Minimum Variance Weights Research Articles

Constructing Portfolio Optimization: Analysis in Indonesia Non-Cyclical Industry (Markowitz Approach and Skewness and Kurtosis)

Purpose: The purpose of this research is to examine the application of portfolio optimization in the context of real-world financial issues, particularly in light of the challenges posed by the COVID-19 pandemic. Traditional portfolio optimization strategies, such as those proposed by Markowitz, often rely on the assumption of normally distributed returns, which may not accurately capture the risks associated with extreme events like the COVID-19 crisis. This study aims to shed new light on portfolio optimization methods by exploring various approaches and considering the implications of non-normally distributed returns on portfolio construction. Methods: This research employs a quantitative approach to analyze portfolio optimization techniques in the face of non-normally distributed returns. Using data from financial markets impacted by the COVID-19 pandemic, the study investigates different portfolio construction methods, including risk-free rate for equally weighted portfolios, optimal risk portfolios, minimum variance weights, and maximum expected returns. Various risk metrics such as variance, Sharpe ratio, and standard deviation are considered to evaluate portfolio performance under different constraints. Results and discussion: The empirical findings highlight the limitations of traditional portfolio optimization techniques, particularly in accurately assessing and managing risk in the presence of non-normally distributed returns. Assets exhibit heavily tailed returns, leading to an underestimation of risk when using standard approaches. The study identifies that certain assets offer high returns but also entail significant risks, necessitating a nuanced approach to portfolio construction. By considering stable distribution models and optimizing for both maximum expected return and minimum variance weight, investors can build more profitable and diversified portfolios while managing risk effectively. Implications of the research: The research findings have important implications for investors and financial practitioners, particularly in navigating uncertain market conditions such as those brought about by the COVID-19 pandemic. By recognizing the limitations of traditional portfolio optimization methods and embracing more sophisticated approaches that account for non-normally distributed returns, investors can make more informed decisions and better manage portfolio risk. These insights can inform the development of robust investment strategies tailored to mitigate the impact of extreme events on portfolio performance. Originality/value: This research contributes to the literature by offering a fresh perspective on portfolio optimization under the backdrop of the COVID-19 pandemic. By systematically evaluating different portfolio construction methods and considering the implications of non-normally distributed returns, the study advances understanding in the field of financial risk management. The identification of stable distribution models and the emphasis on balancing maximum expected return with minimum variance weight provide practical guidance for investors seeking to build resilient and profitable portfolios in turbulent market environments. Overall, this research underscores the importance of adapting portfolio optimization strategies to address the realities of contemporary financial markets.

Market risk analysis by Linking the Portfolio of the Beta Factor Model (BFM) Modified from the BARRA Model

