Pitman Estimator Research Articles

Pitman estimator for the parameter and reliability function of the exponential distribution

The easiest and most essential distribution in reliability studies is the exponential distribution. An important issue in this area is to seek for a good estimator to estimate the parameter and reliability function of this distribution. So in this article, The Pitman method is used to derive an estimator for the parameter and reliability function of the exponential distribution, then a comparison was made with the usual methods such as maximum likelihood estimator and Bayes estimator through simulation studies with different sizes of samples. The results showed that the Pitman estimator has a good performance for the parameter of exponential distribution compared with other methods by relying on the statistical measure Mean square error. Also it showed that the Bayes estimator has a good performance compared with the other methods in estimating the reliability function by relying on the statistical measure integral mean square error.

Translation invariant statistical experiments with independent increments

We provide a full description of the class of translation invariant experiments with independent increments. Necessary and sufficient conditions for the weak convergence and the comparison of experiments within this class are given. Finally, we prove exponential boundedness of Pitman estimators in these models.

Contribution to the Theory of Pitman Estimators

New inequalities are proved for the variance of the Pitman estimators (minimum variance equivariant estimators) of θ constructed from samples of fixed size from populations F(x − θ). The inequalities are closely related to the classical Stam inequality for the Fisher information, its analog in small samples, and a powerful variance drop inequality. The only condition required is finite variance of F; even the absolute continuity of F is not assumed. As corollaries of the main inequalities for small samples, one obtains alternate proofs of known properties of the Fisher information, as well as interesting new observations like the fact that the variance of the Pitman estimator based on a sample size n scaled by n monotonically decreases in n. Extensions of the results to polynomial versions of the Pitman estimators and a multivariate location parameter are given. Also, the search for the characterization of equality conditions for one of the inequalities leads to a Cauchy-type functional equation for independent random variables, and an interesting new behavior of its solutions is described. Bibliography: 21 titles.

On Moments of Pitman Estimators: The Case of Fractional Brownian Motion

In some nonregular statistical estimation problems, the limiting likelihood processes are functionals of fractional Brownian motion (fBm) with Hurst's parameter $H$, $0<H\le 1$. In this paper we present several analytical and numerical results on the moments of Pitman estimators represented in the form of integral functionals of fBm. We also provide Monte Carlo simulation results for variances of Pitman and asymptotic maximum likelihood estimators.

Robust Coherent Noise Suppression by Point Estimation of the Cauchy Location Parameter

This paper introduces a new point estimation algorithm, with particular focus on coherent noise suppression, given several measurements of the device under test where it is assumed that 1) the noise is first-order stationery and 2) the device under test is linear and time-invariant. The algorithm exploits the robustness of the Pitman estimator of the Cauchy location parameter through the initial scaling of the test signal by a centred Gaussian variable of predetermined variance. It is illustrated through mathematical derivations and simulation results that the proposed algorithm is more accurate and consistently robust to outliers for different tailed density functions than the conventional methods of sample mean (coherent averaging technique) and sample median search. Keywords—Central limit theorem, Fisher-Cramer Rao, gamma function, Pitman estimator.

Pitman Estimators: An Asymptotic Variance Revisited

We provide an analytic expression for the variance of ratio of integral functionals of fractional Brownian motion which arises as an asymptotic variance of Pitman estimators for a location parameter of independent identically distributed observations. The expression is obtained in terms of derivatives of a logarithmic moment of the integral functional of limit likelihood ratio process (LLRP). In the particular case when the LLRP is a geometric Brownian motion, we show that the established expression leads to the known representation of the asymptotic variance of Pitman estimator in terms of Riemann zeta-function.

Generalized confidence interval and p-value in location and scale family

The concept of generalized pivotal quantity (GPQ) and generalized p-value (GPV) have recently become popular in small-sample inferences for complex problems such the Behrens–Fisher problem. These techniques have been shown to be efficient in specific distributions by using MLEs. In this paper, we extend the current literature in two directions; first, we propose a unified approach for constructing GPQ and GPV for the parameters of location-scale families and secondly, we use such approach along with Pitman estimators of location and scale parameters to make inferences based on GPQ and GPV. The Pitman estimators are known to be Minimum Risk equivariant (MRE) estimators which may exist even when the MLEs may not. Thus, this paper extends the generalized inference methodology that has so far been applied to specific distributions and by use of MLEs and conditional inference, to general location-scale families and to cases where the MLE may not exist. The performance of the approach is illustrated by Monte Carlo simulations and by application to air lead levels data set, collected by the National Institute of Occupational Safety and Health (NIOSH).

