This report deals with procedures for random-size subset selection fromk(> 2) given populations where the distribution of ir^(i = l, ..., k)has a density f^(x;0^). Let ••• -®[k] denote unknown values ofthe parameters, and let ^[i]» ***'ïï[k] denote the corresponding populations.First, we have considered the problem of selection for consider the/sprocedure that selects TT. if sup L(0;x) > c L(0;x), where L(*;x) is the1 e e u . - - - - -itotal likelihood function, where is the region m the parameter space foriA9= (0^, ..., 0^) having 0^ as the largest component, where 9 is the maximum likelihood estimate of 0 , and where c is a given constant with 0 < c < l .With the densities satisfying seme reasonable requirements given in this report,we have shown that for each i, the probability of includingthe selected subset is decreasing in ®[j] f°r j t i anc* increasing inWe have then derived some results on selection for the t(> 1) best populations,thereby generalizing the results for t = 1. For this problem, we haveconsidered a) selection of a set whose elements consist of subsets of thegiven populations having t members, and requiring that the set of the t• » • • •best populations is included with probability at least P , b) selection ofa subset of the populations so as to include all the t best populationswith probability at least P'*, and c) selection of a subset of the populationssuch that TT[j ^ is included with probability at least P*, j=k-t+l,.•., k. In the final section, we have discussed the relation between thetheories of subset selection based on likelihood ratios and statistical inferenceunder order restrictions, and have considered the complete rankingproblem.
This thesis deals with random-size subset selection and ranking procedures• • • )|(derived through likelihood ratios, mainly in terms of the P -approach.Let IT , . .. , IT, be k(> 2) populations such that IR.(i = l, . . . , k) hasJ_ K. — 12the normal distribution with unknwon mean 0. and variance a.a , where a.i i i2 . . is known and a may be unknown; and that a random sample of size n^ istaken from . To begin with, we give procedure (with tables) whichselects IT. if sup L(0;x) >c SUD L(0;X), where SÎ is the parameter space1for 0 = (0-^, 0^) ; where (with c: ß) is the set of all 0 with0. = max 0.; where L(*;x) is the likelihood function based on the total1sample; and where c is the largest constant that makes the rule satisfy theP*-condition. Then, we consider other likelihood ratios, with intuitivelyreasonable subspaces of ß, and derive several new rules. Comparisons amongsome of these rules and rule R of Gupta (1956, 1965) are made using differentcriteria; numerical for k=3, and a Monte-Carlo study for k=10.For the case when the populations have the uniform (0,0^) distributions,and we have unequal sample sizes, we consider selection for the populationwith min 0.. Comparisons with Barr and Rizvi (1966) are made. Generalizai<j<k Jtions are given.Rule R^ is generalized to densities satisfying some reasonable assumptions(mainly unimodality of the likelihood, and monotonicity of the likelihoodratio). An exponential class is considered, and the results are exemplifiedby the gamma density and the Laplace density. Extensions and generalizationsto cover the selection of the t best populations (using various requirements)are given. Finally, a discussion oil the complete ranking problem,and on the relation between subset selection based on likelihood ratios andstatistical inference under order restrictions, is given.
Let π_{1}, π_{2}, ... π be k (>_2) populations. Let π_{i} (i = 1, 2, ..., k) be characterized by the uniform distributionon (a_{i}, b_{i}), where exactly one of a_{i} and b_{i} is unknown. With unequal sample sizes, suppose that we wish to select arandom-size subset of the populations containing the one withthe smallest value of 0_{i} = b_{i} - a_{i}. Rule R_{i} selects π_{i} iff a likelihood-based k-dimensional confidence region for the unknown (0_{1},..., 0_{k}) contains at least one point having 0_{i} as its smallest component. A second rule, R, is derived through a likelihood ratio and is equivalent to that of Barr and Rizvi (1966) when the sample sizes are equal. Numerical comparisons are made. The results apply to the larger class of densities g(z; 0_{i}) = M(z)Q(0_{i}) iff a(0_{i}) < z < b(0_{i}). Extensions to the cases when both a_{i} and b_{i} are unknown and when 0_{max} is of interest are i i indicated.