최근 글로벌 금융시장은 COVID 충격이후 저금리와 양적완화로 인한 유동성 장세 및 변동성 증가로 주가가 높은 상관관계를 보이는 것으로 나타났다. 따라서 주가가 극단치 꼬리 의존성(tail dependence)을 가질 것으로 보아 이를 더 잘 설명하는 Copula 모형을 시장리스크 측정에 적용하였다. 실증분석 데이터는 벤치마크 시장포트폴리오로 미국 주식시장의 Vanguard S&P500 ETF와 GICKS의 글로벌 산업분류 표준에 의한 11개 산업의 Vanguard ETF 섹터를 분석에 사용하였으며, 분석기간은 2018년 1월 3일~2021년 3월 17일까지이다. 본 연구의 분석모형은 MSCI의 BARRA 산업요인모형(IFM)을 수정한 베타요인모형(BFM)을 최초로 적용하였으며, 공격적 베타와 방어적 베타의 최소분산 가중치를 Student’s t-Copula-VaR 및 ES를 산출하는 데 적용하였다. 분석 결과를 요약하면, BFM모형의 연간 수익률은 2.688%로서 공매도를 제약한 공격적 베타 41.72%와 방어적 베타 39.44%에 비해 현저히 낮았으며, 이 같은 결과는 공매도에 기인된 것으로 추정된다. 또한 모수적 및 비모수적 방법에 의한 VaR, Student’s t-VaR, 그리고 Student’s t-ES는 신뢰수준이 클수록 크며, BFM의 시장 리스크는 모수적 및 비모수적 방법에 관계없이 베타 투자전략에 비해 더 낮았다. 또한 Student’s t-Copula VaR 및 ES는 베타 투자전략에서 모수적 방법이 비모수적 방법보다 작고, 비모수적 방법의 시장 리스크는 공격적 베타 투자전략이 방어적 베타 투자전략 보다 더 큰 것으로 나타났다. 본 연구의 공헌도는 BFM을 시장 리스크 측정에 최초로 도입하였으며, 리스크 매니저가 모델 시장위험을 선제적으로 대응할 수 있는 방향성을 제시하였다. 또한 변동성 충격을 반영하기 위한 꼬리 의존성의 적용은 주변확률(Marginal) t-분포에 의한 통계량(, , )을 계산한 후 모수적(parametric) Student’s t-Copula VaR 및 ES에 대한 분석적(analytic)산출이 가능하도록 하였다. 본 연구의 한계점은 변동성 충격 시기 전·후의 시장 리스크의 차이점을 분석하지 못한 한계점이 존재한다. 추후 연구과제로는 이 같은 한계점을 극복하고 시변(time-varying)변동성이 고려된 동태적(dynamics) Copula를 적용하는 데 대한 연구가 필요하다.Recently, the global financial market has been shown to show a high correlation with the stock price, along with the increase in liquidity market and volatility due to low interest rates and quantitative easing after the COVID shock. Therefore, the Copula model, which explains this behavior better, was applied to the market risk measurement, considering that stock prices have extreme tail dependence of high kurtosis and thick skewness. For the empirical analysis data, the US stock market was analyzed, and the Vanguard S&P 500 ETF and the Vanguard ETF sector of 11 industries according to the global industry classification standard of GICKS were used for the analysis as the benchmark market portfolio. The analysis period is from January 3, 2018 to March 17, 2021. For the analysis model of this study, the beta factor model (BFM) modified from MSCI s BARRA industry factor model (IFM) was applied to this study for the first time, and the minimum variance weight calculated as a beta investment strategy divided into aggressive beta and defensive beta was applied to calculate Student s t-Copula-VaR and ES. To summarize the analysis results, VaR, Student’s t-VaR, and Student’s t-ES by parametric and non-parametric methods increased as the confidence level increased, and the market risk of BFM was lower than that of beta investment strategies regardless of parametric and non-parametric methods. Also, in Student s t-Copula VaR and ES, the parametric method was smaller than the non-parametric method in the beta investment strategy, and the market risk of the non-parametric method was greater in the aggressive beta investment strategy than the defensive beta investment strategy. The contribution of this study introduced BFM for the first time to calculate market risk, and suggested a direction for risk managers to preemptively respond to market risk in the model portfolio. In addition, in order to apply the tail dependence to reflect the volatility shock, statistics(,,) by marginal t-distribution were calculated, and then parametric Student s t-Copula VaR and ES were analyzed analytically. calculation was made possible. A limitation of this study is that it does not analyze the difference between the same portfolio rebalancing strategy and market risk analysis results before and after the volatility shock period. As a future research task, it is necessary to overcome this limitation and to apply a dynamics Copula that considers time-varying variability.

Minimum Variance Combined With Modified Delay Multiply-and-Sum Beamforming for Plane-Wave Compounding.

