Gamma Function Research Articles

Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γze−sz cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function f is mapped to f(0). This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of q-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients.

Inactivation of catalytic and non-catalytic PI3-kinase gamma function enhances hypoxia-induced pulmonary hypertension in mice

Abstract Rationale: Pulmonary hypertension (PH) is a pulmonary vascular disease characterised by chronically elevated pulmonary arterial mean pressure, increased pulmonary vascular resistance and right ventricular (RV) dysfunction and hypertrophy. The pathogenesis is characterised by vasoconstriction of pulmonary vessels and pulmonary vascular remodelling. PI 3-kinase γ (PI3Kγ) is activated via G protein-coupled receptors and abundantly expressed in all cell types involved in the pathogenesis of PH including leukocytes, cardiomyocytes, and endothelial cells (EC). The catalytic function of PI3Kγ is involved in numerous processes that are important for both vascular remodelling and maladaptive cardiac hypertrophy, including leukocyte recruitment and EC proliferation and survival. On the other hand, its non-catalytic function affects cyclic adenosine monophosphate (cAMP) production and thus cardiac contractility. Furthermore, inactivation of PI3Kγ impairs nitric oxide (NO) production by endothelial NO synthase (eNOS), which may lead to increased vascular resistance. Therefore, the aim of our study was to investigate the role of both catalytic and non-catalytic functions of PI3Kγ in the pathogenesis of PH. Methods PI3Kγ knockout mice (PI3Kγ-/-), as well as mice with a catalytically inactive form of PI3Kγ (PI3KγKD/KD) were analysed in vivo using the hypoxia-induced mouse model of PH (21 days at 10% O2 hypoxia (HOX)). To specifically inhibit the non-catalytic function of PI3Kγ, C57BL/6N mice were treated with a mimetic peptide (MP) via intratracheal instillation and studied in the same model. Systolic right ventricular pressure (RVSP) was measured using a Millar pressure catheter inserted via the jugular vein. RV hypertrophy was determined by Fulton's index (RV/LV+S). Results PI3Kγ-/- mice showed significantly increased RVSP after three weeks of hypoxia compared to WT littermate controls (HOX WT 35.64±3.21mmHg vs HOX PI3Kγ-/- 37.04±2.31mmHg;p=0.0049). A significant increase in RVSP was also detected in PI3KγKD/KD and MP-treated mice compared to respective WT controls (HOX WT 34.67±3.83mmHg vs. HOX PI3KγKD/KD 37.9±2.19mmHg;p=0.0404 and HOX vehicle 34.19±2.74mmHg vs. HOX MP 37.61±2.46mmHg;p=0.0111). Under normoxic conditions (NOX), an increased RVSP was already measured in PI3Kγ-/- mice (NOX WT 26.13±1.2mmHg vs. NOX PI3Kγ-/- 28.07±0.88;p=0.0157). Heart rate and systemic blood pressure remained unchanged. A significant increase in RV hypertrophy was only evident for PI3KγKD/KD compared to WT controls (HOX WT 0.38±0.06 vs. HOX PI3KγKD/KD 0.46±0.09mmHg;p=0.0037). Conclusion The results show that both catalytic and non-catalytic inactivation of PI3Kγ in vivo do not counteract the pathogenesis of PH, but conversely enhance it. In this context, reduced phosphorylation of eNOS may play a crucial role, leading to increased vasoconstriction, whereas increased cAMP levels may help the RV to adapt to the increased pressure. The exact mechanisms will be analysed in the future.

Abstract 18479: Lack of Catalytic and Non-Catalytic Function of PI3 Kinase Gamma in Mice Increases Hypoxia-Induced Pulmonary Hypertension

