Toll free: 1- 877- 252 - 7763 | Fax: 1- 425- 458- 9358

# Get customized homework help now!

Sufficient Estimator

Introduction:

The most important target of Statistics is to draw inferences about a population from the analysis of a sample drawn from that population. The theory of estimation was set up by Prof. R. A. Fisher.

What is a parameter space?

Consider X = random variable with p.d.f f (x ,θ ), where θ∈Θ. The set Θ, which is the set of all possible values of θ, is called the parameter space.

Definitions:

• Statistic - Any function of the random sample x1, x2.... xn that are being observed, say Pn (x1, x2xn) is termed as a statistic. Precisely, a statistic is a random variable.
• An Estimator - If a statistic is used to estimate an unknown parameter a of the distribution, it is termed as an estimator.
• An Estimate - A particular value of the estimator, say, Pn (x1, x2....xn) is called an estimate of α.

Characteristics of Estimators:

A good estimator should satisfy the following criteria:

• Unbiased ness
• Consistency
• Efficiency
• Sufficiency

Let us now explain briefly the first criteria Consistency estimator:

Sufficient Estimator:

An estimator is said to be sufficient for a parameter, if it contains all the information in the sample regarding the parameter.

Definition:

If S= s(x1, x2.... xn) is an estimator of a parameter θ, based on a sample x1, x2.... xn of size n from the population with density function f (x, θ) such that the conditional distribution of x1, x2.... xn given S is independent of θ, then S is sufficient estimator for θ.

Theorems Based on Sufficiency:

The necessary and sufficient condition for a distribution to provide sufficient statistic is provided by the following theorem:

Factorization Theorem (Neymann): S = s(x) is sufficient for θ iff the joint density function M (say), of the sample values can be symbolically written as:

M = gθ [s(x)] h(x)

Where gθ [s(x)] h(x) depends on θ) and x only through the value of s(x) and h(x), which is independent of θ).

25% OFF on Homework Help
Name* :
Email* :
Country* :
Phone* :
Subject* :
Due Date*
Time
AM/PM
Timezone
Type Your Questions OR Instructions Below
 (Type Security Code - case sensitive)
Note: We will not do your assignment for you. We will only help you understand the steps to solve it.
 Courses/Topics we help on Quantitative Reasoning for Business Applied Business Research and Statistics Graphs & Diagrams Confidence Interval for Mean & Proportions Average Random Variables - Discrete & Continuous Distributions Correlation Binomial & Poisson Distribution Time Series Quality control - R-chart - p-chart - Mean chart Exponential Smoothing Probability - Conditional Probability - Bayes' Theorem Sampling Distribution Moment Generating Function - Central Limit Theorem Point Estimate & Interval Estimate Normal, Uniform & Exponential Distribution Chi-Square Test - Independence of Attributes F-test - ANOVA Distributions - Bernoulli Geometric t-test Multiple Regression Statistical Methods for Quality Control Sampling Distribution Non Parametric Tests Analysis of Variance Correlation Analysis Regression Analysis Descriptive Statistics Moving Averages Dispersion Sampling Techniques Estimation Theory Testing of Hypothesis - Mean and Proportion Test Data Analysis Numerical Methods Forecasting Goodness-of-Fit Test Inferential Statistics IB Statistics Applied socialogocal research skills Longitudinal study