Plane-wave compounding is an active topic of research in ultrasound imaging because it is a promising technique for ultrafast ultrasound imaging. Unfortunately, due to the data-independent nature of the traditional compounding method, it imposes a fundamental limit on image quality. To address this issue, adaptive beamformers have been implemented in the compounding procedure. In this article, a new adaptive beamformer for the 2-D data set obtained from multiple plane-wave transmissions is investigated. In the proposed scheme, the minimum variance (MV) weights are applied to the backscattered echoes. Then, the final image is obtained by employing a modified version of the delay multiply-and-sum (DMAS) beamformer in the coherent compounding. The results demonstrate that the presented MV-DMAS scheme outperforms the conventional coherent compounding in both terms of resolution and contrast. It also offers improvements over the 2-D-DMAS and some MV-based methods presented in the literature, such that it achieves at least 20.9% enhancement in sidelobe reduction compared with the best result of MV-based methods. Also, by the proposed method, the in vivo study shows an improved generalized contrast-to-noise ratio (GCNR) that implies a higher probability of lesion detection.

High Resolution, High Contrast Beamformer Using Minimum Variance and Plane Wave Nonlinear Compounding with Low Complexity.

The plane wave compounding (PWC) is a promising modality to improve the imaging quality and maintain the high frame rate for ultrafast ultrasound imaging. In this paper, a novel beamforming method is proposed to achieve higher resolution and contrast with low complexity. A minimum variance (MV) weight calculated by the partial generalized sidelobe canceler is adopted to beamform the receiving array signals. The dimension reduction technique is introduced to project the data into lower dimensional space, which also contributes to a large subarray length. Estimation of multi-wave receiving covariance matrix is performed and then utilized to determine only one weight. Afterwards, a fast second-order reformulation of the delay multiply and sum (DMAS) is developed as nonlinear compounding to composite the beamforming output of multiple transmissions. Simulations, phantom, in vivo, and robustness experiments were carried out to evaluate the performance of the proposed method. Compared with the delay and sum (DAS) beamformer, the proposed method achieved 86.3% narrower main lobe width and 112% higher contrast ratio in simulations. The robustness to the channel noise of the proposed method is effectively enhanced at the same time. Furthermore, it maintains a linear computational complexity, which means that it has the potential to be implemented for real-time response.

Eigenspace-based beamformer using oblique signal subspace projection for ultrasound plane-wave imaging.

BackgroundThe Eigenspace-based beamformers, by orthogonal projection of signal subspace, can remove a large part of the noise, and provide better imaging contrast upon the minimum variance beamformer. However, wrong estimate of signal and noise component may bring dark-spot artifacts and distort the signal intensity. The signal component and noise and interference components are considered uncorrelated in conventional eigenspace-based beamforming methods. In ultrasound imaging, however, signal and noise are highly correlated. Therefore, the oblique projection instead of orthogonal projection should be taken into account in the denoising procedure of eigenspace-based beamforming algorithm.MethodsIn this paper, we propose a novel eigenspace-based beamformer based on the oblique subspace projection that allows for consideration of the signal and noise correlation. Signal-to-interference-pulse-noise ratio and an eigen-decomposing scheme are investigated to propose a new signal and noise subspaces identification. To calculate the beamformer weights, the minimum variance weight vector is projected onto the signal subspace along the noise subspace via an oblique projection matrix.ResultsWe have assessed the performance of proposed beamformer by using both simulated software and real data from Verasonics system. The results have exhibited the improved imaging qualities of the proposed beamformer in terms of imaging resolution, speckle preservation, imaging contrast, and dynamic range.ConclusionsResults have shown that, in ultrasound imaging, oblique projection is more sensible and effective than orthogonal subspace projection. Better signal and speckle preservation could be obtained by oblique projection compare to orthogonal projection. Also shadowing artifacts around the hyperechoic targets have been eliminated. Implementation the new subspace identification has enhanced the imaging resolution of the minimum variance beamformer due to the increasing the signal power in direction of arrival. Also it has offered better sidelobe suppression and a higher dynamic range.