Rationale: Pulmonary hypertension (PH) is characterised by vasoconstriction and remodelling of pulmonary vessels, leading to elevated pulmonary arterial pressure and right ventricular (RV) hypertrophy. PI 3-kinase γ (PI3Kγ) is expressed in leukocytes, cardiomyocytes, and endothelial cells (EC) which are all involved in the pathogenesis of PH. The catalytic function of PI3Kγ is involved in leukocyte recruitment and EC proliferation which are important for vascular remodelling and maladaptive cardiac hypertrophy whereas its non-catalytic function affects cardiac contractility via cAMP signaling. Furthermore, inactivation of PI3Kγ impairs nitric oxide production by eNOS, which may lead to increased vascular resistance. Therefore, we investigated the role of both catalytic and non-catalytic functions of PI3Kγ in the pathogenesis of PH. Methods: PI3Kγ knockout mice (PI3Kγ -/- ), as well as mice with a catalytically inactive form of PI3Kγ (PI3Kγ KD/KD ) were exposed to hypoxia-induced PH (21 days at 10% O 2 hypoxia (HOX)). Systolic right ventricular pressure (RVSP) was measured using a Millar pressure catheter inserted via the jugular vein. RV hypertrophy was determined by Fulton's index. Results: PI3Kγ -/- mice showed significantly increased RVSP after hypoxia compared to WT controls (HOX WT 34.16±3.21 mmHg vs HOX PI3Kγ -/- 37.04±2.31 mmHg; p=0.0049). Already under normoxia (NOX), an increased RVSP was measured in PI3Kγ -/- mice (NOX WT 26.13±1.2mmHg vs. NOX PI3Kγ -/- 28.07±0.88 mmHg, p=0.0157). A significant increase in RVSP was also detected in PI3Kγ KD/KD (HOX WT 34.67±2.02 mmHg vs. HOX PI3Kγ KD/KD 37.95±1.39 mmHg; p=0.0228). Here, a significant increase in RV hypertrophy was evident for PI3Kγ KD/KD mice (HOX WT 0.38±0.06 vs. HOX PI3Kγ KD/KD 0.46±0.09; p=0.0037) which was not detected in PI3Kγ -/- compared to WT controls after hypoxia. Conclusion: The results show that both catalytic and non-catalytic inactivation of PI3Kγ in vivo do not counteract the pathogenesis of PH, but conversely enhance it. In this context, reduced phosphorylation of eNOS may play a crucial role, leading to increased vasoconstriction, whereas increased cAMP levels in PI3Kγ -/- may help the RV to adapt to the increased pressure. The exact mechanisms will be analysed in the future.

An optimal system for increasing the contrast resolution qualities of histopathology images in the wavelet domain

AbstractIn the field of histopathology, it has been observed that most images are of poor quality as a result of inadequate luminance and contrast. The first step in identifying and analyzing them is to enhance their quality while maintaining their inherent characteristics. This paper presents a novel method for enhancing histopathology images' important features, brightness preservation, and contrast by utilizing the adaptive intensity transformation gamma function in the wavelet domain. The RGB histopathological color image is first converted to the HSV color model. The V component is then stretched in order to compensate for any color distortion. In the proposed scheme, the stretched V component is split into four sub‐bands using discrete wavelet transform (DWT). Next, it determines the optimal value of the gamma correction function in the DWT lower sub‐band. A correction factor is applied using the singular value decomposition (SVD) technique to determine the optimal gamma function value using the nature‐inspired particle swarm optimization method (NIPSO). The corrected LL sub‐band is then applied to the inverse DWT (IDWT) together with the other unprocessed sub‐bands. In addition, the contrast of each feature is enhanced by employing an adaptive histogram equalization model that is contrast‐limited. The proposed method was evaluated through experiments and validations on a variety of histopathology images retrieved from multiple databases. Several quantitative studies have demonstrated that the proposed method outperforms existing enhancement techniques in terms of entropy, edge preservation index, contrast ratio, and Universal Image Quality Index (UIQI). This proposed technique increases contrast, while preserving the inherent characteristics of the original image, resulting in better quality histopathology images that can be used in disease diagnosis and inspection.