On Bayes’s theorem for improper mixtures

Although Bayes's theorem demands a prior that is a probability distribution on the parameter space, the calculus associated with Bayes's theorem sometimes generates sensible procedures from improper priors, Pitman's estimator being a good example. However, improper priors may also lead to Bayes procedures that are paradoxical or otherwise unsatisfactory, prompting some authors to insist that all priors be proper. This paper begins with the observation that an improper measure on Theta satisfying Kingman's countability condition is in fact a probability distribution on the power set. We show how to extend a model in such a way that the extended parameter space is the power set. Under an additional finiteness condition, which is needed for the existence of a sampling region, the conditions for Bayes's theorem are satisfied by the extension. Lack of interference ensures that the posterior distribution in the extended space is compatible with the original parameter space. Provided that the key finiteness condition is satisfied, this probabilistic analysis of the extended model may be interpreted as a vindication of improper Bayes procedures derived from the original model.

A smoothing principle for the Huber and other location M-estimators

A smoothing principle for M-estimators is proposed. The smoothing depends on the sample size so that the resulting smoothed M-estimator coincides with the initial M-estimator when n → ∞ . The smoothing principle is motivated by an analysis of the requirements in the proof of the Cramér–Rao bound. The principle can be applied to every M-estimator. A simulation study is carried out where smoothed Huber, ML-, and Bisquare M-estimators are compared with their non-smoothed counterparts and with Pitman estimators on data generated from several distributions with and without estimated scale. This leads to encouraging results for the smoothed estimators, and particularly the smoothed Huber estimator, as they improve upon the initial M-estimators particularly in the tail areas of the distributions of the estimators. The results are backed up by small sample asymptotics.

Some inequalities related to the Stam inequality

Zamir showed in 1998 that the Stam classical inequality for the Fisher information (about a location parameter) $$ 1/I(X + Y) \geqslant 1/I(X) + 1/I(Y) $$ for independent random variables X, Y is a simple corollary of basic properties of the Fisher information (monotonicity, additivity and a reparametrization formula). The idea of his proof works for a special case of a general (not necessarily location) parameter. Stam type inequalities are obtained for the Fisher information in a multivariate observation depending on a univariate location parameter and for the variance of the Pitman estimator of the latter.

The Pitman estimator of the Cauchy location parameter

This paper examines the estimation of the Cauchy location parameter when the scale parameter is known. Using the squared error loss function, a closed form of the minimum risk equivariant (MRE) estimator is derived. While several properties of the estimator are discussed, this article focuses particularly on the efficiency of the MRE (or the Pitman) estimator in finite samples. A simulation study indicates that the gain in efficiency when using the MRE estimator rather than the MLE is particularly important in small samples. We also compare the performance of the Pitman estimator with that of other equivariant estimators, including M- and L-estimators and some approximations to the Pitman estimator. In addition, we study the unconditional and conditional inference of the Cauchy location parameter based on both the MLE and the Pitman estimator. We further assess the small sample robustness properties of the estimators included in the simulation by contaminating the Cauchy distribution at three different levels of contamination.

On Estimation of a Location Parameter in the Presence of an Ancillary Component

If $(X, Y)$ is an observation with distribution function $F(x-\theta,y)$, $\sigma^{2}={\rm var}(X)$, $\rho={\rm corr}(X,Y)$ and I is the Fisher information on $\theta$ in $(X,Y)$, then $I\ge \{\sigma^{2}(1-\rho^{2})\}^{-1}$. The equality sign holds under conditions closely related to the conditions for linearity of the Pitman estimator of $\theta$ from a sample from $F(x-\theta,y)$. The results are extensions of earlier results for the case when only the informative component X is observed.

Pitman Estimator to Pareto Scale Parameters

ABSTRACT The Pitman estimators for ordered scale parameters of two Pareto distributions are obtained here. In case of known ordering that have been modified using more informative priors, and found to be better in terms of risks. Some new estimators are also proposed and show that they improve upon Pitman estimators. The results were also verified by simulation study.

Sub- and superadditivity à la Carlen of matrices related to the Fisher information

Let Z = ( Z ( 1 ) , Z ( 2 ) ) be an s-variate random vector partitioned into r- and q-variate subvectors whose distribution depends on an s-variate location parameter θ = ( θ ( 1 ) , θ ( 2 ) ) partitioned in the same way as Z. For the s × s matrix I of Fisher information on θ contained in Z and r × r and q × q matrices I 1 and I 2 of Fisher information on θ ( 1 ) and θ ( 2 ) in Z ( 1 ) and Z ( 2 ) , it is proved that trace ( I - 1 ) ⩽ trace ( I 1 - 1 ) + trace ( I 2 - 1 ) . The inequality is similar to Carlen's superadditivity but has a different statistical meaning: it is a large sample version of an inequality for the covariance matrices of Pitman estimators. If the distribution of Z depends also on an m-variate nuisance parameter η (of a general nature) and I ^ , I ^ ( 1 ) and I ^ ( 2 ) are the efficient matrices of information on θ , θ ( 1 ) , θ ( 2 ) in Z , Z ( 1 ) and Z ( 2 ) , respectively, then trace ( I ^ ) ⩾ trace ( I ^ ( 1 ) ) + trace ( I ^ ( 2 ) ) .