Let π_{1}, ..., π_{k} be k(>_2) populations such that π_{i}, i = 1, 2, ..., k, is characterized by the normal distribution with unknown mean and u_{i} variance a_{i}o^{2} , where a_{i} is known and o^{2} may be unknown. Suppose that on the basis of independent samples of size n_{i} from π (i=1,2,...,k), we are interested in selecting a random-size subset of the given populations which hopefully contains the population with the largest mean.Based on likelihood ratios, several new procedures for this problem are derived in this report. Some of these procedures are compared with the classical procedure of Gupta (1956,1965) and are shown to be better in certain respects.
General methods for the estimation of distributions can be derived from approximations of certain information measures. For example, both the maximum likelihood (ML) method and the maximum spacing (MSP) method can be obtained from approximations of the Kuliback-Leibler information. The ideas behind the MSP method, whereby an estimation method for continuous univariate distributions is obtained from an approximation based on spacings of an information measure, were used by Ranneby and Ekström (1997) (using simple spacings) and Ekström (1997) (using high order spacings) to obtain a class of estimation methods, called generalized maximum spacing (GMSP) methods. In the present paper, GMSP methods will be shown to give consistent estimates under general conditions, comparable to those of Bahadur (1971) for the ML method, and those of Shao and Hahn (1996) for the MSP method. In particular, it will be shown that GMSP methods give L^{l} consistent estimates in any family of distributions with unimodal densities, without any further conditions on the distributions.
The maximum spacing (MSP) method, introduced by Cheng and Amin (1983) and independently by Ranneby (1984), is a general method for estimating parameters in univariate continuous distributions and is known to give consistent and asymptotically efficient estimates under general conditions. This method can be derived from an approximation based on simple spacings of the Kullback-Leibler information.
In the present paper, we introduce a class of estimation methods, derived from approximations based on mth order spacings of certain information measures, i.e. the ^-divergences introduced by Csiszâr (1963). The introduced class of methods includes the MSP method as a special case. A subclass of these methods was considered earlier in Ranneby and Ekström (1997), i.e. those based on first order spacings. Here it is found that such methods can be improved by using high order spacings. We also show that the suggested methods give consistent estimates under general conditions. ^{[1]}
^{[1]}Research was supported by The Bank of Sweden Tercentenary Foundation.
The maximum spacing (MSP) method, introduced by Cheng and Amin (1983) and independently by Ranneby (1984), is a general estimation method for continuous univariate distributions. The MSP method, which is closely related to the maximum likelihood (ML) method, can be derived from an approximation based on simple spacings of the Kullback-Leibler information. It is known to give consistent and asymptotically efficient estimates under general conditions and works also in situations where the ML method fails, e.g. for the three parameter Weibull model.
In this thesis it is proved under general conditions that MSP estimates of parameters in the Euclidian metric are strongly consistent. The ideas behind the MSP method are extended and a class of estimation methods is introduced. These methods, called generalized MSP methods, are derived from approximations based on sum-functions of rath order spacings of certain information measures, i.e. the ^-divergences introduced by Csiszår (1963). It is shown under general conditions that generalized MSP methods give consistent estimates. In particular, it is proved that generalized MSP methods give L^{1} consistent estimates in any family of distributions with unimodal densities, without any further conditions on the distributions. Other properties such as distributional robustness are also discussed. Several limit theorems for sum-functions of rath order spacings are given, for ra fixed as well as for the case when ra is allowed to increase to infinity with the sample size. These results provide a strongly consistent nonparametric estimator of entropy, as well as a characterization of the uniform distribution. Further, it is shown that Cressie's (1976) goodness of fit test is strongly consistent against all continuous alternatives.
The main result of this paper is a consistency theorem for the maximum spacing method, a general method of estimating parameters in continuous univariate distributions, introduced by Cheng and Amin (J. Roy. Statist. Soc. Ser. A45 (1983) 394–403) and independently by Ranneby (Scand. J. Statist.11 (1984) 93–112). This main result generalizes a theorem of Ranneby (Scand. J. Statist.11 (1984) 93–112). Also, some examples are given, which shows that this estimation method works also in cases where the maximum likelihood method breaks down.