Height-Fixed UAV Enabled Energy-Efficient Data Collection in RIS-Aided Wireless Sensor Networks

Wireless sensor nodes (SNs) are usually energy-limited, a low energy efficiency of SN may be caused by the occlusion of transmission link in the unmanned aerial vehicle (UAV)-enabled data collection. Reconfigurable intelligent surfaces (RISs) provide potential communication opportunities by reflecting and enhancing the signal, thus increasing SNs’ energy efficiency. This paper investigates energy-efficient data collection aided by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> RIS elements, where a height-fixed and flight time-constrained UAV is considered as the collector, a tractable air-to-ground channel model is established considering the cascaded RIS channel and time-varying trajectory of UAV, and energy consumption of SNs is minimized. Specifically, coupling two intractable fourth-order moment variables, the air-to-ground channel model is processed into an easily computable expression through the Gamma function. Furthermore, based on the proposed two RIS deployment schemes, maximum energy consumption of SNs minimization problem is formulated as a non-convex mixed-integer nonlinear programming (NMNP) problem with optimization of UAV trajectory and SNs wake-up schedules. Additionally, through the slack technique and Taylor expansion, the NMNP problem is decoupled as linear programming and quadratically constrained quadratic programming, and an effective algorithm is designed to obtain the sub-optimal solution. Simulations prove the validity of our algorithm and RIS deployment insights are given for SNs energy saving.

Genetic aspects of Wood's lactation curve parameters in Jersey crossbred cattle using Bayesian approach.

The study was undertaken to estimate the genetic parameters of lactation curve parameters of Wood's function in Jersey crossbred cattle using the Bayesian approach. Data on 33,906 fortnightly test day milk yields of 1,718 lactation records of Jersey crossbred cows, maintained at the ICAR-National Dairy Research Institute in West Bengal, were collected over a period of 40 years. The lactation curve parameters including 'a' (initial milk yield after calving), 'b' (ascending slope up to peak yield) and 'c' (descending slope after peak yield) and lactation curve traits, peak yield (ymax), time of peak yield (tmax) and persistency of milk yield (P) of individual cow for each lactation were estimated using the incomplete gamma function (Wood's model) by fitting the Gauss-Newton algorithm as an iteration method using PROC NLIN procedure of SAS 9.3. Variance components and genetic parameters of lactation curve parameters/traits were estimated by a repeatability animal model using the Bayesian approach. Estimates of heritabilities were found to be 0.18 ± 0.05, 0.09 ± 0.03 and 0.11 ± 0.04 for parameters 'a', 'b' and 'c', respectively and 0.24 ± 0.05, 0.12 ± 0.04, and 0.15 ± 0.05 for ymax, tmax and P, respectively. Repeatability estimates were 0.31 ± 0.03, 0.21 ± 0.04 and 0.30 ± 0.04 for parameters 'a', 'b' and 'c' respectively and 0.39 ± 0.03, 0.24 ± 0.03 and 0.37 ± 0.03 for ymax, tmax and p, respectively. Genetic correlations among lactation curve parameters/traits ranged from -0.75 to 0.95. Existence of genetic correlations among lactation curve parameters/traits indicated substantial genetic and physiological relationships among lactation curve parameters/traits of Jersey crossbred cattle.

A Bi−directional method for evaluating integrals involving higher transcendental functions. HyperRAF: A Julia package for new hyper−radial functions

