Minimum Chi-square Method Research Articles

Minimum chi-square method for estimating population size in capture-recapture experiments.

Closed population capture-recapture estimation of population size is difficult under heterogeneous capture probabilities. We introduce the minimum chi-square method which can handle multi-occasion capture-recapture data. It complements likelihood methods with elements that can lead to confidence intervals and assessment of goodness-of-fit. We conduct a comprehensive study on the minimum chi-square method for estimating the size of a closed population using multiple-occasion capture-recapture data under heterogeneous capture probability. We also develop two different bootstrap techniques that can be combined with any underlying estimator, be it the minimum chi-square estimator or a likelihood estimator, to perform useful inference for estimating population size. We present a simulation study on the minimum chi-square method and apply it to analyze white stork multiple capture-recapture data. Under certain conditions, the chi-square method outperforms the likelihood based methods.

Age-and sex-specific reference intervals of total cholesterol, LDL cholesterol, HDL cholesterol and non-HDL cholesterol. Comparison of two algorithms for the indirect estimation of reference intervals

Abstract Objectives Reference intervals of total cholesterol concentrations in plasma and of their fractions low-density lipoprotein (LDL)-, high-density lipoprotein (HDL)-and non-HDL concentrations are seldom studied with respect to the relevance of age and sex. Therefore, the effect of age and sex on the reference intervals was reinvestigated with 2 indirect procedures. Methods As an indirect approach, the truncated minimum chi-square method was applied. All analyses were performed by computer programs available. The script published on the homepage of the German Society of Clinical Chemistry and Laboratory Medicine (DGKL) allows to derive a continuous age dependency of reference intervals together with their confidence and equivalence limits. The results of this approach were compared with those obtained by an indirect method developed more recently, the refineR algorithm. Results In the present study, the upper reference limits of total cholesterol varied from 5.1 to 7.8 mmol/L (197–302 mg/dL) depending on various biological variables (as age, sex, inpatients versus outpatients). These upper limits increased with age. Differences between sexes can be neglected except for the age above 80 years. The pattern of reference limits of LDL cholesterol and non-HDL cholesterol paralleled those of total cholesterol. The reference limits of HDL cholesterol were higher in women than in men but were independent of age. Conclusions Reference limits for the concentrations of total cholesterol and their fractions LDL-, HDL-and non-HDL concentrations should be stratified for age and sex.

Research on Evaluation Method of Storage Reliability of a Control Cabin

A problem that is difficult to assess for storage cabin storage reliability. According to the historical data of the control cabin, the life distribution function was assumed, and the model parameters were estimated by the maximum likelihood estimation method and the minimum chi-square estimation method respectively, and different storage life distribution functions were obtained. Then compare and analyze different storage life distribution functions to determine the storage life distribution function of the control cabin. Finally, the Person’s chi-square goodness-of-fit test was used to verify the results. The life distribution function of the control cabin is the I-type maximum value distribution, and the storage life is 18.12 years when the confidence is 0.90 and the reliability is 0.95. It provides a basis for subsequent life extension maintenance work and compares different storage reliability assessment methods.

Simulated Minimum Quadratic Distance Methods Using Grouped Data for Some Bivariate Continuous Models

Quadratic distance methods based on a special distance which make use of survival functions are developed for inferences for bivariate continuous models using selected points on the nonegative quadrant. A related version which can be viewed as a simulated version is also developed and appears to be suitable for bivariate distributions with no closed form expressions and numerically not tractable but it is easy to simulate from these distributions. The notion of an adaptive basis is introduced and the estimators can be viewed as quasilikelihood estimators using the projected score functions on an adaptive basis and they are closely related to minimum chi-square estimators with random cells which can also be viewed as quasilikeliood estimators using a projected score functions on a special adaptive basis but the elements of such a basis were linearly dependent. A rule for selecting points on the nonnegative quadrant which make use of quasi Monte Carlo (QMC) numbers and two sample quantiles of the two marginal distributions is proposed if complete data is available and like minimum chi-square methods; the quadratic distance methods also offer chi-square statistics which appear to be useful in practice for model testing.