Several strong limit theorems are proved for sums of logarithms of mth order spacings from general distributions. In all given results, the order mof the spacings is allowed to increase to infinity with the sample size. These results provide a nonparametric strongly consistent estimator of entropy as well as a characterization of the uniform distribution on [0,1]. Furthermore, it is shown that Cressie's (1976) goodness of fit test is strongly consistent against all continuous alternatives.
Several strong limit theorems axe proved for sums of logarithms of mth order spacings from general distributions. In all given results, the order of the spacings is allowed to increase to infinity with the sample size. These results provide a nonparametric strongly consistent estimator of entropy as well as a characterization of the uniform distribution on [0,1]. Furthermore, it is shown that Cressie's (1976) goodness of fit test is strongly consistent against all continuous alternatives. ^{[1]}
^{[1]} Research was supported by The Bank of Sweden Tercentenary Foundation.
A new method is proposed, based on the pole phase angle (PPA) of a second-order autoregressive (AR) model, to track spectral alteration during localised muscle fatigue when analysing surface myo-electric (ME) signals. Both stationary and non-stationary, simulated and real ME signals are used to investigate different methods to track spectral changes. The real ME signals are obtained from three muscles (the right vastus lateralis, rectus femoris and vastus medialis) of six healthy male volunteers, and the simulated signals are generated by passing Gaussian white-noise sequences through digital filters with spectral properties that mimic the real ME signals. The PPA method is compared, not only with spectra-based methods, such as Fourier and AR, but also with zero crossings (ZCs) and the first AR coefficient that have been proposed in the literature as computer efficient methods. By comparing the deviation (dev), in percent, between the linear regression of the theoretical and estimated mean frequencies of the power spectra for simulated stationary (s) and non-stationary (ns) signals, in general, it is found that the PPA method (dev(s) = 4.29; dev(ns) = 1.94) gives a superior performance to ZCs (dv(s) = 8.25) and the first AR coefficient (4.18<dev(s)<21.8; 0.98<dev(ns)<4.36) but performs slightly worse than spectra-based methods (0.33<dev(s)<0.79; 0.41<dev(ns)<1.07). However, the PPA method has the advantage that it estimates spectral alteration without calculating the spectra and therefore allows very efficient computation.
In this paper, we introduce the nonstationary signal analysis methods to analyze the myoelectric (ME) signals during dynamic contractions by estimating the time-dependent spectral moments. The time-frequency analysis methods including the short-time Fourier transform, the Wigner–Ville distribution, the Choi–Williams distribution, and the continuous wavelet transform were compared for estimation accuracy and precision on synthesized and real ME signals. It is found that the estimates providedby the continuous wavelet transform have better accuracy and precision than those obtained with the other time-frequency analysis methods on simulated data sets. In addition, ME signals from four subjects during three different tests (maximum static voluntary contraction, ramp contraction, and repeated isokinetic contractions) were also examined.
In this paper, we introduce wavelet packets as an alternative method for spectral analysis of surface myoelectric(ME) signals. Both computer synthesized and real ME signals are used to investigate the performance. Our simulation results show that wavelet packet estimate has slightly less mean squareerror (MSE) than Fourier method, and both methods perform similarly on the real data. Moreover, wavelet packets give us some advantages over the traditional methods such as multiresolutionof frequency, as well as its potential use for effecting time-frequency decomposition of the nonstationary signals such as the ME signals during dynamic contractions. We also introduce wavelet shrinkage method for improving spectral estimates bysignificantly reducing the MSE’s for both Fourier and wavelet packet methods.
Suppose that a device is subjected to shocks governed by a counting processN = {N(t): t > 0}. The probability that the device survives beyond t is00then H(t) = E P(N(t) = k)P, , where P, is the probability of survivingk=0 _k shocks. In this paper we prove that H(t) is HNBUE (HNWUE), i.e.00 00/ H(x)dx < (>) y exp(-t/y) for t > 0, where y = / H(x)dx, under semet " 0 — 00different conditions on N and ^^^=0' ^or ^nstance we stuc^y the casewhen the interarrivai times between the shocks are independent and HNBUE(HNWUE). This situation includes the cases when N is a Poisson processor a stationary birth process. Further a certain cumulative damage modelis studied.
