The subject of this paper is the higher-order spectra or polyspectra of multivariate stationary time series. The intent is to derive (i) certain mathematical properties of polyspectra, (ii) estimates of polyspectra based on an observed stretch of time series, (iii) certain statistical properties of the proposed estimates and (iv) several applications of the results obtained. As might be expected, in lower order cases the polyspectrum reduces to spectra already considered. If one is considering a single time series, the first order polyspectrum is the usual power spectrum considered in [2], [14], [22], while the second order polyspectrum is the bispectrum considered in [12], [23], [28]. Also, if one is considering a pair of time series the first order polyspectrum is the cross-spectrum considered in [6], [10], [15]. For the case of a single time series the idea of a higher-order spectrum occurs in [3]. The idea has since been developed to a higher level of algebraic and analytic detail in [24]. Also in [24] the notion of considering a spectral representation for a cumulant rather than for a product moment occurs and is acknowledged to be due to Kolmogorov. Another related early paper is [18]. The present paper generalizes the definitions of these papers in the sense that $k$-dimensional time series are considered. Another contribution is a theorem indicating that for a broad class of processes one is wise to restrict consideration to cumulants rather than product moments. Finally it should be noted that the term polyspectrum is due to J. W. Tukey. I have perhaps used the term in a more restricted sense than he would wish in that I have reserved it for the Fourier transform of a cumulant (at the expense of other functions of moments).

This paper studies the sequential decision model known as the two-armed-bandit with finite memory. It was introduced by Robbins [8] in 1956 and studied further by Isbell [5] in 1959. In this paper, a set of rules is defined which are uniformly better than those given in [5] and [8]. A much larger class of rules is then defined, one member of which is conjectured to be a uniformly best rule.

Given a lot of size $N$ whose items are obtained from a statistically controlled process with an unknown probability $p, 0 < p < 1$, of an item being defective, a rectifying sampling inspection problem arises when the loss involved in sending out the lot with $d, 0 \leqq d \leqq N$, defectives in it is $kd$ where $k > 0$ is the loss involved in sending out a defective item instead of replacing it with a good one. The cost of inspecting $n$ items (defective items detected during inspection are replaced by good ones) is $cn, c > 0$. An inspection plan is to be devised that minimizes (in some suitable sense) the risk (expected loss plus inspection cost). Let $\psi$ denote any sequential inspection plan, which determines sequentially the (random) number $\mathscr{R}$ of items to be inspected. If the lot is sent out after inspecting $n$ items, the cost of inspection incurred is $nc$ and the expected loss due to the defectives remaining in the lot is $(N - n)pk$. Then the risk is $(N - n)pk + nc = Npk + nk(c/k - p),$ and hence the risk associated with the plan $\psi$ is given by \begin{equation*}\tag{1.1}R(p, \psi) = Npk + k(c/k - p)E_p(\mathscr{R}).\end{equation*} Since $E_p(\mathscr{R}) \geqq 0$ for every $p \varepsilon (0, 1)$, it follows from (1.1) that if $p$ were known then the optimal plan (which we denote by $\psi_p$) would be to inspect the whole lot or not to inspect at all according as $p >$ or $\leqq p_0 \equiv c/k$. The risk $R(p, \psi_p)$ of this plan is given by $R(p, \psi_p) = Npk,\quad\text{for} p \leqq p_0, \\ = Nc,\quad\text{for} p > p_0$ (see Figure 1). Let $d$ be the number of defectives observed in the first $n$ items inspected when $p$ is unknown. Since $d$ is a sufficient statistic for $p$ based on "the past history," the decision to stop or continue inspection may be made to depend on $d$. Thus we may restrict our attention to plans defined by two mutually exclusive and exhaustive sets in the $(n, d)$ plane called the continuation and stopping sets. If at any stage the current position $(n, d)$ belongs to the continuation set we inspect another item from the lot. If it belongs to the stopping set we stop inspection and send out the lot. Obviously all the points with $n \geqq N$ or with $d \geqq N$ are stopping points. We now define the boundary set as the set of stopping points which are accessible from some continuation point. Because of the monotonic nature of the two types of cost associated with a plan one would expect that for an admissible plan, the boundary set consists of points $\{(n, d_n)\}$ with $d_n$ monotonically nondecreasing in $n$ and the points $(n, d)$ are stopping or continuation points according as $d \leqq d_n$ or $d > d_n$. Such plans are thus defined by a set of boundary points $\{(n, d_n)\}$ or equivalently $\{(d, n_d)\}$. Anscombe [1] considers a class of linear plans where $n_d$ is defined by \begin{equation*}\tag{1.2}n_d = n_0 + d\omega,\quad d = 0, 1, 2, \cdots, \lbrack (N - n_0)/(\omega + 1)\rbrack,\end{equation*} ([ ] denoting the greatest integer part). He considers the problem of choosing the integers $n_0$ and $\omega$ so that the following conditions are satisfied: (1) Whatever the unknown number of defectives in the lot, there is at most a preassigned risk that after inspection the lot will contain more than a specified number of defectives. (2) The average number of items inspected is as small as possible for some range of values (or simply for a fixed value) of the unknown number of defectives. It is to be noted that Anscombe's problem as formulated above, is typical of the classical Neyman-Pearson theoretic formulation and as such differs from our decision theoretic formulation of minimizing (in some suitable sense) the risk associated with a plan. In fact, Wurtele, was first to consider this formulation in her thesis [7]. She gives a characterization of the Bayes sequential plan $\{(d, \tilde n_d)\}$ as a multiple sampling plan which is a consequence of the fact that $\tilde n_d$ is an increasing function of $d$. She presents algorithms for obtaining the large sample behavior of the optimal (Bayes) boundary in the limiting case appropriate for the Poisson approximation. We shall concern ourselves with the Bayes sequential plans for beta prior distributions. To study the large sample behavior of the optimal boundary and associated risk we formulate the following associated normal version of the sampling inspection problem. We observe $X_t$, a Wiener process with unknown drift $\mu$ per unit time and variance one per unit time. Let $T$ be the time at which observation is stopped. Subject to $T \leqq t_0$, it is desired to locate a stopping set which minimizes $E(T\mu)$ (where $E$ denotes the expectation operator over the marginal distribution of the data with a normal prior distribution for $\mu$). If $\mu$ were known, an optimal procedure would be to stop immediately if $\mu \geqq 0$ and to continue as long as possible if $\mu > 0$. The associated problem is then tackled with the help of techniques developed by Chernoff [3], [4], [5] in connection with the sequential testing for the sign of $\mu$. The results are then related to the original sampling inspection problem. Numerical computations are given comparing the boundary and the risk calculated using the asymptotic formulae with the exact ones. Lastly, the optimal (Bayes) plan is compared with the optimal plan among the class of linear plans (1.2). Here the optimum $n_0$ and $\omega$ are approximated by the solution of the corresponding Wiener process problem.