Minimum Quadratic Distance Methods Using Grouped Data for Parametric Families of Copulas

Minimum quadratic distance (MQD) methods are used to construct chi-square test statistics for simple and composite hypothesis for parametric families of copulas. The methods aim at grouped data which form a contingency table but by defining a rule to group the data using Quasi-Monte Carlo numbers and two marginal empirical quantiles, the methods can be extended to handle complete data. The rule implicitly defines points on the nonnegative quadrant to form quadratic distances and the similarities of the rule with the use of random cells for classical minimum chi-square methods are indicated. The methods are relatively simple to implement and might be useful for applied works in various fields such as actuarial science.

Least-squares and minimum chi-square estimation in a discrete Weibull model

ABSTRACTIn this work, we investigate the properties of least-squares and minimum chi-square methods for the point estimation of the two parameters characterizing a discrete Weibull distribution. The first method, inflected into three variants, is based on the empirical cumulative distribution function and provides a closed analytical expression for each estimate. The second method is based on the minimization of the well-known chi-square statistic, which provides a numerical solution. A Monte Carlo simulation study empirically assesses the performance of the methods; two applications on real data show how the inferential techniques practically work.

Dirac coupled channel analyses of the 2− gamma vibrational band excitation in 20Ne

Dirac coupled channel analyses are performed using optical potential model for the high-lying excited states that belong to the 2$^-$ gamma vibrational band at the 800 MeV unpolarized proton inelastic scatterings from $^{20}$Ne. The first order vibrational collective models are used to obtain the transition optical potentials to describe the high-lying excited vibrational collective states and Lorentz-covariant scalar and time-like vector potentials are used as direct optical potentials. The complicated Dirac coupled channel equations are solved phenomenologically to reproduce the differential cross sections data by varying the optical potential and deformation parameters using minimum chi-square method. It is found that relativistic Dirac coupled channel calculation could describe the excited states of the 2$^-$ gamma vibrational band in $^{20}$Ne much better than the nonrelativistic coupled channel calculation, especially for the 2$^-$ and 3$^-$ states of the band. It is shown that the multistep excitation process via channel coupling with the $3^-$ state is essential to describe the $2^-$ state excitation and pure direct transition from the ground state is dominant for the 3$^-$ state excitation of the 2$^-$ gamma vibrational band in $^{20}$Ne.

Root-N consistent semiparametric estimators of a dynamic panel-sample-selection model

Abstract This paper considers the problem of identification and estimation in panel-data sample-selection models with a binary selection rule when the latent equations contain possibly predetermined variables, lags of the dependent variables, and unobserved individual effects. The selection equation contains lags of the dependent variables from both the latent and the selection equations as well as other possibly predetermined variables relative to the latent equations. We derive a set of conditional moment restrictions that are then exploited to construct a three-step sieve estimator for the parameters of the main equation including a nonparametric estimator of the sample-selection term. In the second step the unknown parameters of the selection equation are consistently estimated using a transformation approach in the spirit of Berkson's minimum chi-square sieve method and a first-step kernel estimator for the selection probability. This second-step estimator is of interest in its own right. It can be used to semiparametrically estimate a panel-data binary response model with a nonparametric individual-specific effect without making any other distributional assumptions. We show that both estimators (second and third stage) are n -consistent and asymptotically normal.

