improved likelihood ratio tests on the von mises fisher distribution summary explicit expressions for the directed likelihood statistic r and two modified directed likelihood statistics r and rt are given for several standard hypotheses on concentrations in von misesfisher distributions and for several standard hypotheses on mean directions in von mises distributions

modified signed log likelihood ratio summary the signed log likelihood ratio r for a one dimensional interest parameter can be modified as r r r log u r so that r is asymptotically standard normally distributed with error of order o n 3 2 u being interpretable as a test statistic a new more direct derivation of this result is given which leads to a more explicit expression for u and provides a better handle on the error term the relation of the tail area approximation determined by r to a generalization of the lugannani rice approximation is discussed and the applicability of r to the analysis of residual variation is indicated

saddlepoint approximations for the bartlett nanda pillai trace statistic in multivariate analysis summary the null distribution of the bartlett nanda pillai trace statistic is shown to be equivalent to the conditional distribution of canonical sufficient statistics in a multiparameter exponential family setting as such sequential and double saddlepoint approximations are used to approximate the distribution the existence of tabulated percentiles for the distribution allows for comparison of the accuracy of the various saddlepoint approximations the ability of the approximations to accommodate large numbers of nuisance parameters is also revealed with numerical studies the bartlett nanda pillai trace statistic is one of the four principal test statistics used in the multivariate analysis of variance p values of tests based on it are computed from its null distribution we show in 2 that this null distribution is equivalent to a conditional distribution involving components of the canonical sufficient statistic in the context of a certain multiparameter regular exponential family such equivalence allows for the possibility of saddlepoint approximation for both the null density and distribution function so that highly accurate approximations for p values can be easily obtained section 3 reviews the various saddlepoint methods that we shall base approximations upon sequential saddlepoint methods introduced by fraser reid wong 1991 are used to compute a sequential density approximation as well as distributional approximations based on the lugannani rice 1980 formula and the r approximation of bamdorff nielsen 1986 1991 double saddlepoint analogues to the above methods are also computed a double saddlepoint density approximation bamdorff nielsen cox 1979 takes the explicit form in 3 8 below as a modification to a rescaled beta density this explicit form is due to the explicit solution 4 11 that arises from the double saddlepoint equation double saddlepoint distributional approximations based on skovgaard 1987 and the r approximation bamdorff nielsen 1986 1991 are also computed the numerical comparisons of 6 reveal that the double saddlepoint distributional approximations of skovgaard 1987 and bamdorff nielsen 1986 1991 are grossly

approximations of marginal tail probabilities and inference for scalar parameters summary in many situations inference for a scalar parameter in the presence of nuisance parameters requires integration of either a joint density of pivotal quantities or a joint posterior density for such inference accurate approximations of marginal tail probabilities are useful to avoid high dimensional integrals two tail probability approximations are developed in this paper numerical results given for conditional inference in location scale and linear regression models show the approximations to be generally accurate even for small sample sizes

conditional properties of unconditional parametric bootstrap procedures for inference in exponential families higher order inference about a scalar parameter in the presence of nuisance parameters can be achieved by bootstrapping in circumstances where the parameter of interest is a component of the canonical parameter in a full exponential family the optimal test which is approximated is a conditional one based on conditioning on the sufficient statistic for the nuisance parameter a bootstrap procedure that ignores the conditioning is shown to have desirable conditional properties in providing third order relative accuracy in approximation of p values associated with the optimal test in both continuous and discrete models the bootstrap approach is equivalent to third order analytical approaches and is demonstrated in a number of examples to give very accurate approximations even for very small sample sizes

maximum likelihood estimation of a multi dimensional log concave density fn is shown to have smaller mean integrated squared error compared with kernel based methods even when we allow the use of a theoret ical optimal fixed bandwidth for the kernel estimator that would not be available in practice we also present a real data clustering example which shows that our methodology can be used in conjunction with the expectation maximization algorithm to fit finite mixtures of log concave densities

stable and invariant adjusted directed likelihoods summary for the purpose of making inferences for a one dimensional interest parameter or constructing approximate complementary ancillaries or residuals the directed likelihood or signed square root of the likelihood ratio statistic can be adjusted so that the resulting modified directed likelihood is under ordinary repeated sampling approximately standard normal with error of o n 312 conditional on a suitable ancillary statistic and hence unconditionally in general suitable specification of the ancillary statistic may be difficult we introduce two adjusted directed likelihoods which are similar to the modified directed likelihood but do not require the specification of the ancillary statistic the error of the standard normal approximation to the distribution of these new adjusted directed likelihoods is o n 1 conditional on any reasonable ancillary statistic which is still an improvement over the unadjusted directed likelihoods