Suppose that a device is subjected to shocks and that P^, k • 0, 1, 2,00denotes the probability of surviving k shocks. Then H(t) = E P(N(t) = k)P,k=0is the probability that the device will survive beyond t, where N = (N(t): t > 0} is the counting process which governs the arrival of shocks. A-Hameed and Proschan (1975) considered the survival function H(t) under what they called the Pure Birth Shock Model. In this paper we shall prove that H(t) is IFRA and DMRL under conditions which differ from those used by A-Hameed and Proschan (1975).
During the last years efforts have been made in order to define suitable bivariate and multivariate extensions of the univariate IFR, IFRA, NBU NBUE and DMRL classes (with duals) of life distributions. In this paper we suggest two new bivariate NBUE (NWUE) and several bivariate HNBUE (HNWUE) definitions. Furthermore, we discuss some of the classes of multivariate life distributions proposed by Buchanan and Singpurwalla (1977). We also study two bivariate shock models. Suppose that two devices are subjected to shocks of some kind. Let P(k^,k2), k^,k2 = 0,1,2,..., denote the probability that the devices survive k^ and k2 shocks, respectively, and let T. denote the time to failure of device number j, j = 1,2, and let H(t^,t2) = P(T^ > t^,T2 > t2)• We study the shock models by Marshall and Olkin and by Buchanan and Singpurwalla and give sufficient conditions, containing P(k^,k2), k^,k2 = 0,1,2,..., under which H.(t^,t2) is bivariate NBU (NWU), bivariate NBUE (NWUE) and bivariate HNBUE (HNWUE) of different forms.
A life distribution and its survival function F = 1 - F with finitemean y = /q F(x)dx are said to be HNBUE (HNWUE) if F(x)dx < (>)U exp(-t/y) for t > 0. The major part of this thesis deals with the classof HNBUE (HNWUE) life distributions. We give different characterizationsof the HNBUE (HNWUE) property and present bounds on the moments and on thesurvival function F when this is HNBUE (HNWUE). We examine whether theHNBUE (HNWUE) property is preserved under some reliability operations andstudy some test statistics for testing exponentiality against the HNBUE(HNWUE) property.The HNBUE (HNWUE) property is studied in connection with shock models.Suppose that a device is subjected to shocks governed by a counting processN = {N(t): t > 0}. The probability that the device survives beyond t isthen00H(t) = S P(N(t)=k)P, ,k=0where P^ is the probability of surviving k shocks. We prove that His HNBUE (HNWUE) under different conditions on N and * ^orinstance we study the situation when the interarrivai times between shocksare independent and HNBUE (HNWUE).We also study the Pure Birth Shock Model, introduced by A-Hameed andProschan (1975), and prove that H is IFRA and DMRL under conditions whichdiffer from those used by A-Hameed and Proschan.Further we discuss relationships between the total time on test transformHp^(t) = /q ^F(s)ds , where F \t) = inf { x: F(x) > t}, and differentclasses of life distributions based on notions of aging. Guided by propertiesof we suggest test statistics for testing exponentiality agains t IFR,IFRA, NBUE, DMRL and heavy-tailedness. Different properties of these statisticsare studied.Finally, we discuss some bivariate extensions of the univariate properties NBU, NBUE, DMRL and HNBUE and study some of these in connection with bivariate shock models.
The HNBUE (HNWUE) class of life distributions (i.e. for which f F (x)dx< (>)00 t< (>) y exp(-t/y) for t > 0, where y = / F(x)dx) is studied. We prove0that the HNBUE (HNWUE) class is larger than the NBUE (NWUE) class. We alsopresent some characterizations of the HNBUE (HNWUE) property by using theTotal Time on Test (TTT-) transform and the Laplace transform. Further weexamine whether the HNBUE (HNWUE) property is preserved under the reliabilityoperations (1) formation of coherent structure, (2) convolution and(3) mixture. Some bounds on the moments and on the survival function of aHNBUE (HNWUE) life distribution are also presented. The class of distributionswith the discrete HNBUE (discrete HNWUE) property (i.e. for which00 00 00I P. < (>) y(l-l/y)k for k = 0,1,2j=k J "where yi=0 JI p. and P. = E p, )J k=j+l kis also studied.