The purpose of this article is to present a uniform method for the evaluation of a large class $S_D$ of Dodge-type continuous sampling inspection plans. The class of Dodge-type plans includes, among others, CSP-1, 2, 3, 4, and 5, MLP-1, $r$, and $T$, and H-106 plans. The evaluation of any plan $S \varepsilon S_D$ is in terms of its average outgoing quality limit (AOQL). The AOQL for $S$ may be defined as an upper bound to the long run proportion of defective items that remains in the output after inspection, given certain assumptions about Nature's (the processes') ability to control process quality. In particular, a specific method of evaluation involving linear programming as its computational tool is developed for the case where Nature is assumed to be unrestricted in her ability to produce and submit defectives. The problem of determining unrestricted AOQL's for the plans in $S_D$ is viewed in terms of two Markovian decision models where Nature is taken to be the decision maker. These decision models are abstractly described in Section 2. Their relation to the problem of evaluating continuous sampling plans is specified in Section 3. The linear programs corresponding to the two decision models are derived in Sections 4 and 5. In Section 6 the linear programming approach is illustrated with an example, and in the last section a row reduction theorem is given for one of the linear programs.

To obtain the limit distribution of a sequence $T_n$ of random vectors, the $j$th component of $T_n$ being the sum of a random number $N_n^{(j)}$ of $j$th components of independent, identically distributed chance vectors $X_n$, it is first necessary to treat the special case where the $N_n^{(j)}$ are degenerate random variables. This is done in Section 2 and generalized to infinitely divisible limits in Section 4. The basic problem is treated in Section 3 and generalizations of theorems of Doeblin [3], Anscombe [1] and Renyi [7] are obtained.