On the Estimation for the Weibull Distribution

Here, we consider estimation of the pdf and the CDF of the Weibull distribution. The following estimators are considered: uniformly minimum variance unbiased, maximum likelihood (ML), percentile, least squares and weight least squares. Analytical expressions are derived for the bias and the mean squared error. Simulation studies and real data applications show that the ML estimator performs better than others.

Enhance contrast in PCA based beamformers using smoothing kernel.

The contrast and resolution have trade-off in medical ultrasound imaging. Most of adaptive beamformer can enhance the imaging resolution significantly but not improve the contrast at the same time. The principal component analysis (PCA) based beamformers such as the eigenspace-based minimum variance (ESBMV) beamformer provide a good imaging resolution. Neighbors of the focal point include the common noise, interface and signal components. Echo signal of the neighbor points can be used to suppress the noise and extract the signal component of the focal point. Based on this idea, in order to improve the quality of PCA based beamformers both in the imaging contrast and resolution, a novel beamforming method is proposed. This proposed beamformer utilizes a kernel to select neighbor points. The number of eigenvectors is estimated by using any PCA method. Then the number of selected eigenvectors for each focal point is compared with the number of selected eigenvectors of its neighbor points and is changed to a new value. The selected eigenvectors of the covariance matrix is used to construct the signal subspace. The estimated signal subspace is projected onto the minimum variance (MV) weight vector to calculate the desire weight vector. Results of experiments show that the proposed beamformer can improve the imaging contrast significantly while keeping the resolution quality similar to ESBMV beamformer.

Power spectrum estimation from peculiar velocity catalogues

The peculiar velocities of galaxies are an inherently valuable cosmological probe, providing an unbiased estimate of the distribution of matter on scales much larger than the depth of the survey. Much research interest has been motivated by the high dipole moment of our local peculiar velocity field, which suggests a large-scale excess in the matter power spectrum and can appear to be in some tension with the Λ cold dark matter (ΛCDM) model. We use a composite catalogue of 4537 peculiar velocity measurements with a characteristic depth of 33 h−1 Mpc to estimate the matter power spectrum. We compare the constraints with this method, directly studying the full peculiar velocity catalogue, to results by Macaulay et al., studying minimum variance moments of the velocity field, as calculated by Feldman, Watkins & Hudson. We find good agreement with the ΛCDM model on scales of k > 0.01 h Mpc−1. We find an excess of power on scales of k < 0.01 h Mpc−1 with a 1σ uncertainty which includes the ΛCDM model. We find that the uncertainty in excess at these scales is larger than an alternative result studying only moments of the velocity field, which is due to the minimum variance weights used to calculate the moments. At small scales, we are able to clearly discriminate between linear and non-linear clustering in simulated peculiar velocity catalogues and find some evidence (although less clear) for linear clustering in the real peculiar velocity data.

Contrast enhancement and robustness improvement of adaptive ultrasound imaging using forward-backward minimum variance beamforming

In adaptive ultrasound imaging, accurate estimation of the array covariance matrix is of great importance, and biases the performance of the adaptive beamformer. The more accurately the covariance matrix can be estimated, the better the resolution and contrast can be achieved in the ultrasound image. To this end, in this paper, we have used the forward-backward spatial averaging for array covariance matrix estimation, which is then employed in minimum variance (MV) weights calculation. The performance of the proposed forward-backward MV (FBMV) beamformer is tested on simulated data obtained using Field II. Data for two closely located point targets surrounded by speckle pattern are simulated showing the higher amplitude resolution of the FBMV beamformer in comparison to the forward-only (F-only) MV beamformers, without the need for diagonal loading. A circular cyst with a diameter of 6 mm and a phantom containing wire targets and two cysts with different diameters of 8 mm and 6 mm are also simulated. The simulations show that the FBMV beamformer, in contrast to the F-only MV, could estimate the background speckle statistics without the need for temporal smoothing, resulting in higher contrast for the FBMV-resulted image in comparison to the MV images. In addition, the effect of steering vector errors is investigated by applying an error of the sound speed estimate to the ultrasound data. The simulations show that the proposed FBMV beamformer presents a satisfactory robustness against data misalignment resulted from steering vector errors, outperforming the regularized F-only MV beamformer. These improvements are achieved without compromising the good resolution of the MV beamformer and resulted from more accurate estimation of the covariance matrix and consequently, the more accurate setting of the MV weights.