objective bayes conditional inference and the signed root likelihood ratio statistic barndorffnielsen oe 1986 biometrika v73 p307 barndorff nielsen o e 1994 inference asymptotic datta g s 2004 probability matching diciccio tj 1993 j roy stat soc b met v55 p305 diciccio tj 1993 biometrika v80 p731 doi 10 1093 biomet 80 4 731 diciccio tj 1994 j roy stat soc b met v56 p397 diciccio tj 1994 scand j stat v21 p447 diciccio tj 2010 biometrika v97 p497 doi 10 1093 biomet asq002 hall p 1992 bootstrap edgeworth lawless jf 1982 stat models methods mukerjee r 1993 biometrika v80 p499 doi 10 1093 biomet 80 3 499 mukerjee r 1997 biometrika v84 p970 doi 10 1093 biomet 84 4 970 nicolaou a 1993 j roy stat soc b met v55 p377 peers hw 1965 j roy stat soc b v27 p9 tibshirani r 1989 biometrika v76 p604 doi 10 1093 biomet 76 3 604 tierney l 1986 j am stat assoc v81 p82 doi 10 2307 2287970 welch b l 1963 j r stat soc b v62 p159

comparisons of approximate confidence intervals for distributions used in life data analysis this article evaluates the accuracy of approximate confidence intervals for parameters and quantiles of the smallest extreme value and normal distributions the findings also apply to the weibull and the lognormal distributions the interval estimates are based on a the asymptotic normality of the maximum likelihood estimator b the asymptotic x 2 distribution of the likelihood ratio lr statistic c a mean and variance correction to the signed square roots of the lr statistic and d the bartlett correction to the lr statistic the extensive monte carlo results about true error probabilities and average lengths under various degrees of censoring show advantages of the lr based intervals for complete or moderately censored samples the mean and variance correction to the lr statistic gives nearly exact and symmetric error probabilities in small samples with heavy censoring the bartlett correction tends to give conservative error probabilities whereas the uncorrected lr interval is often antico

a simple method for the adjustment of profile likelihoods summary we propose a simple adjustment for profile likelihoods the aim of the adjustment is to alleviate some of the problems inherent in the use of profile likelihoods such as bias inconsistency and overoptimistic variance estimates the adjustment is applied to the profile loglikelihood score function at each parameter value so that its mean is zero and its variance is the negative expected derivative matrix of the adjusted score function for cases in which explicit calculation of the adjustments is difficult we give two methods to simplify their computation an automatic simulation method that requires as input only the profile loglikelihood and its first few derivatives first order asymptotic expressions some examples are provided and a comparison is made with the conditional profile log likelihood of cox and reid

first order saddlepoint approximation for reliability analysis in the approximation methods of reliability analysis nonnormal random variables are transformed into equivalent standard normal random variables this transformation tends to increase the nonlinearity of a limit state function and hence results in less accurate reliability approximation the first order saddlepoint approximation for reliability analysis is proposed to improve the accuracy of reliability analysis by the approximation of a limit state function at the most likelihood point in the original random space and employment of the accurate saddlepoint approximation the proposed method reduces the chance of an increase in the nonlinearity of the limit state function this approach generates more accurate reliability approximation than the first order reliability method without an increase in the computational effort the effectiveness of the proposed method is demonstrated with two examples and is compared with the first and second order reliability methods

saddlepoint approximations to the cdf of some statistics with nonnormal limit distributions abstract in standard saddlepoint approximations to the cumulative distribution function of a random variable the normal distribution has appeared to play a special role in this article we consider what happens when the normal base distribution is replaced by an arbitrary base distribution generalized versions of several standard formulas are presented the choice of a chi squared base or an inverse gaussian base is then considered in detail the generalized approximations are compared in two examples a linear combination of chi squared variables and the first passage time distribution for a random walk the former example considers approximations using the chi squared base that are slightly more accurate than their normal based counterparts in the latter example approximations based on the inverse gaussian are considerably more accurate than their normal based counterparts