A solution is given to the problem of where to choose $k + 1$ points and what weights to assign them in order to obtain the minimum variance of what may be called an interior extrapolated value of a polynomial of degree $k$. It is assumed that observations can be taken in the interval $\lbrack -1, 1\rbrack$, except for a subinterval $(\alpha, \beta)$ located in its interior and within which the extrapolation occurs. This one dimensional solution is then used to solve the corresponding two dimensional problem, for a certain class of polynomials, in which it is desired to extrapolate inside a rectangular region that lies inside a larger rectangular region and within which observations can be taken, exclusive of the interior region. In addition, the two dimensional exterior extrapolation problem is solved for the same class of polynomials as those used for interior extrapolation.

Sequential estimation for the particular stopping rule, inverse or Haldane sampling, has been developed for the hypergeometric [4], [13], [15], binomial [10], [12], Poisson [17], and exponential [6], [16], distributions; more general stopping rules have been discussed for the binomial distribution by Girshick, Mosteller, and Savage [11] and for the exponential distribution by the author [14]. It is the purpose of this article to present a unified treatment of these distributions and to extend the known results for inverse sampling to a more general class of stopping rules called monotone. Estimates of parameters are obtained following [11], estimates of the variances of these estimates obtained, their distribution related to that of the fixed sample size stopping rule, and confidence methods suggested. Most of the methods described do not require complete knowledge of the stopping rule for their application.

When the means of samples drawn from normal populations are employed in performing tests of hypothesis, the experimenter usually assumes that the sample was drawn from a complete normal population. When all members of the population within three standard deviations of the mean admit to sampling, assumption that the sample was drawn from a complete population does not seriously affect the power of the tests. However, there arise situations where the sampling interval is restricted to less than three standard deviations of the mean. Aggarwal and Guttman [1], [2] have examined the loss of power when using tests based on the assumption that the variable being sampled has a complete normal distribution when, in fact, the distribution is a symmetrically truncated normal distribution. They derived the distribution of means based on samples of size $n \leqq 4$ using convolutions of \begin{align*}\tag{1.1}f(x) &= (C/(2\pi)^{\frac{1}{2}}) \exp(-x^2/2), |x| < a, \\ &= 0,\quad\text{otherwise},\end{align*} where $C$ is given by \begin{equation*}\tag{1.2}C^{-1} = (2\pi)^{-\frac{1}{2}} \int^a_{-a} e^{-\frac{1}{2}t^2} dt.\end{equation*} Birnbaum and Andrews [2] have pointed out that sums of symmetrically truncated normal random variables have a limiting normal distribution. Thus for large $n$ one can obtain for an approximate cumulative distribution of $\bar X = (1/n) \sum^n_{i = 1} X_i$. For arbitrary $n$, however, no general formula giving the distribution of means (or sums) of samples of size $n$ drawn from a truncated normal population is available. In this paper we extend the work of Aggarwal and Guttman [1] to non-symmetric truncation and arbitrary sample size. An asymptotic series for the distribution of sums of samples of size $n$ drawn from a truncated normal population is presented.