Enhancement of MCMV Capability for Multiuser Detection under Spreading Code Mismatch

This letter deals with multiuser detection under imprecise knowledge of the received signature codes of all active users for multicarrier code division multiple access (MC-CDMA) systems. The weight vector of the modified multiple constrained minimum variance (MMCMV) is found by projecting the multiple constrained minimum variance (MCMV) weight vector onto a vector subspace constructed from the eigenstructure of the correlation matrix. However, MMCMV still cannot handle the large code-mismatch. Shaping the noise subspace with all estimated active spreading codes, we present an effective approach to achieve more robust capabilities than the MMCMV. Computer simulations show the effectiveness of the proposed detector.

Parameterized approximation of fuzzy number with minimum variance weighting functions

In the parameterized fuzzy number expectation and fuzzy number interval approximation with the weighting function method, we once considered the weighting function form with maximum entropy, which has some interesting properties. In the present paper, we will propose another kind of parameterized weighting function with minimum variance, and apply it to the fuzzy number expectation and fuzzy number interval approximation problems. It shows that the minimum variance weighting function also has some similar interesting properties, and can be used to express the decision maker’s preference information in the fuzzy number defuzzification process.

Planning a cluster randomized trial with unequal cluster sizes: practical issues involving continuous outcomes

BackgroundCluster randomization design is increasingly used for the evaluation of health-care, screeening or educational interventions. At the planning stage, sample size calculations usually consider an average cluster size without taking into account any potential imbalance in cluster size. However, there may exist high discrepancies in cluster sizes.MethodsWe performed simulations to study the impact of an imbalance in cluster size on power. We determined by simulations to which extent four methods proposed to adapt the sample size calculations to a pre-specified imbalance in cluster size could lead to adequately powered trials.ResultsWe showed that an imbalance in cluster size can be of high influence on the power in the case of severe imbalance, particularly if the number of clusters is low and/or the intraclass correlation coefficient is high. In the case of a severe imbalance, our simulations confirmed that the minimum variance weights correction of the variation inflaction factor (VIF) used in the sample size calculations has the best properties.ConclusionPublication of cluster sizes is important to assess the real power of the trial which was conducted and to help designing future trials. We derived an adaptation of the VIF from the minimum variance weights correction to be used in case the imbalance can be a priori formulated such as "a proportion (γ) of clusters actually recruit a proportion (τ) of subjects to be included (γ ≤ τ)".

Safety-first portfolio optimization for US investors in emerging global, Asian and Latin American markets

Risk averse US investors with safety-first objectives in portfolio optimization hold small weights (maximum 10%) in emerging markets when constructing portfolios of the Standard and Poor's 500 (SP), and the Emerging Markets Composite Global (CG), Asia (AS) and Latin American (LA) indexes, respectively. The Composite Global and Asia weights are even smaller than their minimum variance weights. Yet, these optimal safety-first portfolios are dominant in terms of risk and return over the global minimum or higher variance portfolios. In contrast, safety-first optimization for Latin America is hardly different from the minimum variance and not clearly dominant over other mean–variance portfolios. Overall, safety-first limits portfolio losses associated with infrequent catastrophic events and otherwise optimize performance.