likelihood ratio statistics based on an integrated likelihood an integrated likelihood depends only on the parameter of interest and the data so it can be used as a standard likelihood function for likelihood based inference in this paper the higher order asymptotic properties of the signed integrated likelihood ratio statistic for a scalar parameter of interest are considered these results are used to construct a modified integrated likelihood ratio statistic and to suggest a class of prior densities to use in forming the integrated likelihood the properties of the integrated likelihood ratio statistic are compared to those of the standard likelihood ratio statistic several examples show that the integrated likelihood ratio statistic can be a useful alternative to the standard likelihood ratio statistic

bootstrap confidence intervals for a class of parametric problems summary we consider the following class of problems having observed a multivariate normal data vector y with unknown mean vector q covariance matrix the identity find an approximate confidence interval for 0 t q a real valued function of q a simple geometric construction is given which leads to highly accurate solutions this construction shows that the standard approximation based on maximum likelihood theory 04 cz00 can be quite misleading when 0 is nonlinear in q we discuss bootstrap based confidence intervals which remove most of the error in the standard approximation at the expense of considerably more calculation the bootstrap intervals are invariant under transformation of both y and i and so they automatically produce accurate solutions in problems which can be transformed to multivariate normality without requiring knowledge of the normalizing transformation

confidence intervals of effect size in randomized comparative parallel group studies we have presented a new likelihood based approach for constructing confidence intervals of effect size that are applicable to small samples we also conduct a simulation study to compare the coverage probability of the new likelihood based method with other three methods proposed by hedges and olkin and kraemer and paik simulation studies show that the confidence interval generated by the modified signed log likelihood ratio method possesses essentially exact coverage probabilities even for small samples although the coverage probabilities are consistently but slightly less than the nominal level the methods are also applied to two examples

mean loglikelihood and higher order approximations summary higher order approximations to p values can be obtained from the log likelihood function and a reparameterization that can be viewed as a canonical parameter in an exponential family approximation to the model this approach clarifies the connection between skovgaard 1996 and fraser et al 1999a and shows that the skovgaard approximation can be obtained directly using the mean log likelihood function

assessment of uncertainty in high frequency data the observed asymptotic variance the availability of high frequency financial data has generated a series of estimators based on intra day data improving the quality of large areas of financial econometrics however estimating the standard error of these estimators is often challenging the root of the problem is that traditionally standard errors rely on estimating a theoretically derived asymptotic variance and often this asymptotic variance involves substantially more complex quantities than the original parameter to be estimated standard errors are important they are used to assess the precision of estimators in the form of confidence intervals to create feasible statistics for testing to build forecasting models based on say daily estimates and also to optimize the tuning parameters the contribution of this paper is to provide an alternative and general solution to this problem which we call observed asymptotic variance it is a general nonparametric method for assessing asymptotic variance avar it provides consistent estimators of avar for a broad class of integrated parameters t dt where the spot parameter process can be a general semimartingale with continuous and jump components the observed avar is implemented with the help of a two scales method its construction works well in the presence of microstructure noise and when the observation times are irregular or asynchronous in the multivariate case the methodology is valid for a wide variety of estimators including the standard ones for variance and covariance and also for more complex estimators such as of leverage effects high frequency betas and semivariance

effects of the reference set on frequentist inferences summary we employ second order likelihood asymptotics to investigate how ideal frequentist inferences depend on the probability model for the data through more than the likelihood function referring to this as the effect of the reference set there are two aspects of higher order corrections to first order likelihood methods namely i that involving effects of fitting nuisance parameters and leading to the modified profile likelihood and ii another part pertaining to limitation in adjusted information generally each of these involves a first order adjustment depending on the reference set however we show that for some important settings likelihood irrelevant model specifications have a second order effect on both of these adjustments this result includes specification of the censoring model for survival data on the other hand for sequential experiments the likelihood irrelevant specification of the stopping rule has a second order effect on adjustment i but a first order effect on adjustment ii these matters raise the issue of what are ideal frequentist inferences since consideration of exact frequentist inferences will not suffice we indicate that to second order ideal frequentist inferences may be based on the distribution of the ordinary likelihood ratio statistic without commonly considered adjustments thereto

improving on exact tests by approximate conditioning summary the often conservative nature for discrete data of so called exact tests seems usually the result of unnecessarily precise conditioning we consider avoiding this by conditioning only approximately on the sufficient statistics for nuisance parameters modest relaxation of conditioning results in small loss in terms of the rationale for conditional inference but can greatly reduce the difficulties caused by discreteness exact calculation of p values based on approximate conditioning is possible but unattractive both in terms of the amount of calculation involved and in requiring explicit specification of the extent to which conditioning is to be relaxed it is shown that there is a highly accurate easily computed and very natural asymptotic approximation that avoids these difficulties