Let $\mathbf{A}_1$ and $\mathbf{A}_2$ be two symmetric matrices of order of $p, \mathbf{A}_1$, positive definite and having a Wishart distribution [2], [12] with $f_1$ degrees of freedom, and $\mathbf{A}_2$, at least positive semi-definite and having a (pseudo) non-central (linear) Wishart distribution ([1], [3], [4], [12], [13]) with $f_2$ degrees of freedom. Now let $\mathbf{A}_2 = \mathbf{CYY'C'}$ where $\mathbf{C}$ is a lower triangular matrix such that $\mathbf{A}_1 + \mathbf{A}_2 = \mathbf{CC}'$ and the density function of $\mathbf{Y} : p \times f_2$ is given by \begin{equation*}\tag{1.1}k_1e^{-\lambda^2} \sum^\infty_{j = 0} (2\lambda y_{11})^j\Gamma\lbrack\frac{1}{2}(f_1 + f_2 + j)\rbrack |\mathbf{I}_p - \mathbf{YY}'|^{\frac{1}{2}}(f_{1 - p - 1)}/j!\end{equation*} where $\mathbf{I}_p$ is an identity matrix of order $p$, $k_1 = \prod^p_{i = 2} \Gamma\lbrack\frac{1}{2}(f_1 + f_2 - i + 1)\rbrack/ \pi^{\frac{1}{2}pf_2} \prod^p_{i = 1} \Gamma\lbrack f_1 - i = 1)/2\rbrack,$ $\lambda$ is the only non-centrality parameter in the linear case and $y_{11}$ is the element in the top left corner of the $\mathbf{Y}$ matrix. Now $V^{(s)}$ criterion suggested by Pillai and $U^{(s)}$ (a constant times Hotelling's $T_0^2$), [7], [8], [9], [10] are the sums of the non-zero characteristic roots of the matrix $\mathbf{YY}'$ and $(\mathbf{I}_p - \mathbf{YY}')^{-1} - \mathbf{I}_p$ respectively. Here $s$ is minimum $(f_2, p)$. Also we may note that $V^{(s)} = \text{trace} \mathbf{YY}' = \text{trace} \mathbf{Y'Y}$ and $U^{(s)} = \mathrm{tr} (\mathbf{I}_p - \mathbf{YY}')^{-1} - p = \mathrm{tr} (\mathbf{I}_{f_2} - \mathbf{Y'Y})^{-1} - f_2$. It can be shown that the density function of the characteristic roots of the matrix $\mathbf{Y'Y}$ for $f_2 \leqq p$ can be obtained from that of the characteristic roots of $\mathbf{YY}'$ for $f_2 \geqq p$ if in the latter case the following changes are made: ([12], [5]) \begin{equation*}\tag{1.2}(f_1, f_2, p) \rightarrow (f_1 + f_2 - p, p, f_2).\end{equation*} Hence, for the criterion $V^{(s)}$, (and similarly for $U^{(s)}$), we shall only consider the density function of $\mathbf{L} = \mathbf{YY}'$ for $f_2 \geqq p$ which is given by [6] \begin{equation*}\tag{1.3}f(\mathbf{L}) = ke^{-\lambda^2}_1F_1\{\frac{1}{2} (f_1 + f_2), \frac{1}{2}f_2, \lambda^2l_{11}\}|\mathbf{L}|^{ (f_2 - p - 1)/2} |\mathbf{I}_p - \mathbf{L}|^{(f_1 - p - 1)/2}, \end{equation*} where $k = \pi^{-p(p - 1)/4} \prod^p_{i = 1} \Gamma\lbrack\frac{1}{2}(f_1 + f_2 + 1 - i)\rbrack/\{\Gamma\lbrack \frac{1}{2}(f_1 + 1 - i)\rbrack\Gamma\lbrack\frac{1}{2}(f_2 + 1 - i)\rbrack\}.$ $l_{11}$ is the element in the top left corner of the matrix $\mathbf{L}$ and $_1F_1$ denotes the confluent hypergeometric function. We shall call the distribution of $\mathbf{L}: p \times p$ the non-central (linear) multivariate beta distribution with $f_2$ and $f_1$ degrees of freedom. Pillai [11] had noted that the elements of the matrix $\mathbf{L}$ can be transformed into independent beta variables which he showed for $p = 2, 3, 4$ and $5$. In this paper we give a theorem which proves the general case. In addition, when $\lambda = 0$ the first and second order moments of $l_{ij}$ are obtained and used to derive the first two moments of $V^{(s)}$ in the non-central case when $f_2 \geqq p$. The moments of $V^{(s)}$ for $f_2 \leqq p$ can be written down with the help of (1.2). Similar results are obtained for $U^{(s)}$.

Hajek (1962) introduced a generalized $t$-statistic for testing the difference of two means of normal populations having unknown, and possibly unequal variances. While the actual distribution of this statistic was unknown, he was able to obtain bounds for the type I error. The present paper is concerned with extending Hajek's result, and obtaining bounds for the power curve, as well as type I error. The result is then a test for the Behrens-Fisher problem which guarantees that the type I error will not exceed $\alpha_0$, while, at the same time, the power against a specified alternative is at least $\beta_0$. Similar results are also obtained for the modified $t$-test, introduced by Lord (1947), in which the sample range $W$ replaces the root-mean-square $s$ as an estimate of standard deviation.