Minimum follow-up time required for the estimation of statistical cure of cancer patients: verification using data from 42 cancer sites in the SEER database

BackgroundThe present commonly used five-year survival rates are not adequate to represent the statistical cure. In the present study, we established the minimum number of years required for follow-up to estimate statistical cure rate, by using a lognormal distribution of the survival time of those who died of their cancer. We introduced the term, threshold year, the follow-up time for patients dying from the specific cancer covers most of the survival data, leaving less than 2.25% uncovered. This is close enough to cure from that specific cancer.MethodsData from the Surveillance, Epidemiology and End Results (SEER) database were tested if the survival times of cancer patients who died of their disease followed the lognormal distribution using a minimum chi-square method. Patients diagnosed from 1973–1992 in the registries of Connecticut and Detroit were chosen so that a maximum of 27 years was allowed for follow-up to 1999. A total of 49 specific organ sites were tested. The parameters of those lognormal distributions were found for each cancer site. The cancer-specific survival rates at the threshold years were compared with the longest available Kaplan-Meier survival estimates.ResultsThe characteristics of the cancer-specific survival times of cancer patients who died of their disease from 42 cancer sites out of 49 sites were verified to follow different lognormal distributions. The threshold years validated for statistical cure varied for different cancer sites, from 2.6 years for pancreas cancer to 25.2 years for cancer of salivary gland. At the threshold year, the statistical cure rates estimated for 40 cancer sites were found to match the actuarial long-term survival rates estimated by the Kaplan-Meier method within six percentage points. For two cancer sites: breast and thyroid, the threshold years were so long that the cancer-specific survival rates could yet not be obtained because the SEER data do not provide sufficiently long follow-up.ConclusionThe present study suggests a certain threshold year is required to wait before the statistical cure rate can be estimated for each cancer site. For some cancers, such as breast and thyroid, the 5- or 10-year survival rates inadequately reflect statistical cure rates, and highlight the need for long-term follow-up of these patients.

Long-term survival rate of stages I-III small cell lung cancer patients in the SEER database-application of the lognormal model

Many published studies for small cell lung cancer have short follow-up. Although published meta-analyses favor the use of prophylactic cranial irradiation (PCI) in complete responders, the degree of long-term benefit is not well established. Some studies show convergence of overall and disease-free survival curves with long-term follow-up of patients who received (PCI+) and did not receive PCI (PCI-), while other studies do not. We studied the long-term survival of small cell lung cancer patients in the Surveillance, Epidemiology and End Results (SEER) database, which is a population database and covers 26% of the United States population. A parametric statistical model, Boag’s lognormal model, was retrospectively validated, using the SEER data and the actuarial Kaplan-Meier method of calculation. The basis of the lognormal model is that the logarithm of survival times of cancer patients who died of their disease followed a normal distribution. From 1988–1991, there were 1060 incident cases of stages I-III small cell lung cancer in the 12 registry areas that comprise the SEER database, of whom 132 received (PCI+) and 928 did not receive PCI (PCI-). Two 2-year periods of diagnosis (1988–1989 and 1990–1991) were combined and then patients were followed-up for an additional two years. The first phase in the validation of the lognormal prediction model tested the goodness of fit of the survival times of those cancer patients, who died with their disease, to a lognormal distribution using a minimum chi-square method. The second phase used a maximum likelihood method to predict the long-term survival rate from short-term follow-up data and confirmed (validated) these results by Kaplan-Meier actuarial calculations using long-term available data. The survival time of uncured patients who received (PCI+) and did not receive PCI (PCI-) followed two different lognormal distributions. For PCI+ and PCI- patients respectively, the five-year cancer-specific survival rates, calculated using the Kaplan-Meier method and actual follow-up data to the year 2000, were 18 ± 3% and 12 ± 1%. The corresponding predictions, using the lognormal model were 15% and 11% respectively. The respective 10-year cancer-specific survival rates were 10 ± 3% and 9 ± 1%, when calculated by the Kaplan-Meier method, and 12% and 10% as predicted by the lognormal model. The available follow-up was not long enough to provide 15-year cancer-specific survival rates by the Kaplan-Meier method, but the lognormal model predicted them to be 12% and 10% respectively. The PCI+ and PCI- cancer-specific survival curves almost converged in long-term follow-up. The difference in the two survival curves, as calculated by the logrank test (which takes into account the whole duration of follow-up), was highly statistically significant (p < 0.001). The survival benefit with PCI is more apparent in the short-term, and diminishes with longer-term follow-up in the present study. The lognormal model has the potential to predict the results of on-going prospective PCI trials earlier than would be possible with the use of the Kaplan-Meier method. This may become a useful tool in outcome research