higher order likelihood inference in meta analysis and meta regression this paper investigates the use of likelihood methods for meta analysis within the random effects models framework we show that likelihood inference relying on first order approximations while improving common meta analysis techniques can be prone to misleading results this drawback is very evident in the case of small sample sizes which are typical in meta analysis we alleviate the problem by exploiting the theory of higher order asymptotics in particular we focus on a second order adjustment to the log likelihood ratio statistic simulation studies in meta analysis and meta regression show that higher order likelihood inference provides much more accurate results than its first order counterpart while being of a computationally feasible form we illustrate the application of the proposed approach on a real example

plausibility functions and exact frequentist inference in the frequentist program inferential methods with exact control on error rates are a primary focus the standard approach however is to rely on asymptotic approximations which may not be suitable this article presents a general framework for the construction of exact frequentist procedures based on plausibility functions it is shown that the plausibility function based tests and confidence regions have the desired frequentist properties in finite samples no large sample justification needed an extension of the proposed method is also given for problems involving nuisance parameters examples demonstrate that the plausibility function based method is both exact and efficient in a wide variety of problems

median bias reduction of maximum likelihood estimates for regular parametric problems we show how median centring of the maximum likelihood estimate can be achieved by a simple modification of the score equation for a scalar parameter of interest the estimator is equivariant under interest respecting reparameterizations and is third order median unbiased with a vector parameter of interest componentwise equivariance and third order median centring are obtained like the implicit method of firth 1993 for bias reduction the new method does not require finiteness of the maximum likelihood estimate and is effective in preventing infinite estimates simulation results for continuous and discrete models including binary and beta regression confirm that the method succeeds in achieving componentwise median centring and in solving the boundary estimate problem while keeping comparable dispersion and the same approximate distribution as its main competitors

approximately unbiased tests of regions using multistep multiscale bootstrap resampling approximately unbiased tests based on bootstrap probabilities are considered for the exponential family of distributions with unknown expectation parameter vector where the null hypothesis is represented as an arbitraryshaped region with smooth boundaries this problem has been discussed previously in efron and tibshirani ann statist 26 1998 1687 1718 and a corrected p value with second order asymptotic accuracy is calculated by the two level bootstrap of efron halloran and holmes proc natl acad sci u s a 93 1996 13429 13434 based on the abc bias correction of efron j amer statist assoc 82 1987 171 185 our argument is an extension of their asymptotic theory where the geometry such as the signed distance and the curvature of the boundary plays an important role we give another calculation of the corrected p value without finding the nearest point on the boundary to the observation which is required in the two level bootstrap and is an implementational burden in complicated problems the key idea is to alter the sample size of the replicated dataset from that of the observed dataset the frequency of the replicates falling in the region is counted for several sample sizes and then the p value is calculated by looking at the change in the frequencies along the changing sample sizes this is the multiscale bootstrap of shimodaira systematic biology 51 2002 492 508 which is third order accurate for the multivariate normal model here we introduce a newly devised multistep multiscale bootstrap calculating a third order accurate p value for the exponential family of distributions in fact our p value is asymptotically equivalent to those obtained by the double bootstrap of hall the bootstrap and edgeworth expansion 1992 springer new york and the modified signed likelihood ratio of barndorff nielsen biometrika 73 1986 307 322 ignoring o n 3 2 terms yet the computation is less demanding and free from model specification the algorithm is remarkably simple despite complexity of the theory behind it the differences of the pvalues are illustrated in simple examples and the accuracies of the bootstrap methods are shown in a systematic way

more accurate confidence intervals in exponential families summary fisher s theory of maximum likelihood estimation routinely provides approximate confidence intervals for a parameter of interest 0 the standard intervals 0 za5 s where 0 is the maximum likelihood estimator 5 is an estimate of standard error based on differentiation of the log likelihood function and za is a normal percentile point recent work has produced systems of better approximate confidence intervals which look more like exact intervals when exact intervals exist and in general have coverage probabilities an order of magnitude more accurate than the standard intervals this paper develops an efficient and dependable algorithm for calculating highly accurate approximate intervals on a routine basis for parameters 0 defined in the framework of a multiparameter exponential family the better intervals require only a few times as much computational effort as the standard intervals a variety of numerical and theoretical arguments are used to show that the algorithm works well and that the improvement over the standard intervals can be striking in realistic situations

bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator for rather general parametric models a simple connection is established between the bartlett adjustment factor of the log likelihood ratio statistic and the normalizing constant c of the formula c i i 1 2l for the conditional distribution of a maximum likelihood estimator as applied to the full model and the model of the hypothesis tested this leads to a relatively simple demonstration that division of the likelihood ratio statistic by a suitable constant or estimated factor improves the chi squared approximation to its distribution various expressions for these quantities are discussed in particular for the case of a one dimensional parameter an approximation to the constants involved is derived which does not require integration over the sample space