Investigations of association schemes for three or more associate class PBIB designs have been limited mainly to the works of Vartak [17], Raghavarao [5], Roy [9], Singh and Shukla [15], Raghavarao and Chandrasekhararao [8] and Tharthare [16]. In this article we generalize the right angular association scheme introduced by the author in [16] to study the combinatorial properties, construction and nonexistence of further class of four associate class designs. In Section 4, a simplified method of analysis of these designs is given, while Section 5 deals with the method of constructing $v \times s^m$ balanced asymmetrical factorial design in $vs^{m - 1}$ plots, where $s$ is a prime or a prime power. It can be seen that this balanced asymmetrical design is nothing but a particular case of the four associate class design introduced in this paper and hence its analysis can be completed by the method of Section 4. Some methods of construction discussed in Section 6 give a new way of arranging an $s^3$ factorial experiment in blocks of sizes different from $s$ and $s^2$, with better efficiency than the usual three dimensional lattice designs [4]. For convenience we denote PBIB designs with the generalized right angular association scheme by "GRAD".

Let $(\Omega, \mathscr{F}, P)$ be a probability space, and let $X$ be a random variable defined on $(\Omega, \mathscr{F}, P)$. If $\mathscr{A}$ is a sub $\sigma$-field of $\mathscr{F}$, then $E(X \mid \mathscr{A})$ is the a.s. unique $\mathscr{A}$ measurable function such that, for all $A \varepsilon \mathscr{A}$, \begin{equation*}\tag{1}\int_A X dP = \int_A E(X \mid \mathscr{A}) dP,\end{equation*} provided $EX$ is defined. ([2], p. 341). If $EX$ is not defined, that is, if $EX^+ = EX^- = \infty$, we may then define $E(X \mid \mathscr{A}) = E(X^+ \mid \mathscr{A}) - E(X^- \mid \mathscr{A})$, provided the difference is defined almost surely ([2], p. 342). We show that this is the only reasonable definition of $E(X \mid \mathscr{A})$ (Lemma 2), and exhibit several apparent pathologies, akin to the fact that a conditionally convergent series of real numbers may be re-ordered to give any sum. If $X$ is any random variable with a continuous distribution such that $EX$ is not defined, then we can find an $\mathscr{A} \subset \mathscr{F}$ such that $E(X \mid \mathscr{A}) = 0$ a.s. (Theorem 1), and if $Y$ is any random variable independent of $X$, we can find an $\mathscr{A} \subset \mathscr{F}$ such that $E(X \mid \mathscr{A}) = Y$ a.s. (Theorem 2). In fact, if $X_1, X_2, \cdots$ is a sequence of independent random variables such that for $n \geqq 2, EX_n$ is not defined and $X_n$ has a continuous distribution, we can find a sequence of $\sigma$-fields $\mathscr{A}_1 \subset \mathscr{A}_2 \subset \cdots \subset \mathscr{F}$ such that $X_1, \cdots, X_n$ are $\mathscr{A}_n$ measurable and $E(X_{n + 1} \mid \mathscr{A}_n) = X_n$ a.s. (Theorem 3). The sequence $\{X_n, \mathscr{F}_n, n = 1, 2, \cdots\}$ is not a martingale however, since for $m > n + 1, E(X_m \mid \mathscr{F}_n)$ is not defined. We remark that the standard theorem on iterated conditional expectations, ([2], p. 350) which says that if $\mathscr{A} \subset \mathscr{B} \subset \mathscr{F}$ then $E(X \mid \mathscr{A}) = E(E(X \mid \mathscr{B}) \mid \mathscr{A})$ a.s. is valid only if $E(X \mid \mathscr{A})$ is defined a.s.