A Note on the Pitman Estimator of Ordered Normal Means When the Variances Are Unequal

ABSTRACT Independent random samples are drawn from k( ≥ 2) normal populations having means Θ1,…, Θ k ; Θ1 ≤ Θ2 ≤ · · · ≤ Θ k and known variances . Estimation of Θ = (Θ1,…, Θ k ) with respect to the norm squared error loss is considered. Let δ p be the analog of the Pitman estimator of Θ, that is, the generalized Bayes estimator of Θ with respect to the uniform prior on the restricted space Ω = {Θ: Θ1 ≤ Θ2 ≤ · · · ≤ Θ k }. It has been proved earlier that when the ′s are equal, δ p is minimax. Here we exhibit for k = 3, that when ′s are unequal, the minimaxity of δ p may fail to hold. Furthermore, the risk performance of δ p is compared numerically with that of the restricted maximum likelihood estimator δ MLE and the usual estimator X = (X 1, X 2, X 3).

Optimal methods for estimating kinetic isotope effects from different forms of the Rayleigh distillation equation

The interpretation of stable isotope values hinges on precise, accurate estimates of kinetic isotope effects (α), which are equal to 1k/ 2k, where 1k and 2k are the reaction rate constants for the two isotopes. Kinetic isotope effects are commonly determined by monitoring the reactant concentration ( C) and isotope ratio ( R) relative to their initial values ( C 1 and R 1, respectively). Values of α are estimated from the C and R values by using the Rayleigh distillation equation (RDE). (A) (R/R 1)=(C/C 1) 1/α−1 We conducted simulation experiments to evaluate the precision of the many different published approaches of estimating α from linearized versions of the RDE and reached the following conclusions: (a) kinetic isotope effects estimated from the slope of the line ln( R) vs. ln( C) were accurate and precise; (b) regressing ln( R i/ R j) on ln( C i/ C j), where all i datapoints were compared to all preceding datapoints j resulted in inflated 95% confidence intervals; (c) forcing the regression of ln( R/ R 1) on ln( C/ C 1) through the origin resulted in 95% confidence intervals for α that covered the true value less than 90% of the time; and (d) regression methods that compensate for errors in both x and y values need to be used with caution. When combining multiple datasets, values of α were sensitive to the form of the equation and the level of error. If all datasets had the same level of error, the optimal estimate of α was achieved by a linear regression with dummy variables. However, when the three datasets had different levels of error, the optimal estimate of α and much narrower 95% confidence intervals were obtained by using the Pitman estimator. Our study demonstrates that some of the other methods jeopardize the accuracy and precision of empirically determined kinetic isotope effects, thus confounding the interpretation of stable isotope values in the environment.

Higher‐order approximations for pitman estimators and for optimal compromise estimators

AbstractLaplace approximations for the Pitman estimators of location or scale parameters, including terms O(n−1), are obtained. The resulting expressions involve the maximum‐likelihood estimate and the derivatives of the log‐likelihood function up to order 3. The results can be used to refine the approximations for the optimal compromise estimators for location parameters considered by Easton (1991). Some applications and Monte Carlo simulations are discussed.

Estimation of parameters of exponential distribution in the truncated space using asymmetric loss function

This paper is concerned with estimation of location and scale parameters of an exponential distribution when the location parameter is bounded above by a known constant. We propose estimators which are better than the standard estimators in the unrestricted case with respect to the suitable choice of LINEX loss. The admissibility of the modified Pitman estimators with respect to the LINEX loss is proved. Finally the theory developed is applied to the problem of estimating the location and scale parameters of two exponential distributions when the location parameters are ordered.

Exact Computation of the Asymptotic Efficiency of Maximum Likelihood Estimators of a Discontinuous Signal in a Gaussian White Noise

In this paper, the problem of computing the exact value of the asymptotic efficiency of maximum likelihood estimators of a discontinuous signal in a Gaussian white noise is considered. A method based on constructing difference equations for the appropriate moments is presented and used to show that the exact variance of the Pitman estimator is $16\zeta(3)$, where $\zeta$ is the Riemann zeta function.

Simulated deficiencies of robust estimators of a proportionality constant

In linear regression, robust methods are at the beginning of their use in practice. In the small sample case, such robust methods provide a necessary measure of protection against deviations from the assumed error distribution. This paper studies through simulation the deficiencies of bioptimal estimators and compares them with more common methods like Huber's estimator or Tukey's estimator. Polyoptimal estimators are convex combinations of Pitman estimators and are optimally robust for a confrontation containing several shapes. The word confrontation is due to J.W. Tukey. It expresses the situation when compromising two or several error distributions. The paper uses the confrontation containing the Gaussian distribution along with a symmetric heavy-tailed distribution having a tail of order 0(t-2) as t→ ±∞.