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

likelihood based confidence intervals for a log normal mean to construct a confidence interval for the mean of a log normal distribution in small samples we propose likelihood based approaches the signed log likelihood ratio and modified signed log likelihood ratio methods extensive monte carlo simulation results show the advantages of the modified signed log likelihood ratio method over the signed log likelihood ratio method and other methods in particular the modified signed log likelihood ratio method produces a confidence interval with a nearly exact coverage probability and highly accurate and symmetric error probabilities even for extremely small sample sizes we then apply the methods to two sets of real life data

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

conditional inferential models combining information for prior free probabilistic inference summary the inferential model im framework provides valid prior free probabilistic inference by focusing on predicting unobserved auxiliary variables but efficient im based inference can be challenging when the auxiliary variable is of higher dimension than the parameter here we show that features of the auxiliary variable are often fully observed and in such cases a simultaneous dimension reduction and information aggregation can be achieved by conditioning this proposed conditioning strategy leads to efficient im inference and casts new light on fisher s notions of sufficiency conditioning and also bayesian inference a differential equation driven selection of a conditional association is developed and validity of the conditional im is proved under some conditions for problems that do not admit a conditional im of the standard form we propose a more flexible class of conditional ims based on localization examples of local conditional ims in a bivariate normal model and a normal variance components model are also given

tail probabilities from observed likelihoods summary an exponential model not in standard form is fully characterized by an observed likelihood function and its first sample space derivative up to one one transformations of the observable variable this property is used to modify the lugannani rice 1980 tail probability approximation to make it parameterization invariant then for general continuous models a version of tangent exponential model is defined and used to derive a general tail probability approximation that uses only the observed likelihood and its first sample space derivative the analysis extends from density functions to distribution functions the tangent exponential model methods of fraser 1988 a related tail probability approximation has been reported barndorff nielsen 1988b in the discussion to reid 1988

approximate confidence intervals for the parameters of a stationary binary markov chain the power divergence family of test statistics introduced by cressie and read 1984 is used to obtain approximate confidence intervals for the parameters of a stationary first order binary markov chain intervals based on inverting the likelihood ratio pearson and freeman tukey statistics are included in this family small sample comparisons are used to select the best intervals for the different parameters the results show that the best power divergence intervals for the probability of success and in particular the likelihood ratio interval have error rates closer to nominal levels than the sample proportion based intervals suggested by crow 1979 the properties of power divergence joint confidence regions for the transition probabilities are also explored

on the common mean of several inverse gaussian distributions based on a higher order likelihood method an interval estimation method for the common mean of several heterogeneous inverse gaussian ig populations is discussed the proposed method is based on a higher order likelihood based procedure the merits of the proposed method are numerically compared with the signed log likelihood ratio statistic two generalized pivot quantities and the simple t test method with respect to their expected lengths coverage probabilities and type i errors numerical studies show that the coverage probabilities of the proposed method are very accurate and type i errors are close to the nominal level 05 even for very small samples the methods are also illustrated with two examples

many valued lattices and their representations this paper presents an investigation of many valued lattices from the point of view of enriched category theory for a bounded partially ordered set p the conditions for p to become a lattice can be postulated as existence of certain adjunctions reformulating these adjunctions by aid of enriched category theory in many valued setting two kinds of many valued lattices weak w lattices and w lattices are introduced it is shown that the notion of w lattices coincides with that of lattice fuzzy orders of belohlavek and the notion of weak w lattices coincides with that of vague lattices of demirci

likelihood and observed geometries a partir du point de vue de la conditionnalite on etablit un parallele observe a la theorie de chentsov efron amari etc en utilisant l information observee et un tenseur d asymetrie observee

the modified signed likelihood statistic and saddlepoint approximations summary for a number of tests in exponential families we show that the use of a normal approximation to the modified signed likelihood ratio statistic r is equivalent to the use of a saddlepoint approximation this is also true in a large deviation region where the signed likelihood ratio statistic r is of order in