Wavelet designs for estimating nonparametric curves with heteroscedastic error

In this paper, we discuss the problem of constructing designs in order to maximize the accuracy of nonparametric curve estimation in the possible presence of heteroscedastic errors. Our approach is to exploit the flexibility of wavelet approximations to approximate the unknown response curve by its wavelet expansion thereby eliminating the mathematical difficulty associated with the unknown structure. It is expected that only finitely many parameters in the resulting wavelet response can be estimated by weighted least squares. The bias arising from this, compounds the natural variation of the estimates. Robust minimax designs and weights are then constructed to minimize mean-squared-error-based loss functions of the estimates. We find the periodic and symmetric properties of the Euclidean norm of the multiwavelet system useful in eliminating some of the mathematical difficulties involved. These properties lead us to restrict the search for robust minimax designs to a specific class of symmetric designs. We also construct minimum variance unbiased designs and weights which minimize the loss functions subject to a side condition of unbiasedness. We discuss an example from the nonparametric literature.

On Estimating the QSO Transmission Power Spectrum

The Lyα forest has become an important tool for measuring the mass power spectrum at high redshifts (z = 2-4). A crucial intermediate step is the measurement of the transmission power spectrum. We present new methods to minimize the systematic and random errors for such a measurement and discuss their implications for observing strategies. Sources of systematic errors explored include metal line contamination and continuum fitting. We advocate the technique of trend removal in place of traditional continuum fitting: here a spectrum is normalized by its (smoothly varying) mean rather than its continuum; this method is easily automated and removes biases introduced by continuum fitting. Moreover, trend removal can be easily applied to spectra where continuum fitting is difficult, such as when the resolution or signal-to-noise ratio (S/N) is low, or for spectra at high redshifts. We further show that a measurement of the continuum power spectrum (plus a related quantity) using trend removal, from either low-redshift quasar spectra or the red side of Lyα, can be used to constrain the amount of spurious large-scale power introduced by the uncertain continuum and in principle allows the removal of such contamination and thereby expanding scales probed to larger ones. We also derive expressions for the shot noise bias and variance of the power spectrum estimate, taking into account the non-Poissonian nature of the shot noise and the non-Gaussianity of the cosmic fluctuations. An appropriate minimum variance weighting of the data is given. Finally, we give practical suggestions on observing strategy: the desired resolution and S/N for different purposes and instruments, as well as how to distribute one's finite observing time among quasar targets. Also discussed is the quasar spectroscopic study of the Sloan Digital Sky Survey (SDSS), which has the potential to measure the power spectrum accurate to better than 1% per mode (Δk ~ 10^(-4) s km^(-1)). The techniques presented here will be useful for tackling the anticipated issues of shot noise and continuum contamination.

Effects of sampling on statistics of large-scale structure

The effects of sampling are investigated on measurements of counts-in-cells in three-dimensional magnitude-limited galaxy surveys, with emphasis on moments of the underlying smooth galaxy density field convolved with a spherical window. A new estimator is proposed for measuring the kth-order moment 〈ρk〉: the weighted factorial moment F̃k[ω]. Since these statistics are corrected for the effects of the varying selection function, they can extract the moments in one pass without the need to construct a series of volume-limited samples. The cosmic error on the measurement of F̃k[ω] is computed via the formalism of Szapudi & Colombi, which is generalized to include the effects of the selection function. The integral equation for finding the minimum variance weight is solved numerically, and an accurate and intuitive analytical approximation is derived, ωoptimal(r) ∝ 1/Δ(r), where Δ(r) is the cosmic error as a function of the distance from the observer. The resulting estimator is more accurate than the traditional method of counts-in-cells in volume-limited samples, which discards useful information. As a practical example, it is demonstrated that, unless unforeseen systematics prevent it, the proposed method will extract moments of the galaxy distribution in the future Sloan Digital Sky Survey (hereafter SDSS) with accuracy of order a few per cent for k = 2, 3 and better than 10 per cent for k = 4 in the scale range of 1 ≤ ℓ ≤ 50 h−1 Mpc. In the particular case of the SDSS, a homogeneous (spatial) weight ω = 1 is reasonably close to optimal. Optimal sampling strategies for designing magnitude-limited redshift surveys are investigated as well. The arguments of Kaiser are extended to higher order moments, and it is found that the optimal strategy depends greatly on the statistics and scales considered. A sampling rate ƒ ~ 1/3 — 1/10 is appropriate to measure low-order moments with k≤ 4 in the scale range 1 ≲ ℓ ≲ 50 h−1 Mpc. However, the optimal sampling rate increases with the order considered, k,and with l/ℓ. Therefore counts-in-cells statistics in general, such as the shape of the distribution function, high-order moments, cluster selection, etc., require full sampling, especially at small, highly non-linear scales ℓ ~ 1 h−1 Mpc. Another design issue is the optimal geometry of a catalogue, when it covers only a small fraction of the sky. Similarly to Kaiser, we find that a survey composed of several compact subsamples of angular size ΩF spread over the sky on a glass-like structure would do better, with regards to the cosmic error, than the compact or the traditional slice-like configurations, at least at small scales. The required dynamic range of the measurements determines the characteristic size of the subsamples. It is, however, difficult to estimate, since an accurate and cumbersome calculation of edge effects would be required at scales comparable to the size of a subsample.