Parameterization of Particle Size Distributions by Three Methods

While the analysis and description of mass-frequency particle size distributions has long been of academic interest in sedimentology such analysis is of considerable economic importance in the commercial recovery of ores and in the handling of mine tailings. An important step in understanding the process of mineral separation and transport is constructing an accurate parametric model of the analysis of the mass-frequency particle size data. Two methods for estimating parametric model parameters already exist (i) the Kullback–Leibler (KL) “distance” function method and (ii) the minimum chi-square (MC) method. During the course of this investigation a third method was developed, the “maximum asymptotic likelihood (MAL) method.” The three methods were compared by application to a practical data set. This comparison demonstrated that the minimum chi-square method is, the most promising of the three.

Assessment of myocardial blood volume and water exchange: theoretical considerations and in vivo results.

The measured signal response in contrast-enhanced myocardial perfusion imaging has been shown to be affected by the rate of water exchange between the intravascular and extravascular compartments, the effect being particularly significant when intravascular contrast agents are used. In the present study, the T(1) relaxation rates were measured in eight pigs in blood and myocardium using a Look-Locker sequence after repeated injections of the intravascular contrast agent NC100150. The selection of myocardial region of interest was automated based on a minimum chi-square method. The intra- and extravascular water exchange rates and the myocardial blood volume were calculated from the measured relaxation rates by applying a two-compartment water exchange limited model that accounts for biexponential longitudinal relaxation. The following (mean +/- SD) values were obtained for the exchange frequency (f), the extravascular residence time (tau(e)), the intravascular residence time (tau(i)) and blood volume (BV), respectively: f = 1.39 +/- 0.52 s(-1), tau(e) = 708 +/- 264 ms, tau(i) = 107 +/- 63 ms, and BV = 11.2 +/- 2.1 mL/100 g. The mean value of f was found to be about 15% higher if biexponential relaxation was not accounted for, supporting the hypothesis that significant biexponential relaxation in tissues with large blood volume can lead to an overestimation of water exchange rates unless corrected for.

A 20-year follow-up study of the long-term survival rate of limited stage small cell lung cancer patients

A 20-year follow-up study of the long-term survival rate of limited stage small cell lung cancer patients

Goodness-of-fit tests for rating scale data: Applying the minimum chi-square method

Goodness-of-fit tests for rating scale data: Applying the minimum chi-square method

Estimation of Mean Rain Rate through Censoring

Abstract A mixed lognormal distribution is fit to rain-rate data to estimate the space–time average. To mitigate problems associated with biased data, data below a minimum and above a maximum threshold are treated as if they are at the respective threshold, and thus in a sense, are censored. The asymptotic variability of the estimate is computed using Fisher information. Computations of asymptotic variance over a portion of the parameter space show that estimates from censored data are more efficient than those from the minimum chi-square and least squares methods. The censoring, minimum chi-square, and least square methods are applied to simulated data to illustrate their usefulness in estimating rain rate.

Simultaneous estimation of all the parameters of a stepwise mutation model.