Let F be a life distribution with survival function F = 1 - F and00 —finite mean y * j n F(x)dx. The scaled total time on test transform-1F( t ) —<P,,(t) = /n /F(x)dx/y was introduced by Barlow and Campo (1975) as ar Utool in the statistical analysis of life data. The properties IFR, IFRA, NBUE, DMRL and heavy-tailedness can be translated to properties of tp„(t).rWe discuss the previously known of these relationships and present some new results. Guided by properties of <P„(t) we suggest some test statisticsrfor testing exponentiality against IFR, IFRA, NBUE, DMRL and heavy-tailed-ness, respectively. The asymptotic distributions of the statistics are derived and the asymptotic efficiencies of the tests are studied. The power for some of the tests is estimated by simulation for some alternatives when the sample size is n = 10 or n = 20.
Let F be a life distribution with survival function F = 1 - F and00 —finite mean y = /q F(x)dx. Then F is said to be harmonic new better00 —than used in expectation (HNBUE) if / F(x)dx < y exp(-t/y) for t > 0. If the reversed inequality is true F is said to be HNWUE (W = worse). We develop some tests for testing exponentiality against the HNBUE (HNWUE) property. Among these is the test based on the cumulative total time on test statistic which is ordinarily used for testing against the IFR (DFR) alternative. The asymptotic distributions of the statistics are discussed. Consistency and asymptotic relative efficiency are studied. A small sample study is also presented.
This thesis is composed of six papers, all dealing with the issue of sampling from a finite population. We consider two different topics: real time sampling and distributions in sampling. The main focus is on Papers A–C, where a somewhat special sampling situation referred to as real time sampling is studied. Here a finite population passes or is passed by the sampler. There is no list of the population units available and for every unit the sampler should decide whether or not to sample it when he/she meets the unit. We focus on the problem of finding suitable sampling methods for the described situation and some new methods are proposed. In all, we try not to sample units close to each other so often, i.e. we sample with negative dependencies. Here the correlations between the inclusion indicators, called sampling correlations, play an important role. Some evaluation of the new methods are made by using a simulation study and asymptotic calculations. We study new methods mainly in comparison to standard Bernoulli sampling while having the sample mean as an estimator for the population mean. Assuming a stationary population model with decreasing autocorrelations, we have found the form for the nearly optimal sampling correlations by using asymptotic calculations. Here some restrictions on the sampling correlations are used. We gain most in efficiency using methods that give negatively correlated indicator variables, such that the correlation sum is small and the sampling correlations are equal for units up to lag m apart and zero afterwards. Since the proposed methods are based on sequences of dependent Bernoulli variables, an important part of the study is devoted to the problem of how to generate such sequences. The correlation structure of these sequences is also studied.
The remainder of the thesis consists of three diverse papers, Papers D–F, where distributional properties in survey sampling are considered. In Paper D the concern is with unified statistical inference. Here both the model for the population and the sampling design are taken into account when considering the properties of an estimator. In this paper the framework of the sampling design as a multivariate distribution is used to outline two-phase sampling. In Paper E, we give probability functions for different sampling designs such as conditional Poisson, Sampford and Pareto designs. Methods to sample by using the probability function of a sampling design are discussed. Paper F focuses on the design-based distributional characteristics of the π-estimator and its variance estimator. We give formulae for the higher-order moments and cumulants of the π-estimator. Formulae of the design-based variance of the variance estimator, and covariance of the π-estimator and its variance estimator are presented.
The maximum spacing (MSP) method, introduced by Cheng and Amin (1983) and independently by Ranneby (1984), is a general method for estimating parameters in univariate continuous distributions and is known to give consistent and asymptotically efficient estimates under general conditions. This method, which is closely related to the maximum likelihood (ML) method, can be derived from an approximation based on simple spacings of the Kullback-Leibler information. In the present paper, the ideas behind the MSP metod axe extended and a class of estimation methods is derived from approximations of certain information measures, i.e. the ^-divergences introduced by Csiszâr (1963). We call these methods generalized maximum spacing (GMSP) methods, and it will be shown under general conditions that they give consistent estimates. GMSP methods have the advantage that they work also in situations where the ML method breaks down, e.g. due to an unbounded likelihood function. Other properties, such as asymptotic normality and the behaviour of the estimates when the assigned model is only approximately true, will be discussed.^{[1]}
^{[1]} Research was supported by MISTRA, the Foundation for Strategic Environmental Research.
