $I^2(\omega)$ ranges between $L$ and $L' (L < L')$ as $\omega$ traverses the interval $(\underset{\bar}{\omega}, \bar\omega)$. Karlin [1] showed that $x/(\lambda + 1)$ is inadmissible for all $\lambda < L$ and $\lambda > L'$. While criteria for inadmissibility of $x/(\lambda + 1)$ are given in terms of $L$ and $L'$, the conditions for admissibility are in terms of integrability of $\beta^{-\lambda}(\omega)$ near the end points of $\Omega$. The purpose of this paper is to link up these two criteria and characterize Karlin's integrability conditions in terms of the behavior of $I^2(\omega)$ near $\underline{\omega}$ and $\bar\omega$ (Theorem 2.2). It is also shown (Theorem 2.1) that the range of $\lambda$ for which both of Karlin's integrals are infinite form a sub-interval of $(0, \infty)$. While Karlin's conjecture still remains open, the characterizations we obtained serve to settle the conjecture for a class of cases, an example of which is given in Example 2.1. It is also interesting to regard $I^2(\omega)$ as the ratio of the risks of two linear estimates $x/(\lambda + 1)$ corresponding to $\lambda = 0$ and $\lambda = \infty$, the boundary points of the range of $\lambda$.

Let $X$ be a $p$-variate random vector, $p \geqq 3$, having a normal distribution with mean vector $\theta$ and covariance matrix $\sigma^2I$. For estimating $\theta$ with loss function $L_1(\theta,\hat{\theta}) = (\theta - \hat{\theta})'(\theta - \hat{\theta}) = \|\theta - \hat{\theta}\|^2,$ and known $\sigma^2$, Stein [2] has shown the inadmissibility of the usual estimator $X$ by considering an alternative estimator with uniformly smaller risk than that of $X$, the improvement being substantial for $\theta$ close to the origin. The problem of estimating $\theta$ with the same loss function when $\sigma^2$ is unknown has been treated similarly by James and Stein [1], when an observation is available on another random variable which is distributed as $\sigma^2\chi^2_n$ independently of $X$. James and Stein have also demonstrated the inadmissibility of the usual estimator for $\theta$ under the loss function \begin{equation*}\tag{1}L_2(\theta,\hat{\theta}) = (\theta - \hat{\theta})'D(\theta - \hat{\theta})\end{equation*} for the case of known $\sigma^2$, where $D$ is a diagonal matrix with unequal diagonal elements. This result has been proved even without the normality assumption, but no explicit formulas for alternative estimators have been given which improve on the usual estimator in some parts of the parameter space under normality. In this note an estimator for $\theta$ is obtained for the case when $\sigma^2$ is unknown and the loss function is $L_2$. An upper bound for the risk function of this estimator is given, which always remains below the risk function of the usual estimator and is substantially smaller for $\theta$ close to the origin. This estimator coincides with the estimator given by James and Stein when the diagonal elements of $D$ become equal. An application of this estimator gives an improvement on the least squares estimator of the parameter vector in a usual linear observational model with normal errors.

In [2] a central limit theorem [CLT] for sequences of univariate random linear forms was proved. That result is extended in this note to a multivariate CLT for $q$-dimensional linear forms of constant $q$-vectors with real-valued random weight coefficients. Some applications are indicated in Section 3.

It is known that, for the estimation of $p$ individual weights, the optimum weighing design [3] for a chemical balance is given [4] by a Hadamard matrix $X$ of dimensions $p \times p$, when it exists. If $r$ rows of $X$ are used for the weighing operations, the resultant design matrix $X_1$ of dimensions $r \times p$ will be a fraction of the full design matrix $X$, and will necessarily be singular. While it is not possible, with such a fractional weighing design, to furnish unique and unbiased estimates of the individual weights, it may be practicable, however, to afford a unique and unbiased estimate of a linear function of the weights. In a recent paper, Zacks [6] has considered questions of admissibility of "randomization procedures" for such fractional weighing designs and has indicated a few basic results in this direction proceeding on the same lines as followed in [2]. We furnish, in this paper, some results of connected interest with respect to such fractional weighing designs without resorting to any randomization procedure in the selection of rows of the full design matrix $X$. Apart from this connection, the results are expected to have an importance of their own. We have spelled out here the structure of the estimate of the estimable linear function along with its variance, bringing out the connection of this variance with the variance as obtainable with the full design matrix. And, in the process, it has been indicated, in relation to the fraction used, to what extent we can afford to be arbitrary in the selection of the components of $\lambda_p$ which enters into the estimable linear function $\lambda_p'\beta_p$, where $\lambda_p$ and $\beta_p$ are $p \times 1$ column vectors representing the coefficients and the weights respectively. We have shown that, depending on $\lambda_p$, we can obtain, with a fraction, the same precision for the estimate without having to perform all the weighing operations as required in a full design matrix. In such situations, repetitions of the fraction would lead to increased precision as compared to the adoption of the full design matrix.