Let $Z(t)$ denote the number of cells at time $t$ which are progeny of a single cell born at $t = 0, G(t)$ with $G(0) = 0$ be the lifetime distribution function of each cell, and $h(s) = \sum^\infty_{r = 0} s^r$, where $p_r$ are constants, $p_r \geqq 0, \sum^\infty_{r = 0} p_r = 1$ be the generating function of the number of cell progeny which replace each cell on completion of its life. Cells develop and proliferate independently of each other. For general $G(t)$ this process is called an age dependent branching process and for $G(t)$ an exponential distribution, a Markov branching process [3]. When the mean number of progeny per cell, $h^{(1)} (1) = 1$, and $h^{(2)} (1) > 0, h^{(3)} (1) < \infty$, and $G(t)$ is an exponential distribution with parameter $\lambda$, Sevast'yanov [5] showed by study of a differential equation satisfied by $F(s, t) = \sum^\infty_{j = 0} P\lbrack Z(t) = j\rbrack s^j$, that $\lim_{t\rightarrow\infty} tP\lbrack Z(t) > 0\rbrack = 2\lbrack\lambda h^{(2)} (1)\rbrack^{-1}$ and that for $u \geqq 0$, \begin{equation*}\tag{1}\lim_{t\rightarrow\infty} P\lbrack 2(\lambda h^{(2)} (1)t)^{-1}Z(t) > u \mid Z(t) > 0\rbrack = \exp (-u).\end{equation*} Analogous limit theorems for the discrete time case were obtained by Kolmogorov and by Yaglom. See [3], pp. 21-22, 108-109. It is the purpose of this paper to extend the results of Sevast'yanov to the case of general $G(t)$. In Section 1, Theorem 1 gives the form of the asymptotic moments of such an age dependent branching process by study of an integral equation satisfied by $D(s, t) = 1 - E \lbrack\exp (-sZ(t))\rbrack$. Chover and Ney [1] have shown that for mild conditions on $G(t)$ and $h(s)$, that $\lim_{t\rightarrow\infty} tP\lbrack Z(t) > 0\rbrack = b$, where $b$ is a strictly positive constant to be defined. In Section 2, this result, together with Theorem 1 yields a conditional limit theorem which generalizes (1). Section 3 contains remarks on an analogous general discrete time result of Mullikin [4].

In [1] Conolly derived an important result (3.19) for the queueing system $GI/E_k/1$. This result has been used by him to obtain the joint distribution of the number of customers served during a busy period initiated by one customer and its length for the queueing system $GI/E_k/1$. In this note we shall derive their steady state properties by using the above result of Conolly. Earlier Wishart [6] and Conolly [2] had studied these features using different methods. The queueing system $GI/E_k/1$, studied in this note is the one in which (i) the time intervals between arrivals are independent and are identically distributed according to the law $dA(t) (0 < t < \infty)$, with mean $a$ and Laplace transform $\psi(\theta)$; (ii) the queue-discipline is "first come, first served"; (iii) there is only one counter and the service-times are independent and identically distributed having a $\chi^2$ distribution, with mean $b$ and $2k$ degrees of freedom i.e. $dB(t) = \lbrack e^{-kt/b}/(k - 1) !(kt/b)^{k - 1}k dt/b; \text{and}$ (iv) the service-times are independent of the inter-arrival times. Thus we may imagine that the service takes place in $k$ consecutive phases, where the time spent in each phase is distributed negative exponentially with mean $b/k$.

Let $x' = (x_1, x_2, \cdots, x_p)$ be a random vector and let $E\lbrack x\rbrack = 0, E\lbrack xx'\rbrack = \Sigma = (\sigma_{ij})$ where we assume that $\Sigma$ is non-singular. Further let \[ \Sigma = T\Lambda T' = (t_1, t_2, \cdots, t_p) \left[ \begin{array}{c|c|c|c|c|c|c}\lambda_1 & 0 & \cdot & \cdot & \cdot & 0 \\ 0 & \lambda_2 & \cdot & \cdot & \cdot & 0 \\ \vdots & \vdots & \vdots & \vdots \\0 & 0 & \cdot & \cdot & \cdot & \lambda_p\end{array} \right] \] \[ \left[ \begin{array}{|c}t'_1 \\ t'_2 \\ \vdots \\ t'_p\end{array}\right] \] where $TT' = I$ and where we suppose that the eigenvalues of $\Sigma$ are in order of decreasing magnitude, that is $\lambda_1 \geqq \lambda_2 \geqq \cdots \geqq \lambda_p > 0$. The principal components of $x$, namely $u_1 = t'_1 x, u_2 = t'_2 x, \cdots, u_p = t_p' x$ were introduced by Hotelling (1933) who characterised them by certain optimal properties. Since then Girshick (1936), Anderson (1958) and Kullback (1959) have characterised the principal components by slightly different sets of optimal properties. Thus Anderson shows that $u_1$ is the linear function $\alpha_1' x$ having maximum variance subject to $\alpha_1' \alpha_1 = 1; u_2$ is the linear function $\alpha_2'x$ which is uncorrelated with $u_1$ and has maximum variance subject to $\alpha_2'\alpha_2 = 1$; and so on. The above mentioned characterisations have two properties in common; they introduce the principal components one by one and, more importantly, the optimal properties hold only with-in the class of linear functions of $x_1, x_2, \cdots, x_p$. In the following theorem the first $k$ principal components are characterized by an optimal property within the class of all random variables.