The electron repulsion integrals over Slater−type orbitals with non−integer principal quantum numbers are investigated. These integrals are useful in both non−relativistic and relativistic calculations of many−electron systems. They involve hyper−geometric functions that are practically difficult to compute. Relationships free from hyper−geometric functions for expectation values of Coulomb potential (r21−1) are derived. These relationships are new and show that the complication coming from two−range nature of Laplace expansion for the Coulomb potential is removed. This is achieved by utilizing auxiliary functions represented in finite power series. They serve as essential components in deriving straightforward recurrence relationships for electron repulsion integrals. In the context of computing the expectation values of potentials with arbitrary power, the methodology presented here for evaluation of these integrals forms the initial condition. It is also adapted to multi−center integrals. Program summaryProgram Title: HyperRAFCPC Library link to program files:https://doi.org/10.17632/6pbv2y7s42.1Developer's repository link:https://github.com/abagciphys/HyperRAF.gitLicensing provisions: MITProgramming language:Julia Programming Language [1]Supplementary material: An exploratory variant of the software program written in the Mathematica Programming Language [2].External routines/libraries:Nemo, a computer algebra package for the Julia programming language [3], JRAF, a Julia package for computation of relativistic molecular auxiliary functions [4].Nature of problem: Definite integrals involving higher transcendental functions given by,fmn1(a,b,x)=xm−1e−bxΓ[n,ax], andfmn2(a,b,x)=xm−1e−bxγ[n,ax] are frequently encountered in atomic physics, with the electron repulsion integral being a notable illustration.For exclusive solutions to these integrals, one can refer to Erdélyi's [5] or Gradshteyn and Ryzhik's [6] books. The solutions are obtained by using the series representation of incomplete gamma functions. The result is hyper−geometric functions of the form F12[1,b,c;z], where b=m+n, c=m+1, z=b/(a+b) for fmn1(a,b,x) and c=n+1, z=a/(a+b) for fmn2(a,b,x), respectively. Due to the non−trivial structure of infinite series that are used to define them, the computation for hyper−geometric poses challenges. Convergence of their series strictly depends on the values of parameters. Computational issues such as cancellation or round−off error emerge.Solution method: This research introduces novel bi−directional hyper−radial functions that are used to establish fresh recurrence relationships for electron repulsion integrals, eliminating the dependence on hyper−geometric functions. Dual functionality is inherent in the hyper−radial functions as they offer alternative solutions for definite integrals involving higher transcendental functions. Additionally, they transform the representation of the hyper−geometric functions into finite power series.

On Degenerate Laplace-type Integral Transform

This paper is motivated by the work of Taekyun Kim and Dae San Kim on the degenerate Laplace transform and degenerate gamma function, as published in the Russian Journal of Mathematical Physics. We introduce the degenerate Laplace-type integral transform and delve into its properties and interrelations. This paper focuses on the degenerate Laplace-type integral transforms of several fundamental functions, including the degenerate sine, degenerate cosine, degenerate hyperbolic sine, and degenerate hyperbolic cosine functions. Furthermore, we establish crucial connections between the degenerate Laplace-type integral transform and existing degenerate integral transforms. Specifically, we explore its relationships with the degenerate Laplace transform, the degenerate Elzaki transform, and the degenerate Sumudu transforms.

An alternative proof of Tataru’s dispersive estimates

Abstract The aim of this article is to give an alternative proof of Tataru’s dispersive estimates for wave equations posed on the hyperbolic space. Based on the formula for the wave kernel on ℍ n {\mathbb{H}^{n}} , we give the proof from the perspective of Bessel potentials, by exploiting various facts about Gamma functions, modified Bessel functions, and Bessel potentials. This leads to our proof being more self-contained than that in [D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc. 353 2001, 2, 795–807].

Method for determining eigen numbers in heat conduction problems for a cylinder

RELEVANCE. Due to the difficulties of finding eigenvalues and proper functions for bodies with axial (cylinder) and central (ball) symmetries defined in classical methods from the edge Sturm-Liouville problems, including Bessel equations whose exact analytical solutions are not obtained (known only numerical solutions, described by approximation formulas), there is a need to develop analytical methods of their solution.THE PURPOSE. Using orthogonal methods of weighted residuals, an approximate analytical method for determining eigenfunctions and eigenvalues in boundary value problems with axial and central symmetry (cylinder, ball) has been developed.METHODS. The method is based on the use of orthogonal systems of coordinate functions and additional boundary conditions. Latter are in such a form that their fulfillment by the desired solution is equivalent to the fulfillment of the differential equation of the boundary value problem at the boundary points of the region, leading to its fulfillment inside the considered region. Moreover, the accuracy of the equation depends on the number of approximations, which, in turn, depends on the number of additional boundary conditions used. Using the orthogonality property of trigonometric coordinate functions included in a series representing eigenfunctions makes it possible to increase the accuracy of the fulfillment of the differential equation of the Sturm-Liouville boundary value problem and the accuracy of determining the eigenvalues. To satisfy the initial condition, its residual is compiled and the condition of its orthogonality to all coordinate functions is required. The orthogonality of trigonometric systems of coordinate functions with respect to the unknown constants of integration leads to a system of algebraic linear equations, the number of which is equal to the number of approximations. As a result, the fulfillment of the initial condition is simplified and the accuracy of its fulfillment is increased.RESULTS. The advantage of the method is that the resulting solution contains only simple algebraic expressions, excluding special functions (Bessel function, Legendre function, gamma function).CONCLUSION. Thus, bypassing direct integration over a spatial variable, the use of additional boundary conditions makes it possible to find a solution of any complexity of the equations of the Sturm-Liouville boundary value problem, which reduces to the definition of simple integrals.