Error estimates of protein structure coordinates and deviations from standard geometry by full-matrix refinement of gammaB- and betaB2-crystallin.

Faster workstations with larger memories are making error estimation from full-matrix least-squares refinement a more practicable technique in protein crystallography. Using minimum variance weighting, estimated standard deviations of atomic positions have been calculated for two eye lens proteins from the inverse of a least-squares normal matrix which was full with respect to the coordinate parameters. gammaB-crystallin, refined at 1.49 A yielded average errors in atomic positions which ranged from 0.05 A for main-chain atoms to 0.27 A for unrestrained water molecules. The second structure used in this work was that of betaB2-crystallin refined at 2.1 A resolution where the corresponding average errors were 0.08 and 0.35 A, respectively. The relative errors in atomic positions are dependent on the number and kinds of restraints used in the refinements. It is also shown that minimum variance weighting leads to mean-square deviations from target geometry in the refined structures which are smaller than the variances used in the distance weighting.

The combination of forecasts: a ranking and subset selection approach

Forecasts combination has attracted a great deal of interest in recent years and has been discussed from both classical and Bayesian viewpoints. We first present some results on the sampling distribution of the optimum (minimum forecast error variance) weight; only partial could be obtained at present. Similar difficulties with the Bayesian approach (e.g. choice of the prior), together with the availability of a large number of forecast methods in any given situation, point to the need for new directions in this important problem. The ranking and subset selection approach is suggested as a statistical procedure for ranking alternative forecasts. The main objective of this procedure is to screen out “bad” forecasts based on past performances, and at the same time, allow some control by the decision-maker. An adaptive method based on the forecast error ranges is then developed to combine the chosen forecasts. This simple method is shown to compare favorably to those based on other optimality criteria when applied to some real series.

Minimum Variance Approximate Formulas

We discuss the construction of minimum variance approximate formulas for a given linear functional. We first describe an elementary method for obtaining the minimum variance weights; for the purpose, we set up two systems having the advantage that the order of the matrix to be inverted will not exceed half the number of abscissas in the formula. A simple interpretation, existence and characterization of minimum variance approximate formulas is given in the space of certain degree polynomials. With the help of this interpretation, we then obtain the minimum variance weights explicitly in terms of certain orthonormal polynomials. We consider minimum variance formulas with equispaced and Chebyshev abscissas; for these abscissas, the orthonormal polynomials are known explicitly and the corresponding minimum variance formulas are obtained explicitly. We give examples of minimum variance formulas for interpolation, numerical differentiation and integration.

Some Comments on the Combination of Forecasts

This paper examines some of the theoretical implications of combining forecasts using a minimum variance criterion. In particular, the derivation of the exact expression for the minimum variance weight vector is provided, together with a proof that the error variance of the composite forecast is no greater than that of any of the component forecasts. A detailed examination is made of the probability distributions of the weight estimators, and an explanation is given for the occurrence of negative weights.