Exposure to whole-body vibration (WBV) may cause health problems, e.g. lumbago. The risk will depend on intensity and duration. Exposure to WBV in vehicles varies due to several factors as the vehicle type, the terrain condition, the driver, the speed etc. To estimate the health risk, the measurement strategy has to consider this variation. Furthermore, to understand the importance of different preventive strategies, the cause of the variation has to be known. The objective of this study was to describe variation in exposure to seated WBV during occupational operation of forwarder vehicles and to investigate sources for variation. WBV was measured in 10 various terrain types for seven forwarders operated by 11 drivers. For each driver there were between four and 35 measurements. The measurement periods varied between 0.2 and 34 min. The vibration total value (a_{v}) and total vibration dose value (VDV_{t}) were determined. Results showed that WBV exposure varied considerably and that this variation could result in different conclusions regarding health risk assessments. The highest magnitudes were achieved during travelling activities. During travelling empty, variations in a_{v} were significantly dependent upon forwarder model and terrain type. No significant predictor for variation in VDV_{t} was however found for travelling empty. During travelling loaded the forwarder model and operator were the most important predictors for variation in a_{v}. Variation in VDV_{t} was also dependent on the forwarder model during travelling loaded.
We consider the problem of estimating the failure stresses of bundles (i.e. the tensile forces that destroy the bundles), constructed of several statisti-cally similar fibres, given a particular kind of censored data. Each bundle consists of several fibres which have their own independent identically dis-tributed failure stresses, and where the force applied on a bundle at any moment is distributed equally between the unbroken fibres in the bundle. A bundle with these properties is an example of an equal load-sharing sys-tem, often referred to as the Daniels failure model. The testing of several bundles generates a special kind of censored data, which is complexly struc-tured. Strongly consistent non-parametric estimators of the distribution laws of bundles are obtained by applying the theory of martingales, and by using the observed data. It is proved that random sampling, with replace-ment from the statistical data related to each tested bundle, can be used to obtain asymptotically correct estimators for the distribution functions of deviations of non-parametric estimators from true values. In the case when the failure stresses of the fibres are described by a Weibull distribution, we obtain strongly consistent parametric maximum likelihood estimators of the distribution functions of failure stresses of bundles, by using the complexly structured data. Numerical examples illustrate the behavior of the obtained estimators.
We consider the problem of estimating the reliability of bundles constructed of several fibres, given a particular kind of censored data. The bundles consist of several fibres which have their own independent identically dis-tributed failure stresses (i.e.the forces that destroy the fibres). The force applied to a bundle is distributed between the fibres in the bundle, accord-ing to a load-sharing model. A bundle with these properties is an example of a load-sharing system. Ropes constructed of twisted threads, compos-ite materials constructed of parallel carbon fibres, and suspension cables constructed of steel wires are all examples of load-sharing systems. In par-ticular, we consider bundles where load-sharing is described by either the Equal load-sharing model or the more general Local load-sharing model.
In order to estimate the cumulative distribution function of failure stresses of bundles, we need some observed data. This data is obtained either by testing bundles or by testing individual fibres. In this thesis, we develop several theoretical testing methods for both fibres and bundles, and related methods of statistical inference.
Non-parametric and parametric estimators of the cumulative distribu-tion functions of failure stresses of fibres and bundles are obtained from different kinds of observed data. It is proved that most of these estimators are consistent, and that some are strongly consistent estimators. We show that resampling, in this case random sampling with replacement from sta-tistically independent portions of data, can be used to assess the accuracy of these estimators. Several numerical examples illustrate the behavior of the obtained estimators. These examples suggest that the obtained estimators usually perform well when the number of observations is moderate.
In this paper the rates of strong uniform convergence over any compact set for an alternative nearest neighbor density estimator are obtained when the observations satisfy a ø-mixing or an a-mixing condition. In the ø-mixing case we obtain a quite better convergence rate than for a-mixing processes and we do not require a geometric condition on the mixing coefficients. For independent or m-dependent observations, as a special case of ømixing, the result gives us the optimal rate of strong uniform convergence for density estimators.
Strong limit theorems are proved for sums of logarithms of spacings of increasing order when the observations satisfy a phi-mixing or an alpha-mixing condition. Applications of the results in goodness of fit and parametric estimation problems are discussed.