Approximation for The Expectation of A Function of The Sample Mean

Let X¯ n be the mean of a random sample of size n from a distribution with mean μ and variance σ2. Under some conditions it is shown that Ef(X¯ n ) = f(μ) + (σ2/2n) f″(μ) + O(n −2), and var(f(X¯ n )) = (σ2/n) (f′(μ))2 + O(n −2), where f is a continuous function with a suitable growth condition. This complements a result of Lehmann [(1991). Theory of Point Estimation. Wadsworth, California] and Cramér [(1946). Mathematical Methods of Statistics. Princeton University Press, Princeton, N.J.] for wider application. An illustrative example is given to show an application where the usual approximations do not apply

A Remark on Estimating The Mean of A Normal Distribution with Known Coefficient of Variation

Let X1, X2, …, Xn be iid N(μ, aμ2) (a\u3e0) random variables with an unknown mean μ\u3e0 and known coefficient of variation (CV) √a. The estimation of μ is revisited and it is shown that a modified version of an unbiased estimator of μ [cf. Khan RA. A note on estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1968;63:1039–1041] is more efficient. A certain linear minimum mean square estimator of Gleser and Healy [Estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1976;71:977–981] is also modified and improved. These improved estimators are being compared with the maximum likelihood estimator under squared-error loss function. Based on asymptotic consideration, a large sample confidence interval is also mentioned

Approximation for The Expectation of A Function of The Sample Mean

Let X¯ n be the mean of a random sample of size n from a distribution with mean μ and variance σ2. Under some conditions it is shown that Ef(X¯ n ) = f(μ) + (σ2/2n) f″(μ) + O(n −2), and var(f(X¯ n )) = (σ2/n) (f′(μ))2 + O(n −2), where f is a continuous function with a suitable growth condition. This complements a result of Lehmann [(1991). Theory of Point Estimation. Wadsworth, California] and Cramér [(1946). Mathematical Methods of Statistics. Princeton University Press, Princeton, N.J.] for wider application. An illustrative example is given to show an application where the usual approximations do not apply

A Probabilistic Analysis of The Trading The Line Strategy

We provide analytic models for which the appropriate statistics of the trading the line strategy, N h , can be derived in closed form. In particular, we provide closed-form expressions concerning the average duration of the open position, E(N h ), the variance of the open duration, Var(N h ), the average of the stopped log price, E(S N h ), the variance of the stopped log price, Var(S N h ), the correlation, Corr(N h , S N h ), and the Laplace transform, E(e−s N h ). These results are obtained, in discrete time settings, for binomial and other price scenarios. Furthermore, when analytic results are not possible, such as the case of a normal distribution for log returns, we show by simulation that our general conclusions still hold. Using these statistics we point out some of the subtle features of the trailing stops strategy

A Probabilistic Analysis of The Trading The Line Strategy

We provide analytic models for which the appropriate statistics of the trading the line strategy, N h , can be derived in closed form. In particular, we provide closed-form expressions concerning the average duration of the open position, E(N h ), the variance of the open duration, Var(N h ), the average of the stopped log price, E(S N h ), the variance of the stopped log price, Var(S N h ), the correlation, Corr(N h , S N h ), and the Laplace transform, E(e−s N h ). These results are obtained, in discrete time settings, for binomial and other price scenarios. Furthermore, when analytic results are not possible, such as the case of a normal distribution for log returns, we show by simulation that our general conclusions still hold. Using these statistics we point out some of the subtle features of the trailing stops strategy

In vivo Evaluation of a Cosmetic Emulsion Containing Soybean Extract for Anti-Aging

Purpose: To develop and assess the anti-aging potential of a cosmetic W/O emulsion containing an extract of soybean, Glycine max (L.) Merr. Fabaceae.Methods: This single-blind placebo-controlled study was performed in 11 healthy male human volunteers. A formulation comprising of 4 % of concentrated extract of soybean was prepared by loading the extract in the internal aqueous phase of the emulsion. A control (base), consisting of theemulsion without the extract, was also prepared. Both formulations were applied to the cheeks of all volunteers for 12 weeks and their effect on different skin parameters, i.e., moisture contents, elasticity and surface evaluation of living skin (SELS) were assessed.Results: The formulation containing 4 % soybean extract showed significant (p ≤ 0.05) effects on skin elasticity and moisture contents but the base showed insignificant effect (p ≤ 0.05). There was significant (p . 0.05) decline in SELS, i.e., SEsc (skin scaliness, from 1.73 } 0.05 to 1.66 } 0.06), SEw (skin wrinkles, from 71.74 ± 1.52 to 68.51 ± 1.64), SEsm (skin smoothness, from 109.01 ± 4.77 to 102.03 ± 4.23), and SEr (skin roughness, from 4.04 ±0.09 to 3.82± 0.08) parameters after applicationof the extract formulation for 12 weeks.Conclusion: Topical application of the cosmetic emulsion containing soybean extract exerts potential skin anti-aging effects.Keywords: Glycine max, Soybean, Anti-aging, Skin elasticity, Cosmetic emulsion, Surface evaluation of living skin (SELS), Skin moisture conten

A Numerical Model of an Electrostatic Precipitator

This paper presents a Computational Fluid Dynamics (CFD) model for a wire-plate electrostatic precipitator (ESP). The turbulent gas flow and the particle motion under electrostatic forces are modelled using the CFD code FLUENT. Numerical calculations for the gas flow are carried out by solving the Reynolds-averaged Navier-Stokes equations and turbulence is modelled using the k-ε turbulence model. An additional source term is added to the gas flow equation to capture the effect of electric field. This additional source term is obtained by solving a coupled system of the electric field and charge transport equations. The particle phase is simulated by using Discrete Phase Model (DPM). The results of the simulation are presented showing the particle trajectory inside the ESP under the influence of both aerodynamic and electrostatic forces. The simulated results have been validated by the established data. The model developed is useful to gain insight into the particle collection phenomena that takes place inside an industrial ESP