Minisatellite and microsatellite are short tandemly repetitive sequences dispersed in eukaryotic genomes, many of which are highly polymorphic due to copy number variation of the repeats. Because mutation changes copy numbers of the repeat sequences in a generalized stepwise fashion, stepwise mutation models are widely used for studying the dynamics of these loci. We propose a minimum chi-square (MCS) method for simultaneous estimation of all the parameters in a stepwise mutation model and the ancestral allelic type of a sample. The MCS estimator requires knowing the mean number of alleles of a certain size in a sample, which can be estimated using Monte Carlo samples generated by a coalescent algorithm. The method is applied to samples of seven (CA)n repeat loci from eight human populations and one chimpanzee population. The estimated values of parameters suggest that there is a general tendency for microsatellite alleles to expand in size, because (1) each mutation has a slight tendency to cause size increase and (2) the mean size increase is larger than the mean size decrease for a mutation. Our estimates also suggest that most of these CA-repeat loci evolve according to multistep mutation models rather than single-step mutation models. We also introduced several quantities for measuring the quality of the estimation of ancestral allelic type, and it appears that the majority of the estimated ancestral allelic types are reasonably accurate. Implications of our analysis and potential extensions of the method are discussed. SINCE the discovery that a large number of loci with tandemly repeated sequences in human and many eukaryote species are highly polymorphic because of copy number variation of the repeats in different individuals (Jeffreys 1985; Litt and Luty 1989; Weber and May 1989), allele size data from such loci are rapidly becoming the dominant source of genetic markers for genome mapping, forensic testing, and population studies. Loci with repeat sequences longer than 5 bp are generally referred to as minisatellite or variable number tandem repeat loci, and those with repeat sequences between 2 to 5 bp are referred to as microsatellite or short tandem repeat loci (Tautz 1993). Because mutations change the copy number of such loci in a stepwise fashion, rapid accumulation of population samples from minisatellite and microsatellite loci has resurrected the interest of the stepwise mutation model (SMM), which was popular in the 1970s.

New Minimum Chi-Square Methods in Empirical Finance

The paper reviews recently developed simulation-based minimum chi-square estimators for structural models. Particular attention is paid to selection of the auxiliary model that defines the GMM-type criterion used in the minimum chi-square estimation. Considerations of statistical efficiency and behavior under misspecification make a strong case for using a very flexible, nonparametric approach to select the auxiliary model. To avoid a numerically ill-behaved GMM criterion function, the dynamic stability of the auxiliary model must also be verified, though, interestingly, the dynamic stability of the structural model itself is automatically enforced and need not be imposed in estimation. The empirical application involves estimation of a single-factor diffusion model for the 30-day Eurodollar interest rate.

Semiparametric Estimation of Willingness to Pay Distributions

The most popular survey method used in contingent valuations asks open-ended dichotomous choice questions. This method generates grouped or interval-censored data on respondents' willingness to pay. This paper specifies the willingness to pay distribution using the proportional hazard specification in duration analysis. This semiparametric distribution, on the one hand, controls for the effects of observed personal characteristics, and on the other, allows the shape of the distribution to be unspecified. To estimate the willingness to pay distribution from grouped data, we propose both a maximum likelihood estimation method and a minimum Chi-square method. The latter procedure applies to many observations per cell cases where the observable covariates are either categorical or amendable to sensible grouping. Specification tests for the proportionality assumption are proposed. The statistical inference procedures are illustrated using the data set from the San Joaquin Valley contingent valuation survey.

Special Versus General Elections and Composition of the Voters: Evidence From Louisiana School Tax Elections

Using Louisiana school board property tax elections from the past decade, the authors study the question of whether or not special elections tend to produce lower turnout and a greater percentage of yes votes than do general elections. With the problem focusing on the choice of voting yes, voting no, or abstaining from voting, modified minimum chi-square methods are used in the analysis. The authors find that opposition to local school taxes increases with turnout. They also find that turnout is affected by the size of the tax, by the presence or absence of other taxes on the ballot, and by the presence or absence of state or federal matters on the same ballot. Both the inclusion of state or federal issues (or candidates) on the ballot and a higher tax rate lead to increased relative opposition at the polls.

On a generalization of Morisita's model for estimating the habitat preference

The Morisita's model for estimating the habitat preference by the ant lionsGenuroides japonicus is generalized by introducing, in addition to the environmental densitiesa andb, a repulsivity parameter ϑ. The probability function of the numberL n of individuals choosing fine sand to settle when a total ofn ant lions are introduced is examined. A heuristic and the minimum chi-square methods for estimating the parametersa, b and ϑ are discussed.