Ohno relation for regularized multiple zeta values

The Ohno relation for multiple zeta values can be formulated as saying that a certain operator, defined for indices, is invariant under taking duals. In this paper, we generalize the Ohno relation to regularized multiple zeta values by showing that, although the suitably generalized operator is not invariant under taking duals, the relation between its values at an index and at its dual index can be written explicitly in terms of the gamma function.

Fractional distributional representation of gamma function and the generalized kinetic equation

Motivated by the recent research about non-integer order derivatives of Dirac delta function, we use a fractional Taylor series to explore a fractional distributional representation of gamma function. It proved significant to develop a novel fractional kinetic equation, which is an essential mathematical model to measure the degree of transformation in the chemical structure of stars (sun) as a key component of our environment. In addition to the classical solution of the formulated fractional kinetic equation, the distributional solution is also attained which is not possible without using fractional derivatives of Dirac delta function. Novel identities involving the gamma function are obtained using several fractional transforms. Convergence of new series is proved using the theory of distributions leading to further new applications.

Notes on Various Transforms Identified by Some Special Functions with Complex (or Real) Parameters and Some of Related Implications

The fundamental aim of this special research is first to introduce certain essential information in regards to some special functions, which are the Gamma function and the Beta function and play a big role in both (applied) mathematics and most engineering sciences, and then to present both a number of their familiar properties and several relationships between them. Afterward, various possible-undeniable effects of those special functions in the transformation theory, their special implications, and suggestions for the relevant researchers will be also considered as special information.

On the properties of invariant functions

If f(x,y) is a real function satisfying y>0 and ∑r=0n−1f(x+ry,ny)=f(x,y) for n=1,2,3,…, we say that f(x,y) is an invariant function. Many special functions including Bernoulli polynomials, Gamma function and Hurwitz zeta function are related to invariant functions. In this paper we systematically investigate the properties of invariant functions.

Refinement of Gautschi's harmonic mean inequality for the gamma function

In 1974, W. Gautschi proved that 1&lt;\frac{2}{1/\Gamma(x) +1/\Gamma(1/x)} \quad \textrm{for} \quad 0&lt;x\neq 1. Here, we present the following refinement: 1&lt;\Gamma\Bigl( \frac{2}{x+1/x}\Bigr)&lt; \frac{2}{1/\Gamma(x) +1/\Gamma(1/x)}, \quad 0&lt;x\neq 1.

A novel fractional-order grey prediction model: a case study of Chinese carbon emissions.

Objective and accurate prediction of carbon emissions can provide a basis for the country to achieve carbon emission reduction targets and can also comprehensively and scientifically predict the peak of carbon emissions effectively, providing valuable reference information for the implementation of specific emission reduction policies and measures at each stage. In this paper, a novel fractional-order grey multivariate forecasting model is established to analyze and forecast China's carbon emissions, reflecting the principle of new information priority. The model adds fractional-order cumulative sequences to the traditional integer-order cumulative sequences, uses the Gamma function to represent the fractional-order sequences and the time-response equation, and uses the particle swarm algorithm to find the optimal order of the cumulative sequence. Finally, the modeling steps of the model are given. Then, the new model is analyzed for its effectiveness from three different perspectives using 21years of Chinese carbon emission data. The results of the first and second cases show that the newly established particle swarm optimization fractional-order model is superior to the grey multivariate comparison model. The results of the third case show that the new model is superior to the three classical grey prediction comparison models. It has stable characteristics for both simulation and prediction and also shows high accuracy, and all three cases fully illustrate the effectiveness of the new model. Finally, this new model is applied to forecast China's carbon emissions from 2022-2026, analyze the forecast results, and make relevant recommendations.