practical small sample asymptotics for distributions used in life data analysis fraser and lawless discussed exact conditional intervals for the parameters and quantiles of the location scale model when complete data are used moreover lawless extended the exact method to failure censored type ii censored data nevertheless the exact intervals are difficult to obtain in practice and are unavailable under time censoring type i censoring as a consequence approximate large sample intervals are widely used in this article a likelihood based third order procedure is developed the method does not require explicit nuisance parameterization and can be easily implemented into algebraic computational packages numerical examples are presented to show the accuracy of the method even when the sample size is small

cost allocation the traveling salesman binpacking and the knapsack we examine the problem of cost allocation for the traveling salesman problem tsp we note the significance of a home city in the tsp for designing cost allocation schemes focusing on tsps with triangular inequality in addition a similar cost allocation question is stated for the binpacking and the knapsack problems

empirical bayes methods for combining likelihoods abstract suppose that several independent experiments are observed each one yielding a likelihood l k k for a real valued parameter of interest k for example k might be the log odds ratio for a 2 2 table relating to the kth population in a series of medical experiments this article concerns the following empirical bayes question how can we combine all of the likelihoods l k to get an interval estimate for any one of the k s say 1 the results are presented in the form of a realistic computational scheme that allows model building and model checking in the spirit of a regression analysis no special mathematical forms are required for the priors or the likelihoods this scheme is designed to take advantage of recent methods that produce approximate numerical likelihoods l k k even in very complicated situations with all nuisance parameters eliminated the empirical bayes likelihood theory is extended to situations where the k s have a regression structure as well as an empiri

on parameter transformations and interval estimation summary parameterizations which reduce the asymptotic bias and skewness of various pivotal quantities arising in large sample theory are discussed for models depending on an unknown scalar parameter transformation formulae by which such parameterizations can be obtained are derived and these formulae extend those for one dimensional curved exponential families given by hougaard 1982 to assess the accuracy of normal approximations to the distributions of the pivots their second order properties are considered and comparisons with the signed square root of the likelihood ratio statistic are drawn

approximate interval probabilities for models of parametric dimension 1 two parameterization invariant approximations to the conditional distribution function of the maximum likelihood estimator are considered the first is derived by partial integration of a formula for the conditional density given an ancillary statistic a technique which in addition yields a useful expression for the error term the second approximation is based on an adjusted version of the signed log likelihood ratio statistic a third approximation is a modification of the first that avoids the specification of an ancillary

infereni on full or partial parameters based on the standardized signed log likelihood ratio on etudie des tests et des limites de confiance bases sur la statistique du rapport de log vraisemblance signee en ajustant cette statistique de telle sorte qu a un ordre superieur d approximation la distribution normale standard soit valable

interval estimates for the ratio and difference of two lognormal means health research often gives rise to data that follow lognormal distributions in two sample situations researchers are likely to be interested in estimating the difference or ratio of the population means several methods have been proposed for providing confidence intervals for these parameters however it is not clear which techniques are most appropriate or how their performance might vary additionally methods for the difference of means have not been adequately explored we discuss in the present article five methods of analysis these include two methods based on the log likelihood ratio statistic and a generalized pivotal approach additionally we provide and discuss the results of a series of computer simulations finally the techniques are applied to a real example

approximate inference for the generalized gamma distribution methods of approximate conditional inference in location and scale parameter distributions based on normal approximations to the distributions of signed square roots of likelihood ratio statistics are discussed these methods are applied to obtain approximate inference for the quantiles and scale parameter of the log generalized gamma distribution from uncensored samples assuming that the index parameter of the distribution is known as special cases approximate inference in the extreme value and normal distributions from type ii censored samples is considered the accuracy of the approximate methods for small samples is illustrated by comparison with exact results in some examples

default priors for bayesian and frequentist inference summary we investigate the choice of default priors for use with likelihood for bayesian and frequentist inference such a prior is a density or relative density that weights an observed likelihood function leading to the elimination of parameters that are not of interest and then a density type assessment for a parameter of interest for independent responses from a continuous model we develop a prior for the full parameter that is closely linked to the original bayes approach and provides an extension of the right invariant measure to general contexts we then develop a modified prior that is targeted on a component parameter of interest and by targeting avoids the marginalization paradoxes of dawid and co workers this modifies jeffreys s prior and provides extensions to the development of welch and peers these two approaches are combined to explore priors for a vector parameter of interest in the presence of a vector nuisance parameter examples are given to illustrate the computation of the priors