Estimating effective survey duration in camera trap distance sampling surveys.

Among other approaches, camera trap distance sampling (CTDS) is used to estimate animal abundance from unmarked populations. It was formulated for videos and observation distances are measured at predetermined 'snapshot moments'. Surveys recording still images with passive infrared motion sensors suffer from frequent periods where animals are not photographed, either because of technical delays before the camera can be triggered again (i.e. 'camera recovery time') or because they remain stationary and do not immediately retrigger the camera following camera recovery time (i.e. 'retrigger delays'). These effects need to be considered when calculating temporal survey effort to avoid downwardly biased abundance estimates. Here, we extend the CTDS model for passive infrared motion sensor recording of single images or short photo series. We propose estimating 'mean time intervals between triggers' as combined mean camera recovery time and mean retrigger delays from the time interval distribution of pairs of consecutive pictures, using a Gamma and Exponential function, respectively. We apply the approach to survey data on red deer, roe deer and wild boar. Mean time intervals between triggers were very similar when estimated empirically and when derived from the model-based approach. Depending on truncation times (i.e. the time interval between consecutive pictures beyond which data are discarded) and species, we estimated mean time intervals between retriggers between 8.28 and 15.05 s. Using a predefined snapshot interval, not accounting for these intervals, would lead to underestimated density by up to 96% due to overestimated temporal survey effort. The proposed approach is applicable to any taxa surveyed with camera traps. As programming of cameras to record still images is often preferred over video recording due to reduced consumption of energy and memory, we expect this approach to find broad application, also for other camera trap methods than CTDS.

Slash-Weighted Lindley Distribution: Properties, Inference, and Applications

The slash-weighted Lindley model is introduced due to the need to obtain a model with more kurtosis than the weighted Lindley distribution. Several expressions for the pdf of this model are given. Its cumulative distribution function is expressed in terms of a generalized hypergeometric function and the incomplete gamma function. Moments and maximum likelihood estimation were studied. A simulation study was carried out to illustrate the good performance of the estimates. Finally, two real applications are included.

On performance of short-packet ultra-reliable and low-latency communications NOMA MIMO relay networks

This paper proposes and analyzes the performance of a multi-antenna NOMA relay system over the Nakagami-m fading channel, in which the size of signal packets is finite. Based on the application of the Gaussian–Chebyshev quadrature, Riemann integral and incomplete Gamma function, the exact and asymptotic closed-form expressions of the block error rate (BLER), throughput, goodput of the proposed finite block-length multi-antenna NOMA relay system are derived. Moreover, in order to improve the BLER, throughput, and goodput of the proposed system, maximal-ratio combining and maximal-ratio transmission are applied to receivers and transmitters, respectively. The provided analysis and simulation results indicate that the proposed system meets the requirements of ultra-reliable and low-latency communications (URLLC). Especially, the reliability is higher than 99.99%, i.e., the BLER is less than 10−4. The accuracy of analysis results is verified by simulation results. Furthermore, the number of transmission bits and power allocation coefficients are optimized to maximize the throughput, and goodput and minimize the BLER.

Calibration estimation of subpopulation total for direct and indirect situations

Sample surveys are conducted to estimate parameters of subpopulation such as mean, variance and totals, etc. The utility of the supporting information lies in the calibration technique, we use to estimate the subpopulation parameter. In this article, we propose calibration-based estimators for a total of the subpopulations. The defined estimators are based on gamma and exponential models. Since both models have right-skewed or positively skewed forms, the defined estimators are appropriate for right-skewed or positively skewed supportive variables. Two situations are considered in which the total of the supporting variable hold for all subpopulations: (i) the total of subpopulations of the supporting variable is known in advance and (ii) the total subpopulations of the supportive variable are not known. Due to difficulty arising in the calibration weight, we have carried out a simulation study in terms of absolute relative bias and simulated relative standard error using Sweden Municipalities 1984 (Sarndal, Swensson, and Wretman, 1992, Appendix B). A simulation study shows that the estimator based on the gamma function is more efficient than ratio, generalized regression, and exponential estimators for all subpopulations in the considered situations.